8 Similarity 8.1 Similar Polygons 8.2 Proving Triangle Similarity by AA 8.3 Proving Triangle Similarity by SSS and SAS 8.4 Proportionality Theorems

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hsva_geo_pe_08op.indd 378 5/9/17 11:54 AM Maintaining Mathematical Proficiency

Determining Whether Ratios Form a Proportion

2 3 Example 1 Tell whether — and — form a proportion. 8 12 Compare the ratios in simplest form. 2 2 ÷ 2 1 — = — = — 8 8 ÷ 2 4 3 3 ÷ 3 1 — = — = — 12 12 ÷ 3 4 The ratios are equivalent. 2 3 So, — and — form a proportion. 8 12

Tell whether the ratios form a proportion. 5 35 9 24 8 6 1. — , — 2. — , — 3. — , — 3 21 24 64 56 28 18 27 15 55 26 39 4. — , — 5. — , — 6. — , — 4 9 21 77 8 12

Finding a Scale Factor Example 2 Find the scale factor of each dilation.

a. b. A′ 25 B′ 3 P P′ A 10 B 2 C C DE E′ D′ CP′ 2 Because — = — , CP 3 A′B′ 25 Because — = — , the 2 AB 10 the scale factor is k = — . 3 25 5 scale factor is k = — = — . 10 2

Find the scale factor of the dilation.

7. 8. 9. J′ 14 P J P′ C K′ K C 6 P 9 C 28 P′ 24 M 14 M′

10. ABSTRACT REASONING If ratio X and ratio Y form a proportion and ratio Y and ratio Z form a proportion, do ratio X and ratio Z form a proportion? Explain your reasoning.

Dynamic Solutions available at BigIdeasMath.com 379

hsva_geo_pe_08op.indd 379 5/9/17 11:54 AM Mathematical Mathematically profi cient students look for and make use of a pattern Processes or structure. Discerning a Pattern or Structure Core Concept Dilations, Perimeter, Area, and Volume

Consider a fi gure that is dilated by a scale factor of k. Scale factor: k 1. The perimeter of the image is k times the perimeter of the original fi gure. 2. The area of the image is k2 times the area of the original fi gure. 3. If the original fi gure is three dimensional, then the volume of the image is k3 times the volume of the original fi gure. original image

Finding Perimeter and Area after a Dilation

The triangle shown has side lengths of 3 inches, 4 inches, and 5 inches. Find the perimeter and area of the image 5 in. when the triangle is dilated by a scale factor of (a) 2, (b) 3, 3 in. and (c) 4. 4 in. SOLUTION 1 P = + + = A = — = 2 Perimeter: 5 3 4 12 in. Area: 2 (4)(3) 6 in.

Scale factor: k Perimeter: kP Area: k2A a. 2 2(12) = 24 in. (22)(6) = 24 in.2 b. 3 3(12) = 36 in. (32)(6) = 54 in.2 c. 4 4(12) = 48 in. (42)(6) = 96 in.2 Monitoring Progress 1. Find the perimeter and area of the image 2. Find the perimeter and area of the image when the trapezoid is dilated by a scale when the parallelogram factor of (a) 2, (b) 3, and (c) 4. is dilated by a scale factor of (a) 2, 2 cm 1 — (b) 3, and (c) 2 . 5 cm 5 ft 4 ft 3 cm

6 cm 2 ft

3. A rectangular prism is 3 inches wide, 4 inches long, and 5 inches tall. Find the surface area and volume of the image of the prism when it is dilated by a scale factor of (a) 2, (b) 3, and (c) 4.

380 Chapter 8 Similarity

hsva_geo_pe_08op.indd 380 5/9/17 11:54 AM 8.1 Similar Polygons

Essential Question How are similar polygons related?

Comparing Triangles after a Dilation Work with a partner. Use dynamic geometry software to draw any △ABC. Dilate △ABC to form a similar △A′B′C′ using any scale factor k and any center of dilation.

A C

a. Compare the corresponding angles of △A′B′C′ and △ABC. b. Find the ratios of the lengths of the sides of △A′B′C′ to the lengths of the corresponding sides of △ABC. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results?

Comparing Triangles after a Dilation Work with a partner. Use dynamic geometry software to draw any B △ABC. Dilate △ABC to form a similar △A′B′C′ using any LOOKING FOR scale factor k and any center STRUCTURE of dilation. To be profi cient in math, A you need to look closely a. Compare the perimeters of △A′B′C′ C to discern a pattern and △ABC. What do you observe? or structure. b. Compare the areas of △A′B′C′ and △ABC. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results? Communicate Your Answer 3. How are similar polygons related? 4. A △RST is dilated by a scale factor of 3 to form △R′S′T′. The area of △RST is 1 square inch. What is the area of △R′S′T′?

Section 8.1 Similar Polygons 381

hsva_geo_pe_0801.indd 381 5/9/17 11:54 AM 8.1 Lesson What You Will Learn

Use similarity statements. Find corresponding lengths in similar polygons. Core VocabularyVocabulary Find perimeters and areas of similar polygons. Previous Decide whether polygons are similar. similar fi gures similarity transformation corresponding parts Using Similarity Statements Recall from Section 4.6 that two geometric fi gures are similar fi gures if and only if there is a similarity transformation that maps one fi gure onto the other.

LOOKING FOR Core Concept STRUCTURE Corresponding Parts of Similar Polygons In the diagram below, △ABC is similar to △DEF. You can write “△ABC is similar Notice that any two to △DEF ” as △ABC ∼ △DEF. A similarity transformation preserves angle congruent fi gures are also measure. So, corresponding angles are congruent. A similarity transformation also △LMN △WXY similar. In and enlarges or reduces side lengths by a scale factor k. So, corresponding side lengths below, the scale factor is are proportional. 5 6 7 — = — = — = 5 6 7 1. So, you can E write △LMN ∼ △WXY and B ka kc △LMN ≅ △WXY. a c similarity transformation M X C A F D b kb 5 7 5 7 Corresponding angles Ratios of corresponding side lengths NL W Y 6 6 DE EF FD ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F — = — = — = k AB BC CA

Using Similarity Statements READING In the diagram, △RST ∼ △XYZ. T statement of In a Z proportionality, any a. Find the scale factor from △RST to △XYZ. 30 pair of ratios forms b. List all pairs of congruent angles. 25 15 18 a true proportion. c. Write the ratios of the corresponding side XY12 lengths in a statement of proportionality. RS20 SOLUTION XY 12 3 YZ 18 3 ZX 15 3 a. — = — = — — = — = — — = — = — RS 20 5 ST 30 5 TR 25 5 3 So, the scale factor is — . 5 b. ∠R ≅ ∠X, ∠S ≅ ∠Y, and ∠T ≅ ∠Z. XY YZ ZX c. Because the ratios in part (a) are equal, — = — = — . K RS ST TR 6 4 J L Q Help in English and Spanish at BigIdeasMath.com 8 Monitoring Progress 9 6 1. In the diagram, △JKL ∼ △PQR. Find the scale factor from △JKL to △PQR. Then list all pairs of congruent angles and write the ratios of the corresponding P R 12 side lengths in a statement of proportionality.

382 Chapter 8 Similarity

hsva_geo_pe_0801.indd 382 5/9/17 11:54 AM Finding Corresponding Lengths in Similar Polygons Core Concept Corresponding Lengths in Similar Polygons READING If two polygons are similar, then the ratio of any two corresponding lengths in Corresponding lengths in the polygons is equal to the scale factor of the similar polygons. similar triangles include side lengths, altitudes, medians, and midsegments. Finding a Corresponding Length

In the diagram, △DEF ∼ △MNP. Find the value of x. E

SOLUTION x 15 The triangles are similar, so the corresponding side lengths are proportional. DF20 MN NP — = — Write proportion. FINDING AN DE EF N 18 30 — = — Substitute. ENTRY POINT 15 x 30 There are several ways to 18x = 450 Cross Products Property 18 write the proportion. For x = 25 Solve for x. example, you could write MP24 DF EF — = — MP NP. The value of x is 25.

Finding a Corresponding Length

△ ∼ △ X In the diagram, TPR XPZ.— T Find the length of the altitude PS . 6 P 8 S 20 Y SOLUTION 6 8 △ △ R First, fi nd the scale factor from XPZ to TPR. Z TR 6 + 6 12 3 — = — = — = — XZ 8 + 8 16 4 Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. PS 3 — = — Write proportion. PY 4 PS 3 — = — Substitute 20 for PY. 20 4 PS = 15 Multiply each side by 20 and simplify. — The length of the altitude PS is 15.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com 2. Find the value of x. 3. Find KM.

A 12 B K GH40 E Q 6 R 10 x 35 5 4 D C TS JLM F 16 8 48 ABCD ∼ QRST △JKL ∼ △EFG

Section 8.1 Similar Polygons 383

hsva_geo_pe_0801.indd 383 5/9/17 11:54 AM Finding Perimeters and Areas of Similar Polygons Theorem Theorem 8.1 Perimeters of Similar Polygons ANALYZING Q If two polygons are similar, then the ratio of L P RELATIONSHIPS their perimeters is equal to the ratios of their K When two similar polygons corresponding side lengths. have a scale factor of k, the N M ratio of their perimeters is S R k equal to . PQ + QR + RS + SP PQ QR RS SP If KLMN ∼ PQRS, then —— = — = — = — = — . KL + LM + MN + NK KL LM MN NK

Proof Ex. 52, p. 390; BigIdeasMath.com

Modeling with Mathematics

A town plans to build a new swimming pool.p An Olympic pool is rectangular withw a length of 50 meters and a width of 25 m 252 meters. The new pool will be similar in shapes to an Olympic pool but will have a lengthl of 40 meters. Find the perimeters of 50 m ana Olympic pool and the new pool.

SOLUTIONS 1.1 Understand the Problem You are given the length and width of a rectangle and the length of a similar rectangle. You need to fi nd the perimeters of both rectangles. 2. Make a Plan Find the scale factor of the similar rectangles and fi nd the perimeter of an Olympic pool. Then use the Perimeters of Similar Polygons Theorem to write VIDEO and solve a proportion to fi nd the perimeter of the new pool. 3. Solve the Problem Because the new pool will be similar to an Olympic pool, the 40 4 — = — scale factor is the ratio of the lengths, 50 5 . The perimeter of an Olympic pool is STUDY TIP 2(50) + 2(25) = 150 meters. Write and solve a proportion to fi nd the perimeter x You can also write the of the new pool. scale factor as a decimal. x 4 — = — Perimeters of Similar Polygons Theorem In Example 4, you can 150 5 write the scale factor as 0.8 = Multiply each side by 150 and simplify. and multiply by 150 to get x 120 x = = 0.8(150) 120. So, the perimeter of an Olympic pool is 150 meters, and the perimeter of the new pool is 120 meters.

4. Look Back Check that the ratio of the perimeters is equal to the scale factor. 120 4 — = — ✓ 150 5 Gazebo B Gazebo A F 15 m G Monitoring Progress Help in English and Spanish at BigIdeasMath.com AB10 m 9 m C H 4. x 18 m The two gazebos shown are similar pentagons. Find the perimeter of Gazebo A. 12 m DE 15 m JK

384 Chapter 8 Similarity

hsva_geo_pe_0801.indd 384 5/9/17 11:54 AM Theorem Theorem 8.2 Areas of Similar Polygons ANALYZING Q If two polygons are similar, then the ratio of L P RELATIONSHIPS their areas is equal to the squares of the ratios K of their corresponding side lengths. When two similar polygons N M have a scale factor of k, S R the ratio of their areas is k2 equal to . Area of PQRS PQ 2 QR 2 RS 2 SP 2 If KLMN ∼ PQRS, then —— = — = — = — = — . Area of KLMN ( KL ) ( LM) ( MN) ( NK) Proof Ex. 53, p. 390; BigIdeasMath.com

Finding Areas of Similar Polygons

In the diagram, △ABC ∼ △DEF. Find the area of △DEF.

B E 10 cm 5 cm DF AC

Area of △ABC = 36 cm2

SOLUTION Because the triangles are similar, the ratio of the area of △ABC to the area of △DEF is equal to the square of the ratio of AB to DE. Write and solve a proportion to fi nd the area of △DEF. Let A represent the area of △DEF. Area of △ABC AB 2 —— = — Areas of Similar Polygons Theorem Area of △DEF ( DE) 36 10 2 — = — Substitute. A ( 5 ) 36 100 — = — Square the right side of the equation. A 25 36 ⋅ 25 = 100 ⋅ A Cross Products Property 900 = 100A Simplify. 9 = A Solve for A.

The area of △DEF is 9 square centimeters.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. In the diagram, GHJK ∼ LMNP. Find the area of LMNP. PL K G 7 m 21 m HJ N M

Area of GHJK = 84 m2

Section 8.1 Similar Polygons 385

hsva_geo_pe_0801.indd 385 5/9/17 11:54 AM Deciding Whether Polygons Are Similar

Deciding Whether Polygons Are Similar

Decide whether ABCDE and KLQRP are similar. Explain your reasoning.

D 9 E 6 C P 6 R 4 Q 12 9 8 6

B 12 A K 8 L

SOLUTION 2 — Corresponding sides of the pentagons are proportional with a scale factor of 3 . However, this does not necessarily mean the pentagons are similar. A dilation with 2 — center A and scale factor 3 moves ABCDE onto AFGHJ. Then a refl ection moves AFGHJ onto KLMNP.

D 9 E 6 C H 6 4 P N 4 J 4 M G 6 R Q 9 6 8 8 6

B 4 F 8 A K 8 L

KLMNP does not exactly coincide with KLQRP, because not all the corresponding angles are congruent. (Only ∠A and ∠K are congruent.)

Because angle measure is not preserved, the two pentagons are not similar.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com Refer to the fl oor tile designs below. In each design, the red shape is a regular hexagon.

Tile Design 1 Tile Design 2

6. Decide whether the hexagons in Tile Design 1 are similar. Explain. 7. Decide whether the hexagons in Tile Design 2 are similar. Explain.

386 Chapter 8 Similarity

hsva_geo_pe_0801.indd 386 5/9/17 11:55 AM 8.1 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE For two fi gures to be similar, the corresponding angles must be ______, and the corresponding side lengths must be ______.

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

What is the scale factor? A

What is the ratio of their areas? 20 12 D 5 3 What is the ratio of their corresponding side lengths? FE4 CB16 △ABC ∼ △DEF What is the ratio of their perimeters?

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, fi nd the scale factor. Then list 6. E all pairs of congruent angles and write the ratios H of the corresponding side lengths in a statement of proportionality. (See Example 1.) x 20 15 18 3. △ABC ∼ △LMN G 12 J L D 16 F A

6.75 N P 4.5 6 9 7. 12 JK CB 6 6 N M 8 12 13 24 26 4. ∼ DEFG PQRS M x L D 9 E S RQ22 R 4 3 6 1 2 F QP 3 8. L 6 M 12 G H G 4 9 6 15 10 In Exercises 5–8, the polygons are similar. Find the J x (See Example 2.) N value of . 8 x K 5. J P P

x 20 21 14

QR12 LK18

Section 8.1 Similar Polygons 387

hsva_geo_pe_0801.indd 387 5/9/17 11:55 AM In Exercises 9 and 10, the black triangles are similar. 18. MODELING WITH MATHEMATICS Your family has Identify the type of segment shown in blue and fi nd the decided to put a rectangular patio in your backyard, value of the variable. (See Example 3.) similar to the shape of your backyard. Your backyard has a length of 45 feet and a width of 20 feet. 9. The length of your new patio is 18 feet. Find the 27 perimeters of your backyard and of the patio. x 16 In Exercises 19–22, the polygons are similar. The area of one polygon is given. Find the area of the other polygon. 18 (See Example 5.)

19. 10. 3 ft y 18 6 ft A = 27 ft2 16

y − 1 20. 4 cm 12 cm A = 10 cm2 In Exercises 11 and 12, RSTU ∼ ABCD. Find the ratio of their perimeters. 21. 11. R S AB 4 in. 12 8 20 in. A = 2 D C 100 in. U T 14 22. 12. AB24 R 18 S

3 cm 12 cm U T A = 2 36 D C 96 cm

ERROR ANALYSIS In Exercises 13–16, two polygons are similar. 23. Describe and correct the error The perimeter of one polygon and the ratio of the in fi nding the perimeter of triangle B. The triangles corresponding side lengths are given. Find the are similar. perimeter of the other polygon. 10 5 28 —2 ✗ 6 — = — A 13. perimeter of smaller polygon: 48 cm; ratio: 3 10 x

3 12 5x = 280 14. perimeter of smaller polygon: 66 ft; ratio: — 4 5 x = 56 B 1 — 15. perimeter of larger polygon: 120 yd; ratio: 6

2 — 16. perimeter of larger polygon: 85 m; ratio: 5 24. ERROR ANALYSIS Describe and correct the error in fi nding the area of rectangle B. The rectangles MODELING WITH MATHEMATICS 17. A school are similar. gymnasium is being remodeled. The basketball court will be similar to an NCAA basketball court, which A = 24 units2 6 24 has a length of 94 feet and a width of 50 feet. The ✗ A — = — school plans to make the width of the new court 6 18 x 45 feet. Find the perimeters of an NCAA court and of 6x = 432 the new court in the school. (See Example 4.) B x = 72 18

388 Chapter 8 Similarity

hsva_geo_pe_0801.indd 388 5/9/17 11:55 AM In Exercises 25 and 26, decide whether the red and blue 36. DRAWING CONCLUSIONS In table tennis, the table is polygons are similar. (See Example 6.) a rectangle 9 feet long and 5 feet wide. A tennis court is a rectangle 78 feet long and 36 feet wide. Are the 25. 40 two surfaces similar? Explain. If so, fi nd the scale factor of the tennis court to the table. 30 22.5

26.

3 3 3 3 3 3 MATHEMATICAL CONNECTIONS In Exercises 37 and 38, the two polygons are similar. Find the values of x and y.

37. 27 27. REASONING Triangles ABC and DEF are similar. Which statement is correct? Select all that apply. 18 BC AC AB CA x − 6 ○A — = — ○B — = — EF DF DE FE 24 39 AB BC CA BC y C — = — D — = — ○ EF DE ○ FD EF

ANALYZING RELATIONSHIPS In Exercises 28–34, 38. x 116° JKLM ∼ EFGH. (y − 73)° 61° J 4 6 65° 30 116° H 3 G 20 M 5 y 11 z° 8 FE K x L ATTENDING TO PRECISION In Exercises 39– 42, the fi gures are similar. Find the missing corresponding 28. Find the scale factor of JKLM to EFGH. side length. 29. Find the scale factor of EFGH to JKLM. 39. Figure A has a perimeter of 72 meters and one of the side lengths is 18 meters. Figure B has a perimeter 30. Find the values of x, y, and z. of 120 meters.

31. Find the perimeter of each polygon. 40. Figure A has a perimeter of 24 inches. Figure B has a perimeter of 36 inches and one of the side 32. Find the ratio of the perimeters of JKLM to EFGH. lengths is 12 inches.

33. Find the area of each polygon. 41. Figure A has an area of 48 square feet and one of the side lengths is 6 feet. Figure B has an area of 34. Find the ratio of the areas of JKLM to EFGH. 75 square feet.

35. USING STRUCTURE Rectangle A is similar to 42. Figure A has an area of 18 square feet. Figure B rectangle B. Rectangle A has side lengths of 6 and has an area of 98 square feet and one of the side 12. Rectangle B has a side length of 18. What are lengths is 14 feet. the possible values for the length of the other side of rectangle B? Select all that apply. ○A 6 ○B 9 ○C 24 ○D 36

Section 8.1 Similar Polygons 389

hsva_geo_pe_0801.indd 389 5/9/17 11:55 AM CRITICAL THINKING In Exercises 43–48, tell whether the 52. PROVING A THEOREM Prove the Perimeters of polygons are always, sometimes, or never similar. Similar Polygons Theorem (Theorem 8.1) for similar rectangles. Include a diagram in your proof. 43. two isosceles triangles 44. two isosceles trapezoids 53. PROVING A THEOREM Prove the Areas of Similar 45. two rhombuses 46. two squares Polygons Theorem (Theorem 8.2) for similar rectangles. Include a diagram in your proof. 47. two regular polygons

48. a right triangle and an equilateral triangle 54. THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In 49. MAKING AN ARGUMENT Your sister claims spherical geometry, all points are points on the that when the side lengths of two rectangles are surface of a sphere. A line is a circle on the sphere proportional, the two rectangles must be similar. whose diameter is equal to the diameter of the sphere. Is she correct? Explain your reasoning. A plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? Explain your reasoning. 50. HOW DO YOU SEE IT? You shine a fl ashlight directly on an object to project its image onto a parallel screen. Will the object and the image be similar? 55. CRITICAL THINKING In the diagram, PQRS is a Explain your reasoning. square, and PLMS ∼ LMRQ. Find the exact value of x. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles. P Q L

S 1 R M x

MODELING WITH MATHEMATICS 51. During a total 56. MATHEMATICAL CONNECTIONS The equations of the eclipse of the Sun, the moon is directly in line with 4 4 lines shown are y = — x + 4 and y = — x − 8. Show the Sun and blocks the Sun’s rays. The distance DA 3 3 △ ∼ △ between Earth and the Sun is 93,000,000 miles, the that AOB COD. distance DE between Earth and the moon is y 240,000 miles, and the radius AB of the Sun is B 432,500 miles. Use the diagram and the given measurements to estimate the radius EC of the moon. A O C x B C D A E D Sun moon Earth △DEC ∼ △DAB Not drawn to scale

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the value of x. 57. 58. 59. 60. x° 52° x° 24°

76° 41° x° x°

390 Chapter 8 Similarity

hsva_geo_pe_0801.indd 390 5/9/17 11:55 AM 8.2 Proving Triangle Similarity by AA

Essential Question What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent?

Comparing Triangles Work with a partner. Use dynamic geometry software. a. Construct △ABC and △DEF so that m∠A = m∠D = 106°, m∠B = m∠E = 31°, and △DEF is not congruent DE 106° 31° to △ABC. A B 106° 31°

C F

b. Find the third angle measure and the side lengths of each triangle. Copy the table below and record your results in column 1.

1. 2. 3. 4. 5. 6. m∠ A, m∠D 106° 88° 40° m∠B, m∠E 31° 42° 65° m∠C m∠F CONSTRUCTING AB VIABLE ARGUMENTS DE To be profi cient in math, BC you need to understand EF and use stated assumptions, AC defi nitions, and previously established results in DF constructing arguments. c. Are the two triangles similar? Explain. d. Repeat parts (a)–(c) to complete columns 2 and 3 of the table for the given angle measures. e. Complete each remaining column of the table using your own choice of two pairs of equal corresponding angle measures. Can you construct two triangles in this way that are not similar? f. Make a conjecture about any two triangles with two pairs of congruent corresponding angles.

R M 3 N Communicate Your Answer

3 2. What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent? L TS4 3. Find RS in the fi gure at the left.

Section 8.2 Proving Triangle Similarity by AA 391

hsva_geo_pe_0802.indd 391 5/9/17 11:56 AM 8.2 Lesson What You Will Learn

Use the Angle-Angle Similarity Theorem. Core VocabularyVocabulary Solve real-life problems. Previous Using the Angle-Angle Similarity Theorem similar fi gures similarity transformation Theorem Theorem 8.3 Angle-Angle (AA) Similarity Theorem If two angles of one triangle are E congruent to two angles of another B triangle, then the two triangles D A are similar. If ∠A ≅ ∠D and ∠B ≅ ∠E, △ ∼ △ C then ABC DEF. F Proof p. 392

Angle-Angle (AA) Similarity Theorem

Given ∠ ≅ ∠ ∠ ≅ ∠ E A D, B E B D Prove △ABC ∼ △DEF A

C F

DE Dilate △ABC using a scale factor of k = — and center A. The image AB of △ABC is △AB′C′. B′ B A

C C′

Because a dilation is a similarity transformation, △ABC ∼ △AB′C′. Because the AB′ DE ratio of corresponding lengths of similar polygons equals the scale factor, — = — . AB AB ′ = Multiplying— each —side by AB yields AB DE. By the defi nition of congruent segments, AB ′ ≅ DE . By the Refl exive Property of Congruence (Theorem 2.2), ∠A ≅ ∠A. Because corresponding angles of similar polygons are congruent, ∠B′ ≅ ∠B. Because ∠B′ ≅ ∠B and ∠B ≅ ∠E, ∠B′ ≅ ∠E by the Transitive Property of Congruence (Theorem 2.2). — — Because ∠A ≅ ∠D, ∠B′ ≅ ∠E, and AB ′ ≅ DE , △AB′C′ ≅ △DEF by the ASA Congruence Theorem (Theorem 5.10). So, a composition of rigid motions maps △AB′C′ to △DEF. Because a dilation followed by a composition of rigid motions maps △ABC to △DEF, △ABC ∼ △DEF.

392 Chapter 8 Similarity

hsva_geo_pe_0802.indd 392 5/9/17 11:56 AM Using the AA Similarity Theorem

Determine whether the triangles are CDH similar. If they are, write a similarity 26° 64° statement. Explain your reasoning. E SOLUTION GK Because they are both right angles, ∠D and ∠G are congruent. By the Triangle Sum Theorem (Theorem 5.1), 26° + 90° + m∠E = 180°, VISUAL so m∠E = 64°. So, ∠E and ∠H are congruent. REASONING So, △CDE ∼ △KGH by the AA Similarity Theorem.

H C D Using the AA Similarity Theorem

E Show that the two triangles are similar. GKa. △ABE ∼ △ACD b. △SVR ∼ △UVT A T Use colored pencils to S V show congruent angles. E 52° B This will help you write similarity statements. ° R D 52 C U

SOLUTION a. Because m∠ABE and m∠C both equal 52°, ∠ABE ≅ ∠C. By the Refl exive VISUAL Property of Congruence (Theorem 2.2), ∠A ≅ ∠A. REASONING So, △ABE ∼ △ACD by the AA Similarity Theorem.

You may fi nd it helpful b. You know ∠SVR ≅ ∠UVT by the Vertical Angles Congruence Theorem to redraw the triangles — — (Theorem 2.6). The diagram shows RS ʈ UT , so ∠S ≅ ∠U by the Alternate Interior separately. Angles Theorem (Theorem 3.2). A T S ° A V E 52 B R U ° D 52 C So, △SVR ∼ △UVT by the AA Similarity Theorem.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com Show that the triangles are similar. Write a similarity statement.

1. △FGH and △RQS 2. △CDF and △DEF

G R D

58° HF 32° Q S C F E

— — 3. WHAT IF? Suppose that SR ʈ TU in Example 2 part (b). Could the triangles still be similar? Explain.

Section 8.2 Proving Triangle Similarity by AA 393

hsva_geo_pe_0802.indd 393 5/9/17 11:56 AM Solving Real-Life Problems Previously, you learned a way to use congruent triangles to fi nd measurements indirectly. Another useful way to fi nd measurements indirectly is by using similar triangles.

Modeling with Mathematics

A fl agpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby who is 5 feet 4 inches tall casts a shadow that is 40 inches long. How tall is the fl agpole to the nearest foot?

SOLUTION 1. Understand the Problem You are given the length of a fl agpole’s shadow, the height of a Not drawn to scale woman, and the length of the woman’s shadow. You need to fi nd the height of the flagpole. 2. Make a Plan Use similar triangles to write a proportion and solve for the height of the fl agpole. 3. Solve the Problem The fl agpole and the woman form sides of two right triangles with the ground. The Sun’s rays hit the fl agpole and the woman at the same angle. x You have two pairs of congruent angles, ft so the triangles are similar by the 5 ft 4 in. AA Similarity Theorem. 40 in. 50 ft You can use a proportion to fi nd the height x. Write 5 feet 4 inches as 64 inches so that you can form two ratios of feet to inches. x ft 50 ft — = — Write proportion of side lengths. 64 in. 40 in.

40x = 3200 Cross Products Property x = 80 Solve for x. The fl agpole is 80 feet tall.

4. Look Back Attend to precision by checking that your answer has the correct units. The problem asks for the height of the fl agpole to the nearest foot. Because your answer is 80 feet, the units match. Also, check that your answer is reasonable in the context of the problem. A height of 80 feet makes sense for a fl agpole. You can estimate that an eight-story building would be about 8(10 feet) = 80 feet, so it is reasonable that a fl agpole could be that tall.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com 4. WHAT IF? A child who is 58 inches tall is standing next to the woman in Example 3. How long is the child’s shadow? 5. You are standing outside, and you measure the lengths of the shadows cast by both you and a tree. Write a proportion showing how you could fi nd the height of the tree.

394 Chapter 8 Similarity

hsva_geo_pe_0802.indd 394 5/9/17 11:56 AM 8.2 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE If two angles of one triangle are congruent to two angles of another triangle, then the triangles are ______.

2. WRITING Can you assume that corresponding sides and corresponding angles of any two similar triangles are congruent? Explain.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, determine whether the triangles are In Exercises 11–18, use the diagram to copy and similar. If they are, write a similarity statement. Explain complete the statement. your reasoning. (See Example 1.) 7 AG3 B 3. 4. ° 53 45° F RV Q U 4 C K 85° 65° ° 42 12 2 48° G HJ L 35° 35° D F E S T 9

5. M XW 6. D 11. △CAG ∼ 12. △DCF ∼ ° 82° 40 55° △ ∼ ∠ = C E 13. ACB 14. m ECF 25° S Y 73° 15. m∠ECD = 16. CF = LN ° 25 T 17. BC = 18. DE = U 19. ERROR ANALYSIS Describe and correct the error in In Exercises 7–10, show that the two triangles are using the AA Similarity Theorem (Theorem 8.3). similar. (See Example 2.) F 7. N 8. L ✗ B M M 45° A C E G Y D H ° PQ N X 45 Z Quadrilateral ABCD ∼ quadrilateral EFGH by the AA Similarity Theorem. 9. Y 10. R S 85° 20. ERROR ANALYSIS Describe and correct the error in Z V X 45° fi nding the value of x. T 50° W U U 4 5 4 — = — ✗ 6 x 5 6 4x = 30 x x = 7.5

Section 8.2 Proving Triangle Similarity by AA 395

hsva_geo_pe_0802.indd 395 5/9/17 11:56 AM 21. MODELING WITH MATHEMATICS You can measure 28. HOW DO YOU SEE IT? In the diagram, which the width of the lake using a surveying technique, as triangles would you use to fi nd the distance x shown in the diagram. Find the width of the lake, WX. between the shoreline and the buoy? Explain Justify your answer. your reasoning.

V 104 m W L

x X 6 m K M Z Y 8 m 20 m Not drawn to scale J 100 m NP 25 m 22. MAKING AN ARGUMENT You and your cousin are trying to determine the height of a telephone pole. Your cousin tells you to stand in the pole’s shadow WRITING so that the tip of your shadow coincides with the tip 29. Explain why all equilateral triangles of the pole’s shadow. Your cousin claims to be able are similar. to use the distance between the tips of the shadows and you, the distance between you and the pole, and 30. THOUGHT PROVOKING Decide whether each is a your height to estimate the height of the telephone valid method of showing that two quadrilaterals are pole. Is this possible? Explain. Include a diagram in similar. Justify your answer. your answer. a. AAA b. AAAA REASONING In Exercises 23–26, is it possible for △JKL and △XYZ to be similar? Explain your reasoning. 31. PROOF Without using corresponding lengths 23. m∠J = 71°, m∠K = 52°, m∠X = 71°, and m∠Z = 57° in similar polygons, prove that the ratio of two corresponding angle bisectors in similar triangles is 24. △JKL is a right triangle and m∠X + m∠Y= 150°. equal to the scale factor.

25. m∠L = 87° and m∠Y = 94° 32. PROOF Prove that if the lengths of two sides of a triangle are a and b, respectively, then the lengths of the ∠ + ∠ = ° ∠ + ∠ = ° b 26. m J m K 85 and m Y m Z 80 corresponding altitudes to those sides are in the ratio — . a 27. MATHEMATICAL CONNECTIONS Explain how you MODELING WITH MATHEMATICS can use similar triangles to show that any two points 33. A portion of an on a line can be used to fi nd its slope. amusement park ride is shown. Find EF. Justify your answer. y

A x E B 40 ft 30 ft

DCF

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Determine whether there is enough information to prove that the triangles are congruent. Explain your reasoning. 34. G H 35. U 36. Q

T V

F KJ P S R W

396 Chapter 8 Similarity

hsva_geo_pe_0802.indd 396 5/9/17 11:56 AM 8.3 Proving Triangle Similarity by SSS and SAS

Essential Question What are two ways to use corresponding sides of two triangles to determine that the triangles are similar?

Deciding Whether Triangles Are Similar Work with a partner. Use dynamic geometry software. a. Construct △ABC and △DEF with the side lengths given in column 1 of the table below.

1. 2. 3. 4. 5. 6. 7. AB 55615924 BC 888201218 AC 10 10 10 10 8 16 DE 10 15 9 12 12 8 EF 16 24 12 16 15 6 DF 20 30 15 8 10 8 m∠A m∠B m∠C m∠D m∠E m∠F CONSTRUCTING VIABLE ARGUMENTS b. Copy the table and complete column 1. To be profi cient in math, c. Are the triangles similar? Explain your reasoning. you need to analyze d. Repeat parts (a)–(c) for columns 2–6 in the table. situations by breaking them into cases and recognize e. How are the corresponding side lengths related in each pair of triangles that are and use counterexamples. similar? Is this true for each pair of triangles that are not similar? f. Make a conjecture about the similarity of two triangles based on their corresponding side lengths. g. Use your conjecture to write another set of side lengths of two similar triangles. Use the side lengths to complete column 7 of the table.

Deciding Whether Triangles Are Similar Work with a partner. Use dynamic geometry software. Construct any △ABC. a. Find AB, AC, and m∠A. Choose any positive rational number k and construct △DEF so that DE = k ⋅ AB, DF = k ⋅ AC, and m∠D = m∠A. b. Is △DEF similar to △ABC? Explain your reasoning. c. Repeat parts (a) and (b) several times by changing △ABC and k. Describe your results. Communicate Your Answer 3. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar?

Section 8.3 Proving Triangle Similarity by SSS and SAS 397

hsva_geo_pe_0803.indd 397 5/9/17 11:56 AM 8.3 Lesson What You Will Learn Use the Side-Side-Side Similarity Theorem. Use the Side-Angle-Side Similarity Theorem. Core VocabularyVocabulary Prove slope criteria using similar triangles. Previous similar fi gures Using the Side-Side-Side Similarity Theorem corresponding parts slope In addition to using congruent corresponding angles to show that two triangles are similar, you can use proportional corresponding side lengths. parallel lines perpendicular lines Theorem Theorem 8.4 Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of A R two triangles are proportional, then the triangles are similar. ST B C AB BC CA If — = — = — , then △ABC ∼ △RST. RS ST TR

Proof p. 399

Using the SSS Similarity Theorem

Is either △DEF or △GHJ similar to △ABC? FINDING AN B DF12 H ENTRY POINT 12 8 10 8 When using the 6 9 SSS Similarity Theorem, ACE JG 16 16 compare the shortest sides, the longest sides, and then SOLUTION the remaining sides. Compare △ABC and △DEF by fi nding ratios of corresponding side lengths. Shortest Longest Remaining sides sides sides AB 8 CA 16 BC 12 — = — — = — — = — DE 6 FD 12 EF 9 4 4 4 = — = — = — 3 3 3

All the ratios are equal, so △ABC ∼ △DEF.

Compare △ABC and △GHJ by fi nding ratios of corresponding side lengths. Shortest Longest Remaining sides sides sides AB 8 CA 16 BC 12 — = — — = — — = — GH 8 JG 16 HJ 10 6 = 1 = 1 = — 5

The ratios are not all equal, so △ABC and △GHJ are not similar.

398 Chapter 8 Similarity

hsva_geo_pe_0803.indd 398 5/9/17 11:56 AM SSS Similarity Theorem K S RS ST TR Given — = — = — JK KL LJ

Prove △RST ∼ △JKL J L P Q RT — — — — Locate P on RS so that PS = JK. Draw PQ so that PQ ʈ RT . Then △RST ∼ △PSQ by RS ST TR JUSTIFYING the AA Similarity Theorem (Theorem 8.3), and — = — = — . You can use the given STEPS PS SQ QP proportion and the fact that PS = JK to deduce that SQ = KL and QP = LJ. By the The Parallel Postulate SSS Congruence Theorem (Theorem 5.8), it follows that △PSQ ≅ △JKL. Finally, use (Postulate 3.1) allows you the defi nition of congruent triangles and the AA Similarity Theorem (Theorem 8.3) to to draw an auxiliary line conclude that △RST ∼ △JKL. PQ⃖៮៮⃗ in △RST. There is only one line through point P parallel to ⃖ RT៮៮⃗ , so you are Using the SSS Similarity Theorem able to draw it. Find the value of x that makes △ABC ∼△DEF. E B 4 x − 1 12 18 A 8 C FINDING AN D x + F ENTRY POINT 3( 1) You can use either SOLUTION AB BC AB AC — = — — = — DE EF or DE DF Step 1 Find the value of x that makes corresponding side lengths proportional. in Step 1. AB BC — = — Write proportion. DE EF 4 x − 1 — = — Substitute. 12 18 4 ⋅ 18 = 12(x − 1) Cross Products Property 72 = 12x − 12 Simplify. 7 = x Solve for x. Step 2 Check that the side lengths are proportional when x = 7. BC = x − 1 = 6 DF = 3(x + 1) = 24 AB ? BC 4 6 AB ? AC 4 8 — = — — = — ✓ — = — — = — ✓ DE EF 12 18 DE DF 12 24

When x = 7, the triangles are similar by the SSS Similarity Theorem.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com R 24 S X Use the diagram. L 1. Which of the three triangles are similar? 30 33 39 36 Write a similarity statement. 20 26 T 2. The shortest side of a triangle similar M 24 N Y 30 Z to △RST is 12 units long. Find the other side lengths of the triangle.

Section 8.3 Proving Triangle Similarity by SSS and SAS 399

hsva_geo_pe_0803.indd 399 5/9/17 11:56 AM Using the Side-Angle-Side Similarity Theorem Theorem Theorem 8.5 Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an X M angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. P N YZ ZX XY If ∠X ≅ ∠M and — = — , then △XYZ ∼ △MNP. PM MN Proof Ex. 37, p. 406

Using the SAS Similarity Theorem

YouY are building a lean-to shelter starting from a tree branch, as shown. Can you constructc the right end so it is similar to the left end using the angle measure and lengthsl shown?

F A 53° 10 ft 53° 15155ft ft 6 ft H C G 9 ft

SOLUTION Both m∠A and m∠F equal 53°, so ∠A ≅ ∠F. Next, compare the ratios of the lengths of the sides that include ∠A and ∠F. Shorter sides Longer sides AB 9 AC 15 — = — — = — FG 6 FH 10 3 3 = — = — 2 2 The lengths of the sides that include ∠A and ∠F are proportional. So, by the SAS Similarity Theorem, △ABC ∼ △FGH.

Yes, you can make the right end similar to the left end of the shelter.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com Explain how to show that the indicated triangles are similar.

3. △SRT ∼ △PNQ 4. △XZW ∼ △YZX S X P 20 15 24 18 12

NQ W 16 9 YZ R T 21 28

400 Chapter 8 Similarity

hsva_geo_pe_0803.indd 400 5/9/17 11:56 AM Using Coordinate Geometry To prove two triangles are similar in a coordinate plane using the SSS Similarity Theorem, use the Distance Formula to fi nd the lengths of the sides of the triangles REMEMBER and then show the corresponding side lengths are proportional. L x y Recall that if ( 1, 1) and N x y Proving Triangle Similarity in the Coordinate Plane ( 2, 2) are points in the coordinate plane, then the △ △ distance between L and N is Is LMN similar to PQR? y Q(1, 6) ——— 6 LN = x − x 2 + y − y 2 √ ( 2 1) ( 2 1) . SOLUTION 4 Use the Distance Formula to fi nd the lengths of all the sides of the triangles. M(−1, 2) 2 △LMN ANOTHER METHOD LM = ∣ 2 − (−3) ∣ = ∣ 5 ∣ = 5 −2 2 6 x — — — — N − Prove LN ⊥ MN and PR ⊥ QR ——— (1, 2) LN = √ [1 − (−1)] 2 + [−2 − (−3)]2 by showing that their slopes — — L − − are negative reciprocals. = √4 + 1 = √ 5 ( 1, 3) ——— — −2 − (−3) 1 MN = √[1 − (−1)]2 + (−2 − 2)2 R − LN : m = — = — −6 (7, 6) 1 − (−1) 2 — — — = √4 + 16 = √ 20 = 2√ 5 — −2 − 2 MN : m = — −8 1 − (−1) △PQR −4 P(1, −9) = — = −2 2 PQ = ∣ 6 − (−9) ∣ = ∣ 15 ∣ = 15 — −6 − (−9) 3 1 ——— PR m = — = — = — = − 2 + − − − 2 : − PR √ (7 1) [ 6 ( 9)] 7 1 6 2 — — — — −6 − 6 −12 = √36 + 9 = √ 45 = 3 √5 QR : m = — = — = −2 7 − 1 6 —— QR = √(7 − 1)2 + (−6 − 6)2 So, ∠N and ∠R are right — — — = √ + = √ = √ angles and are, therefore, 36 144 180 6 5 congruent. Show that the △ △ ratios of corresponding side Compare LMN and PQR by fi nding ratios of corresponding side lengths. LN MN lengths — and — are equal. Shortest sides Longest sides Remaining sides PR QR — — LN √ 5 LM 5 MN 2√ 5 — = — — = — — = — LN 1 MN 1 PR — PQ 15 QR — — = — and — = — . 3 √5 6 √5 PR 3 QR 3 1 1 1 = — = — = — So, △LMN ∼ △PQR by the SAS 3 3 3 Similarity Theorem. All the ratios are equal, so △LMN ∼ △PQR by the SSS Similarity Theorem.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Is △UVW similar to △XYZ? Explain.

y U(−3, 3) 4 X(15, 3) Z(5, 1)

−4 4 8 16 x W(−2, −2) V(0, 0) Y(9, −3) −6

Section 8.3 Proving Triangle Similarity by SSS and SAS 401

hsva_geo_pe_0803.indd 401 5/9/17 11:57 AM Concept Summary Triangle Similarity Theorems AA Similarity Theorem SSS Similarity Theorem SAS Similarity Theorem

A D A D A D

EF EF EF B C B C B C

AB BC AC AB AC If ∠A ≅ ∠D and ∠B ≅ ∠E, If — = — = — , then If ∠A ≅ ∠D and — = — , DE EF DF DE DF then △ABC ∼ △DEF. △ABC ∼ △DEF. then △ABC ∼ △DEF.

Proving Slope Criteria Using Similar Triangles You can use similar triangles to prove the Slopes of Parallel Lines Theorem (Theorem 3.13). Because the theorem is biconditional, you must prove both parts. 1. If two nonvertical lines are parallel, then they have the same slope. 2. If two nonvertical lines have the same slope, then they are parallel. The fi rst part is proved below. The second part is proved in the exercises.

Part of Slopes of Parallel Lines Theorem (Theorem 3.13)

Given ℓ ʈ n, ℓand n are nonvertical. = Prove mℓ mn First, consider the case whereℓand n are horizontal. Because all horizontal lines are parallel and have a slope of 0, the statement is true for horizontal lines. y For the case of nonhorizontal, nonvertical lines, draw two such parallel lines,ℓand n, n — E and label their x-intercepts A and D, respectively. Draw a vertical segment BC parallel to— the y-axis from point B on lineℓto point C on the x-axis. Draw a vertical segment B EF parallel to the y-axis from point E on line n to point F on the x-axis. Because vertical and horizontal lines are perpendicular, ∠BCA and ∠EFD are right angles.

A D STATEMENTS REASONS C F x 1. ℓ ʈ n 1. Given 2. ∠BAC ≅ ∠EDF 2. Corresponding Angles Theorem (Thm. 3.1) 3. ∠BCA ≅ ∠EFD 3. Right Angles Congruence Theorem (Thm. 2.3) 4. △ABC ∼ △DEF 4. AA Similarity Theorem (Thm. 8.3) BC AC 5. — = — 5. Corresponding sides of similar fi gures are EF DF proportional. BC EF 6. — = — 6. Rewrite proportion. AC DF BC EF 7. m = — , m = — 7. Defi nition of slope ℓ AC n DF BC 8. m = — 8. Substitution Property of Equality n AC = 9. mℓ mn 9. Transitive Property of Equality

402 Chapter 8 Similarity

hsva_geo_pe_0803.indd 402 5/9/17 11:57 AM To prove the Slopes of Perpendicular Lines Theorem (Theorem 3.14), you must prove both parts. 1. If two nonvertical lines are perpendicular, then the product of their slopes is −1. 2. If the product of the slopes of two nonvertical lines is −1, then the lines are perpendicular. The fi rst part is proved below. The second part is proved in the exercises.

Part of Slopes of Perpendicular Lines Theorem (Theorem 3.14) Given ℓ⊥ n,ℓand n are nonvertical. = − Prove mℓmn 1

y n Draw two nonvertical, perpendicular lines,ℓand n, that intersect at point A. Draw a E horizontal line j parallel to the x-axis through point A. Draw a horizontal line k parallel A to the x-axis through point— C on line n. Because horizontal lines are parallel, j ʈ k. Draw a vertical segment AB parallel to the y-axis from point A to point B on line k. D j — Draw a vertical segment ED parallel to the y-axis from point E on lineℓto point D on line j. Because horizontal and vertical lines are perpendicular, ∠ ABC and ∠ ADE are BC k right angles. x STATEMENTS REASONS 1. ℓ⊥ n 1. Given 2. m∠CAE = 90° 2. ℓ⊥ n 3. m∠CAE = m∠DAE + m∠CAD 3. Angle Addition Postulate (Post. 1.4) 4. m∠DAE + m∠CAD = 90° 4. Transitive Property of Equality 5. ∠BCA ≅ ∠CAD 5. Alternate Interior Angles Theorem (Thm. 3.2) 6. m∠BCA = m∠CAD 6. Defi nition of congruent angles 7. m∠DAE + m∠BCA = 90° 7. Substitution Property of Equality 8. m∠DAE = 90° − m∠BCA 8. Solve statement 7 for m∠DAE. 9. m∠BCA + m∠BAC + 90° = 180° 9. Triangle Sum Theorem (Thm. 5.1) 10. m∠BAC = 90° − m∠BCA 10. Solve statement 9 for m∠BAC. 11. m∠DAE = m∠BAC 11. Transitive Property of Equality 12. ∠DAE ≅ ∠BAC 12. Defi nition of congruent angles 13. ∠ABC ≅ ∠ADE 13. Right Angles Congruence Theorem (Thm. 2.3) 14. △ABC ∼ △ADE 14. AA Similarity Theorem (Thm. 8.3) AD DE 15. — = — 15. Corresponding sides of similar fi gures AB BC are proportional. AD AB 16. — = — 16. Rewrite proportion. DE BC DE AB 17. m = — , m = −— 17. Defi nition of slope ℓ AD n BC DE AB 18. m m = — ⋅ −— 18. Substitution Property of Equality ℓ n AD ( BC) DE AD 19. m m = — ⋅ − — 19. Substitution Property of Equality ℓ n AD ( DE ) = − 20. mℓmn 1 20. Simplify.

Section 8.3 Proving Triangle Similarity by SSS and SAS 403

hsva_geo_pe_0803.indd 403 5/9/17 11:57 AM 8.3 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE You plan to show that △QRS is similar to △XYZ by the SSS Similarity Theorem QR QS (Theorem 8.4). Copy and complete the proportion that you will use: — = — = — . YZ

2. WHICH ONE DOESN’T BELONG? Which triangle does not belong with the other three? Explain your reasoning.

8 9 12 6 8 6 4 4 3 6 12 18

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, determine whether △JKL or In Exercises 7−9, verify that △ABC ∼ △DEF. Find the △RST is similar to △ABC. (See Example 1.) scale factor of △ABC to △DEF.

3. B 8 C K 74L TS 7. △ABC: BC = 18, AB = 15, AC = 12 △ = = = 3.5 DEF: EF 12, DE 10, DF 8 6 6 7 11 12 R 8. △ABC: AB = 10, BC = 16, CA = 20 J △DEF: DE = 25, EF = 40, FD = 50 A 9. y C(0, 10) 4. L 8 A(3, 7) B T 25 F(−10, 4) B(2, 6) 16 14 17.5 16 4 10.5 D(−4, −2)

AJ20 C KR20 12 S −10 −2 4 x

−4 E(−6, −4) In Exercises 5 and 6, fi nd the value of x that makes △DEF ∼ △XYZ. (See Example 2.) 5. Y In Exercises 10 and 11, determine whether the two triangles are similar. If they are similar, write a E x − 10 14 5 2 1 similarity statement and fi nd the scale factor of triangle B to triangle A. (See Example 3.) DF11 X5x + 2 Z Y 10. DF12 6. E 8 F A X 8 9 B 10 7.5 3(x − 1) x − 1 E X 6 W ZY4 D 11. S A 112° 8 L 24 J R 18 T 112° 10 B K 404 Chapter 8 Similarity

hsva_geo_pe_0803.indd 404 5/9/17 11:57 AM In Exercises 12 and 13, sketch the triangles using the In Exercises 19 and 20, use △XYZ. given description. Then determine whether the two Y triangles can be similar. 12. In △RST, RS = 20, ST = 32, and m∠S = 16°. In △ = = ∠ = ° FGH, GH 30, HF 48, and m H 24 . 13 12

13. The side lengths of △ABC are 24, 8x, and 48, and the side lengths of △DEF are 15, 25, and 6x. X 10 Z In Exercises 14–18, show that the triangles are 19. The shortest side of a triangle similar to △XYZ is similar and write a similarity statement. Explain 20 units long. Find the other side lengths of the your reasoning. triangle. 14. F 5 20. The longest side of a triangle similar to △XYZ is G 24 39 units long. Find the other side lengths of the 15 triangle. 18 △ ∼ △ △ △ H 16.5 5.5 KJ 21. HIJ KLM. The scale factor of HIJ to KLM 3 — is 4. Find KL. 15. E y H(5, 4) 14 4 27 A D C 18 2 21 B 2 4 6 x

16. XZ30 J −2 J(1, −1) I(5, −2) 47° 21 35 −4 Y 47° GD50 22. ERROR ANALYSIS Describe and correct the error in writing a similarity statement. 17. R 12 S T B PR A 15 ✗ 86° 16 18 24 18 12 2486° 20 Q Q V 9 U C △ABC ∼ △PQR by the SAS Similarity Theorem (Theorem 8.5). 18. y 8 B(1, 6) K(−1, 6) 23. MATHEMATICAL CONNECTIONS Find the value of n that makes △DEF ∼ △XYZ when DE = 4, EF = 5, XY = 4(n + 1), YZ = 7n − 1, and ∠E ≅ ∠Y. Include −8 8 x a sketch. X(−5, −2) C(7, −6) P(−1, −4)

−8 A(1, −9)

Section 8.3 Proving Triangle Similarity by SSS and SAS 405

hsva_geo_pe_0803.indd 405 5/9/17 11:57 AM ATTENDING TO PRECISION In Exercises 24–29, use the 33. WRITING Are any two right triangles similar? Explain. diagram to copy and complete the statement. 34. MODELING WITH MATHEMATICS In the portion of L 3.5 S BC BD 2 M the shuffl eboard court shown, — = — . 2 AC AE 4 44.5° N C 61° 4 8 D

P 91° B E 4 R Q

A 24. m∠LNS = 25. m∠NRQ =

26. m∠NQR = 27. RQ = a. What additional information do you need to show 28. m∠NSM = 29. m∠NPR = that △BCD ∼ △ACE using the SSS Similarity Theorem (Theorem 8.4)? 30. MAKING AN ARGUMENT Your friend claims that △JKL ∼ △MNO by the SAS Similarity Theorem b. What additional information do you need to show (Theorem 8.5) when JK = 18, m∠K = 130°, that △BCD ∼ △ACE using the SAS Similarity KL = 16, MN = 9, m∠N = 65°, and NO = 8. Do you Theorem (Theorem 8.5)? support your friend’s claim? Explain your reasoning. 35. ATTENDING TO PRECISION △ABC has side lengths 3, 31. ANALYZING RELATIONSHIPS Certain sections of 4, and 5. Find the coordinates of a similar triangle in stained glass are sold in triangular, beveled pieces. the coordinate plane. Which of the three beveled pieces, if any, are similar? 36. PROOF Given that △BAC is a right triangle and D, E, 3 in. ∠ = ° 3 in. 5.25 in. and F are midpoints, prove that m DEF 90 . 3 in. B 5 in. E 3 in. D 7 in. A F C 4 in. 4 in. 37. PROVING A THEOREM Write a two-column proof of the SAS Similarity Theorem (Theorem 8.5). 32. ATTENDING TO PRECISION In the diagram, AB AC Given ∠A ≅ ∠D, — = — MN MP — = — . Which of the statements must be true? DE DF MR MQ Prove △ABC ∼ △DEF Select all that apply. Explain your reasoning. B Q R E 2 3 N P 14 D F M A C — — ○A ∠1 ≅ ∠2 ○B QR ʈ NP ○C ∠1 ≅ ∠4 ○D △MNP ∼ △MRQ

406 Chapter 8 Similarity

hsva_geo_pe_0803.indd 406 5/9/17 11:57 AM 38. CRITICAL THINKING You are given two right triangles 44. MODELING WITH MATHEMATICS The dimensions with one pair of corresponding legs and the pair of of an actual swing set are shown. You want to create hypotenuses having the same length ratios. a scale model of the swing set for a dollhouse using similar triangles. Sketch a drawing of your swing a. The lengths of the given pair of corresponding legs set and label each side length. Write a similarity are 6 and 18, and the lengths of the hypotenuses statement for each pair of similar triangles. State the are 10 and 30. Use the Pythagorean Theorem to scale factor you used to create the scale model. fi nd the lengths of the other pair of corresponding legs. Draw a diagram. 14 ft b. Write the ratio of the lengths of the second pair D A of corresponding legs. c. Are these triangles similar? Does this suggest a 6 ft Hypotenuse-Leg Similarity Theorem for right triangles? Explain. 6 ft 8 ft 8 ft F E 39. WRITING Can two triangles have all three ratios of corresponding angle measures equal to a value greater 18 in. than 1? less than 1? Explain. 9 in. B C 4 ft 40. HOW DO YOU SEE IT? Which theorem could you △ ∼ △ use to show that OPQ OMN in the portion of 45. PROOF △ABC has vertices A(−3, 4), B(−1, 5), = = the Ferris wheel shown when PM QN 5 feet and C(3, 2). △CDE has vertices C(3, 2), D(7, 4), = = and MO NO 10 feet? and E(15, −2). Use the SSS Similarity Theorem (Theorem 8.4) to show △ABC ∼ △CDE.

46. PROOF Use the SAS Similarity Theorem (Theorem 8.5) P to show △QRS ∼ △RST. M y 8 Q(−10, 4) R − Q ( 8, 4) O 4 N

−4 4 x S(−8, −2) T(10, −2) −4

41. DRAWING CONCLUSIONS Explain why it is not necessary to have an Angle-Side-Angle Similarity 47. DRAWING CONCLUSIONS Can you use the SAS Theorem. Similarity Theorem to prove that the triangles are similar? Explain. y 42. THOUGHT PROVOKING Decide whether each is a 8 valid method of showing that two quadrilaterals are A(−5, 4) similar. Justify your answer. 4 C(3, 2) a. SASA b. SASAS B(0, 0) c. SSSS d. SASSS −2 4 8 x F(6, −4) −4 D(−10, 0) 43. MULTIPLE REPRESENTATIONS Use a diagram to show why there is no Side-Side-Angle Similarity Theorem. E(0, −8)

48. MAKING AN ARGUMENT Your friend claims that if corresponding sides of two triangles have the same slope, the triangles must be similar. Is your friend correct? If so, is the converse of the statement true? Explain your reasoning.

Section 8.3 Proving Triangle Similarity by SSS and SAS 407

hsva_geo_pe_0803.indd 407 5/9/17 11:57 AM 49. PROVING A THEOREM Copy and complete the paragraph proof of the second part of y B n the Slopes of Parallel Lines Theorem (Theorem 3.13) from page 402. E = Given mℓ mn, ℓand n are nonvertical. Prove ℓ ʈ n BC EF You are given that m = m . By the defi nition of slope, m = — and m = — . By A D ℓ n ℓ AC n DF BC EF C F x ______, — = — . Rewriting this proportion yields ______. AC DF By the Right Angles Congruence Theorem (Thm. 2.3), ______. So, △ABC ∼ △DEF by ______. Because corresponding angles of similar triangles are congruent, ∠BAC ≅ ∠EDF. By ______, ℓ ʈ n.

y n 50. PROVING A THEOREM Copy and complete the two-column proof of the second part of E the Slopes of Perpendicular Lines Theorem (Theorem 3.14) from page 403. A Given m m = −1, ℓand n are nonvertical. ℓ n D j Prove ℓ⊥ n

STATEMENTS REASONS BC k x = − 1. mℓmn 1 1. Given DE AB 2. m = — , m = −— 2. Defi nition of slope ℓ AD n BC DE AB 3. — ⋅ −— = −1 3. ______AD BC DE BC 4. — = — 4. Multiply each side of statement 3 AD AB BC by −— . AB DE 5. — = — 5. Rewrite proportion. BC 6. ______6. Right Angles Congruence Theorem (Thm. 2.3) 7. △ABC ∼ △ADE 7. ______8. ∠BAC ≅ ∠DAE 8. Corresponding angles of similar fi gures are congruent. 9. ∠BCA ≅ ∠CAD 9. Alternate Interior Angles Theorem (Thm. 3.2) 10. m∠BAC = m∠DAE, m∠BCA = m∠CAD 10. ______11. m∠BAC + m∠BCA + 90° = 180° 11. ______12. ______12. Subtraction Property of Equality 13. m∠CAD + m∠DAE = 90° 13. Substitution Property of Equality 14. m∠CAE = m∠DAE + m∠CAD 14. Angle Addition Postulate (Post. 1.4) 15. m∠CAE = 90° 15. ______16. ______16. Defi nition of perpendicular lines

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. 51. A(−3, 6), B(2, 1); 3 to 2 52. A(−3, −5), B(9, −1); 1 to 3 53. A(1, −2), B(8, 12); 4 to 3

408 Chapter 8 Similarity

hsva_geo_pe_0803.indd 408 5/9/17 11:57 AM 8.4 Proportionality Theorems

Essential Question What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides?

Discovering a Proportionality Relationship Work with a partner. Use dynamic geometry software to draw any △ABC. — — — — a. Construct DE parallel to BC with endpoints on AB and AC , respectively.

A LOOKING FOR STRUCTURE b. Compare the ratios of AD to BD and AE to CE. To be profi cient in math, — — — — you need to look closely c. Move DE to other locations parallel to BC with endpoints on AB and AC , to discern a pattern and repeat part (b). or structure. d. Change △ABC and repeat parts (a)–(c) several times. Write a conjecture that summarizes your results.

Discovering a Proportionality Relationship Work with a partner. Use dynamic geometry software to draw any △ABC. a. Bisect ∠B and plot point D at the intersection— of the angle bisector and AC . B

b. Compare the ratios of AD to DC A D C and BA to BC. c. Change △ABC and repeat parts (a) and (b) several times. Write a conjecture that summarizes your results.

Communicate Your Answer B 3. What proportionality relationships exist in a triangle E intersected by an angle bisector or by a line parallel D to one of the sides? 4. Use the fi gure at the right to write a proportion. C A

Section 8.4 Proportionality Theorems 409

hsva_geo_pe_0804.indd 409 5/9/17 11:57 AM 8.4 Lesson What You Will Learn

Use the Triangle Proportionality Theorem and its converse. Core VocabularyVocabulary Use other proportionality theorems. Previous Using the Triangle Proportionality Theorem corresponding angles ratio proportion Theorems Theorem 8.6 Triangle Proportionality Theorem Q If a line parallel to one side of a triangle T intersects the other two sides, then it divides the two sides proportionally. R S U — — RT RU If TU ʈ QS , then — = — . Proof Ex. 27, p. 415 TQ US

Theorem 8.7 Converse of the Triangle Proportionality Theorem Q If a line divides two sides of a triangle T proportionally, then it is parallel to the R third side. S U RT RU — — If — = — , then TU ʈ QS . Proof Ex. 28, p. 415 TQ US

Finding the Length of a Segment — — — In the diagram, QS ʈ UT , RS = 4, ST = 6, and QU = 9. What is the length of RQ ?

R Q 4 9 S U 6 T

SOLUTION RQ RS — = — Triangle Proportionality Theorem QU ST RQ 4 — = — Substitute. 9 6 RQ = 6 Multiply each side by 9 and simplify. — The length of RQ is 6 units.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com — 1. Find the length of YZ . V 35 W 44 X

36 Y Z

410 Chapter 8 Similarity

hsva_geo_pe_0804.indd 410 5/9/17 11:57 AM The theorems on the previous page also imply the following: Contrapositive of the Triangle Inverse of the Triangle Proportionality Theorem Proportionality Theorem RT RU — — — — RT RU If — ≠ — , then TU ʈ QS . If TU ʈ QS , then — ≠ — . TQ US TQ US

Solving a Real-Life Problem

On the shoe rack shown, BA = 33 centimeters, C CB = 27 centimeters, CD = 44 centimeters, and DE = 25 centimeters. Explain why the shelf is not B parallel to the fl oor. D A SOLUTION E Find and simplify the ratios of the lengths. CD 44 CB 27 9 — = — — = — = — DE 25 BA 33 11 44 9 — — Because — ≠ — , BD is not parallel to AE . So, the shelf is not parallel to the fl oor. 25 11 Q P 50 90 Monitoring Progress Help in English and Spanish at BigIdeasMath.com — — N 72 40 RS 2. Determine whether PS ʈ QR .

Recall that you partitioned a directed line segment in the coordinate plane in Section 3.5. You can apply the Triangle Proportionality Theorem to construct a point along a directed line segment that partitions the segment in a given ratio.

Constructing a Point along a Directed Line Segment — Construct the point L on AB so that the ratio of AL to LB is 3 to 1.

SOLUTION Step 1 Step 2 Step 3

C C C G G F F E E D D

A B A B A J LK B

— ∠ Draw a— segment and a ray Draw arcs Place the point of Draw a segment Draw GB . Copy AGB Draw AB of any length. Choose any a compass at A and make an arc and construct congruent angles at D, E, ៮៮៮⃗ — point C not on ⃖AB៮៮⃗ . Draw ៮AC៮៮⃗ . of any radius intersecting AC . Label and F with sides— —that intersect— AB at J, K, the point of intersection D. Using the and L. Sides DJ , EK , and FL are all same compass setting, make three — ៮៮៮⃗ parallel, and they divide AB equally. So, more arcs on AC , as shown. Label the AJ = JK = KL = LB. Point L divides points of intersection E, F, and G and directed line segment AB in the ratio 3 to 1. note that AD = DE = EF = FG.

Section 8.4 Proportionality Theorems 411

hsva_geo_pe_0804.indd 411 5/9/17 11:57 AM Using Other Proportionality Theorems Theorem Theorem 8.8 Three Parallel Lines Theorem If three parallel lines intersect two transversals, tsr then they divide the transversals proportionally. U W Y m ZXV

UW VX — = — Proof Ex. 32, p. 415 WY XZ

Using the Three Parallel Lines Theorem

In the diagram, ∠1, ∠2, and ∠3 are all congruent, = = GF 120 yards, DE 150 yards, and F Main St. 1 E CD = 300 yards. Find the distance HF between Main Street and South Main Street. 120 yd2 150 yd G D Second St. SOLUTION 300 yd Corresponding angles are congruent, so 3 ⃖៮៮⃗ ⃖៮៮⃗ ⃖៮៮⃗ South Main St. FE , GD, and HC are parallel. There are HC different ways you can write a proportion to fi nd HG. Method 1 Use the Three Parallel Lines Theorem to set up a proportion. HG CD — = — Three Parallel Lines Theorem GF DE HG 300 — = — Substitute. 120 150 HG = 240 Multiply each side by 120 and simplify. By the Segment Addition Postulate (Postulate 1.2), HF = HG + GF = 240 + 120 = 360.

The distance between Main Street and South Main Street is 360 yards.

Method 2 Set up a proportion involving total and partial distances. Step 1 Make a table to compare the distances.

⃖CE៮៮⃗ ⃖ HF៮៮⃗

Total distance CE = 300 + 150 = 450 HF

Partial distance DE = 150 GF = 120

Step 2 Write and solve a proportion. 450 HF — = — Write proportion. 150 120 360 = HF Multiply each side by 120 and simplify.

The distance between Main Street and South Main Street is 360 yards.

412 Chapter 8 Similarity

hsva_geo_pe_0804.indd 412 5/9/17 11:57 AM Theorem Theorem 8.9 Triangle Angle Bisector Theorem If a ray bisects an angle of a triangle, then A it divides the opposite side into segments D whose lengths are proportional to the lengths of the other two sides. CB AD CA — = — Proof Ex. 35, p. 416 DB CB

Using the Triangle Angle Bisector Theorem — In the diagram, ∠QPR ≅ ∠RPS. Use the given side lengths to fi nd the length of RS .

Q 7 PR 15 x 13

S SOLUTION Because ៮ PR៮៮⃗ is an angle bisector of ∠QPS, you can apply the Triangle Angle Bisector Theorem. Let RS = x. Then RQ = 15 − x. RQ PQ — = — Triangle Angle Bisector Theorem RS PS 15 − x 7 — = — Substitute. x 13 195 − 13x = 7x Cross Products Property 9.75 = x Solve for x. — The length of RS is 9.75 units.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the length of the given line segment. — — 3. BD 4. JM E M 3 40 J C 16 16 G A 2 1 15 18 HK N 30 FDB

Find the value of the variable.

5. S 6. Y 14 T 4 24 x Z 4 2 4 V U 48 W y X

Section 8.4 Proportionality Theorems 413

hsva_geo_pe_0804.indd 413 5/9/17 11:57 AM 8.4 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE STATEMENT If a line divides two sides of a triangle proportionally, then it is ______to the third side. This theorem is known as the ______. — 2. VOCABULARY In △ABC, point R lies on BC and ៮ AR៮៮⃗ bisects ∠CAB. Write the proportionality statement for the triangle that is based on the Triangle Angle Bisector Theorem (Theorem 8.9).

Monitoring Progress and Modeling with Mathematics — In Exercises 3 and 4, fi nd the length of AB . In Exercises 13–16, use the diagram to complete the (See Example 1.) proportion.

A E A 3. 4. F D B 14 C 12 E B G E DB 12 18 3 4 C D C BD CG BF — — 13. — = — 14. — = — In Exercises 5–8, determine whether KM || JN . BF CG DF (See Example 2.) EG DF CG 15. — = — 16. — = — 5. L 6. J N CE BD CE 8 12 22.5 18 K K M M In Exercises 17 and 18, fi nd the length of the indicated 5 7.5 25 20 (See Example 3.) J line segment. N L — — 17. VX 18. SU 7. L 8. J YZ N P 8 RT 12 24 20 WX15 18 35 N S U UV M 34 8 10 K 15 K 10 M 16 NJ 15 In Exercises 19–22, fi nd the value of the variable. (See Example 4.) L 19. 20. z y 3 CONSTRUCTION In Exercises 9–12, draw a segment with 8 1.5 the given length. Construct the point that divides the 4.5 segment in the given ratio. 46 9. 3 in.; 1 to 4 21. 22. p q 36 10. 2 in.; 2 to 3 16.5 16

11. 12 cm; 1 to 3 11 28 29 12. 9 cm; 2 to 5

414 Chapter 8 Similarity

hsva_geo_pe_0804.indd 414 5/9/17 11:57 AM 23. ERROR ANALYSIS Describe and correct the error in 29. MODELING WITH MATHEMATICS The real estate term solving for x. lake frontage refers to the distance along the edge of a piece of property that touches a lake. ✗ A x D 14 C lake 10 16 174 yd B Lot A AB CD 10 14 Lot B = = — Lot C — — — x BC AD 16 48 yd 55 yd 61 yd 10x = 224 Lakeshore Dr. x = 22.4 a. Find the lake frontage (to the nearest tenth) of each lot shown. 24. ERROR ANALYSIS Describe and correct the error in the student’s reasoning. b. In general, the more lake frontage a lot has, the higher its selling price. Which lot(s) should be ✗ B listed for the highest price? D c. Suppose that lot prices are in the same ratio as lake frontages. If the least expensive lot is $250,000, ACwhat are the prices of the other lots? Explain BD AB your reasoning. Because — = — and BD = CD, CD AC it follows that AB = AC. 30. USING STRUCTURE Use the diagram to fi nd the values of x and y.

2 MATHEMATICAL CONNECTIONS In Exercises 25 and 26, — — 1.5 fi nd the value of x for which PQ ʈ RS . 5 P Q 25. P 2x + 4 26. x S 5 3 T 12 21 y R R 7 S x − 2 2 x − 3x + 5 3 1 Q T 31. REASONING In the construction on page 411, explain why you can apply the Triangle Proportionality 27. PROVING A THEOREM Prove the Triangle Theorem (Theorem 8.6) in Step 3. Proportionality Theorem (Theorem 8.6). — — 32. PROVING A THEOREM Use the diagram with the ʈ Q Given QS TU T auxiliary line drawn to write a paragraph proof of the Three Parallel Lines Theorem (Theorem 8.8). QT = SU R Prove — — ʈ ʈ TR UR Given k1 k2 k3 S U CB DE Prove — = — BA EF 28. PROVING A THEOREM Prove the Converse of the Triangle Proportionality Theorem (Theorem 8.7). t t auxiliary 1 2 line ZY ZX C D Given — = — W Y YW XV k B E 1 — ʈ — Z Prove YX WV k V X AF 2 k 3

Section 8.4 Proportionality Theorems 415

hsva_geo_pe_0804.indd 415 5/9/17 11:58 AM 33. CRITICAL THINKING— In △LMN, the angle bisector of 38. MAKING AN ARGUMENT Two people leave points A ∠M also bisects LN . Classify △LMN as specifi cally as and B at the same time. They intend to meet at possible. Justify your answer. point C at the same time. The person who leaves point A walks at a speed of 3 miles per hour. You and a friend are trying to determine how fast the person HOW DO YOU SEE IT? 34. During a football game, who leaves point B must walk. Your friend claims you the quarterback throws the ball to the receiver. The — need to know the length of AC . Is your friend correct? receiver is between two defensive players, as shown. Explain your reasoning. If Player 1 is closer to the quarterback when the ball is thrown and both defensive players move at the same speed, which player will reach the receiver A B fi rst? Explain your reasoning. 0.6 mi 0.9 mi ED C

39. CONSTRUCTION Given segments with lengths r, s, r t and t, construct a segment of length x, such that — = — . s x

35. PROVING A THEOREM Use the diagram with the t auxiliary lines drawn to write a paragraph proof of the Triangle Angle Bisector Theorem (Theorem 8.9). 40. PROOF Prove Ceva’s Theorem: If P is any point AY CX BZ Given ∠YXW ≅ ∠WXZ inside △ABC, then — ⋅ — ⋅ — = 1. YC XB ZA YW XY Prove — = — WZ XZ N B M Y Z X

X W P

A Z auxiliary lines A Y C — (Hint: Draw segments parallel to BY through A and C, 36. THOUGHT PROVOKING Write the converse of the as shown. Apply the Triangle Proportionality Theorem Triangle Angle Bisector Theorem (Theorem 8.9). (Theorem 8.6) to △ACM. Show that △APN ∼ △MPC, Is the converse true? Justify your answer. △CXM ∼ △BXP, and △BZP ∼ △AZN.)

37. REASONING How is the Triangle Midsegment Theorem (Theorem 6.8) related to the Triangle Proportionality Theorem (Theorem 8.6)? Explain your reasoning.

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Use the triangle. c 41. Which sides are the legs? a 42. Which side is the hypotenuse? b Solve the equation. 43. x2 = 121 44. x2 + 16 = 25 45. 36 + x2 = 85

416 Chapter 8 Similarity

hsva_geo_pe_0804.indd 416 5/9/17 11:58 AM Chapter Review Dynamic Solutions available at 8 BigIdeasMath.com 8.1 Similar Polygons (pp. 381–390)

In the diagram, EHGF ∼ KLMN. Find the scale factor from EHGF to KLMN. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. — — From the diagram, you can see that EH and KL K 18 L are corresponding sides. So, the scale factor of E 12 H KL 18 3 15 EHGF to KLMN is — = — = — . 10 EH 12 2 14 21 F ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ N E K, H L, G M, and F N. 16 G 24 KL LM MN NK — = — = — = — M EH HG GF FE

Find the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. 1. ABCD ∼ EFGH 2. △XYZ ∼ △RPQ A B F G Y PQ6 25 8 9 15 10 8 R D 12 C E H XZ 6 20 3. Two similar triangles have a scale factor of 3 : 5. The altitude of the larger triangle is 24 inches. What is the altitude of the smaller triangle? 4. Two similar triangles have a pair of corresponding sides of length 12 meters and 8 meters. The larger triangle has a perimeter of 48 meters and an area of 180 square meters. Find the perimeter and area of the smaller triangle.

8.2 Proving Triangle Similarity by AA (pp. 391–396)

Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. ° Because they are both right angles, ∠F and ∠B are congruent. D A 29 B By the Triangle Sum Theorem (Theorem 5.1), 61° + 90° + m∠E = 180°, ∠ = ° ∠ ∠ △ ∼ △ 61° so m E 29 . So, E and A are congruent. So, DFE CBA by C the AA Similarity Theorem (Theorem 8.3). F E Show that the triangles are similar. Write a similarity statement. 5. R 6. CB F E ° Q 35 T S 35° 60° 30° A D U

7. A cellular telephone tower casts a shadow that is 72 feet long, while a nearby tree that is 27 feet tall casts a shadow that is 6 feet long. How tall is the tower?

Chapter 8 Chapter Review 417

hsva_geo_pe_08ec.indd 417 5/9/17 11:52 AM 8.3 Proving Triangle Similarity by SSS and SAS (pp. 397–408)

Show that the triangles are similar.

a. A D 35 15 14 6 EF12 B 28 C Compare △ABC and △DEF by fi nding ratios of corresponding side lengths.

Shortest sides Longest sides Remaining sides AB 14 7 AC 35 7 BC 28 7 — = — = — — = — = — — = — = — DE 6 3 DF 15 3 EF 12 3 All the ratios are equal, so △ABC ∼ △DEF by the SSS Similarity Theorem (Theorem 8.4). b. Y 21 20 V X 30 Z 14 W ∠YZX ≅ ∠WZV by the Vertical Angles Congruence Theorem (Theorem 2.6). Next, compare the ratios of the corresponding side lengths of △YZX and △WZV. WZ 14 2 VZ 20 2 — = — = — — = — = — YZ 21 3 XZ 30 3

So, by the SAS Similarity Theorem (Theorem 8.5), △YZX ∼ △WZV.

Use the SSS Similarity Theorem (Theorem 8.4) or the SAS Similarity Theorem (Theorem 8.5) to show that the triangles are similar. 8. C 9. T 4 3.5 4.5 B D U 15 8 7 10 9

A E S 7 R 14 Q

10. Find the value of x that makes △ABC ∼ △DEF.

B E 6 2x 32 9 CA 24

DFx + 6 12

418 Chapter 8 Similarity

hsva_geo_pe_08ec.indd 418 5/9/17 11:52 AM 8.4 Proportionality Theorems (pp. 409–416) — — a. Determine whether MP ʈ LQ . — Begin by fi nding and simplifying ratios of lengths determined byMP . N NM 8 2 — = — = — = 2 ML 4 1 8 24 NP 24 2 M P — = — = — = 2 12 PQ 12 1 4 L Q NM NP — — Because — = — , MP is parallel to LQ by the Converse of the Triangle Proportionality ML PQ Theorem (Theorem 8.7). — — b. In the diagram, AD bisects ∠CAB. Find the length of DB . — Because AD is an angle bisector of ∠CAB, you can apply the Triangle Angle Bisector Theorem (Theorem 8.9). DB AB — = — Triangle Angle Bisector Theorem A DC AC x 15 — = — Substitute. 15 5 8 8 8x = 75 Cross Products Property = x 9.375 x Solve for . C 5 D x B — The length of DB is 9.375 units. — — Determine whether AB ʈ CD . 11. D 12. C D 10 B 13.5 12 16 AB C A 20 28 E 22.5 20 — E 13. Find the length of YB . A 15 7 Z Y 24

B C — Find the length of AB . 14. 15. B B D 4 A 6 10 7 4

CA18

Chapter 8 Chapter Review 419

hsva_geo_pe_08ec.indd 419 5/9/17 11:52 AM