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Conditions of Special Parallelograms 399 Activity Assess Activity Assess 8-6 MODEL & DISCUSS Conditions The sides of the lantern are identical quadrilaterals. of Special Parallelograms A. Construct Arguments How could you check to see whether a side is a parallelogram? Justify your PearsonRealize.com answer. B C B. Does the side appear to be I CAN… identify rhombuses, rectangular? How could rectangles, and squares by you check? the characteristics of their E diagonals. C. Do you think that diagonals of a quadrilateral can be used to determine whether the quadrilateral is a A D rectangle? Explain. Which properties of the diagonals of a parallelogram help you to classify ESSENTIAL QUESTION a parallelogram? CONCEPTUAL UNDERSTANDING EXAMPLE 1 Use Diagonals to Identify Rhombuses Information about diagonals can help to classify A B a parallelogram. In parallelogram ABCD, ​​AC‾ ​​ is perpendicular to ​​BD‾ ​​. What else can you conclude about the parallelogram? E D C STUDY TIP A B Parallelograms have several The diagonals of Any angle at E either properties, and some properties a parallelogram forms a linear pair or bisect each other, may not help you solve a _ E is a vertical angle with particular problem. Here, the fact so ​​AE‾ ​ _≅ ​CE ​ and ∠AEB, so all four angles ​​DE‾​ ​BE ​ . that diagonals bisect each other ≅ D C are right angles. allows the use of SAS. The four triangles are congruent by SAS, so ​​AB‾ ​ ≅ ​CB‾ ​ ≅ ​CD‾ ​ ≅ ​AD‾ ​.​ Since ABCD is a parallelogram with four congruent sides, ABCD is a rhombus. Try It! 1. If ∠JHK and ∠JGK are complementary, H what else can you conclude about L GHJK? Explain. G J K 398 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess THEOREM 8-19 Converse of Theorem 8-16 K If the diagonals of a parallelogram If... J are perpendicular, then the parallelogram is a rhombus. L M PROOF: SEE EXERCISE 9. Then... ​​JK‾ ​ ≅ ​KL‾ ​ ≅ ​LM‾ ​ ≅ ​MJ‾ ​ THEOREM 8-20 Converse of Theorem 8-17 If a diagonal of a parallelogram If... B C bisects two angles of the parallelogram, then the parallelogram is a rhombus. A D PROOF: SEE EXAMPLE 2. Then... ​​AB‾ ​ ≅ ​BC‾ ​ ≅ ​CD‾ ​ ≅ ​DA‾ ​ PROOF EXAMPLE 2 Prove Theorem 8-20 Write a proof of Theorem 8-20. G Given: Parallelogram FGHJ with 1 ​ 1 2​ and ​ 3 4​ F 3 ∠ ≅ ∠ ∠ ≅ ∠ 2 4 H Prove: FGHJ is a rhombus. Proof: J G G G STUDY TIP Drawing diagonals in 1 1 1 parallelograms can help you see F 3 F 3 F 3 2 2 2 additional information that is 4 H 4 H 4 H useful in solving problems. J J J By ASA,_ △​ FHJ ≅ △FHG​. By the Alternate By the Converse of Thus, ​​FJ ​ ≅ ​FG‾ ​ . Interior Angles the Isosceles Triangle Theorem, ∠​ 1 ≅ ∠4​, so Theorem,_ ​​FG‾ ​ ≅ ​HG‾ ​ ​∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4​. and ​​FJ ​ ≅ ​HJ‾ ​​. _ Using the Transitive Property of Congruence, ​​FG‾ ​ ≅ ​HG‾ ​ ≅ ​FJ ​ ≅ ​HJ‾ ​ . Since FGHJ is a parallelogram with congruent sides, it is a rhombus. Try It! 2. Refer to the figure FGHJ in Example 2. Use properties of parallelograms to show that if ∠​ 1 ≅ ∠2​ and ∠​ 3 ≅ ∠4​, then the four angles are congruent. LESSON 8-6 Conditions of Special Parallelograms 399 Activity Assess EXAMPLE 3 Use Diagonals to Identify Rectangles Ashton measures the diagonals for his deck frame and finds that they are congruent. Will the deck be rectangular? 10 ft 8 ft Since opposite sides are congruent, the supports form a parallelogram. To show that 8 ft the structure is rectangular, 10 ft show that the angles are right angles. A B Opposite sides and the diagonals are congruent, so △​ ​ ACD ≅ △BDC​ by SSS. Therefore, ∠​ ADC ≅ ∠BCD​. D C In a parallelogram, consecutive angles are supplementary. Angles that are congruent and supplementary are right angles. Similarly, ∠DAB and ∠CBA are also right angles. The frame forms a parallelogram with four right angles, which is a rectangle. Try It! 3. If the diagonals of any quadrilateral are congruent, is the quadrilateral a rectangle? Justify your answer. THEOREM 8-21 Converse of Theorem 8-18 If the diagonals of a parallelogram If... X Y are congruent, then the XZ ​ WY parallelogram is a rectangle. ≅​ W Z PROOF: SEE EXERCISE 11. Then... ∠XWZ, ∠WZY, ∠XYZ, and ∠WXY are right angles EXAMPLE 4 Identify Special Parallelograms Can you conclude whether each parallelogram is a rhombus, a square, or a rectangle? Explain. A. Parallelogram ABCD B CONSTRUCT ARGUMENTS There are often multiple ways By SAS, ​ ABD CBD​. to prove something. How △ ≅ △ A C could you use properties of parallelograms to show the figure ADB CDB by CPCTC. is a rhombus without the congruent ​∠ ≅ ∠ ​ D angles shown? Diagonal ​​BD‾ ​​ bisects ∠ABC and ∠ADC, so parallelogram ABCD is a rhombus. CONTINUED ON THE NEXT PAGE 400 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess EXAMPLE 4 CONTINUED B. Parallelogram PQRS Q Diagonals are perpendicular, so PQRS is a rhombus. P R Diagonals are congruent, so PQRS is a rectangle. S Since the parallelogram is a rhombus and a rectangle, it is a square. Try It! 4. Is each parallelogram a rhombus, a square, or a rectangle? Explain. a. H J b. U V L X T G K W EXAMPLE 5 Use Properties of Special Parallelograms Quadrilateral STUV is a rhombus. T (65 y) (5y 5) What are the values of x and y? MAKE SENSE AND PERSEVERE Consider the information given S U in the diagram. How_ can you determine whether ​​TV ​​ bisects the angles? (4x 3) (5x 10) V If a parallelogram is a rhombus then each diagonal bisects opposite angles. So, ​​TV‾ ​​ bisects ∠​SVU​ and ∠​STU​. Solve for x. Solve for y. _ ​m∠SVT = m∠UVT​ ​​TV ​​ bisects ∠SVU ​m∠STV = m∠UTV​ and ∠STU. ​4x + 3 = 5x −10​ ​65 − y = 5y + 5​ ​−x = −13​ ​−6y = −60​ ​x = 13​ ​y = 10​ Try It! 5. In parallelogram ABCD, ​AC = 3w − 1​ and ​BD = 2(w + 6)​. What must be true for ABCD to be a rectangle? LESSON 8-6 Conditions of Special Parallelograms 401 Activity Assess APPLICATION EXAMPLE 6 Apply Properties of Special Parallelograms A group of friends set up a kickball field with bases 60 ft apart. How can COMMON ERROR they verify that the field is a square? The order in which the quadrilateral is identified matters. Opposite sides are congruent, Be sure to first show that the so the field is a parallelogram. quadrilateral is a parallelogram before applying the theorems 60 ft to identify the quadrilateral as a 60 ft rhombus or a rectangle. 60 ft 60 ft All sides are congruent, so the parallelogram is a rhombus. Home Plate The field is a rhombus. To show that the rhombus is a square, show that it is also a rectangle. A parallelogram is a rectangle if the diagonals are congruent. Second Base Measure the distances from first to third base Third Base First Base and from home plate to second base. Home Plate The group of friends can verify the field is a square if they find that the distances from first base to third base and from second base to home plate are equal. Try It! 6. Is MNPQ a rhombus? Explain. N M 32 R 58 58 P Q 402 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Concept Summary Assess CONCEPT SUMMARY Conditions of Special Parallelograms RHOMBUS RECTANGLE SQUARE A parallelogram is a rhombus if A parallelogram is a rectangle if A parallelogram is a square if • diagonals are perpendicular • diagonals are congruent • diagonals are perpendicular and congruent • a diagonal bisects angles • a diagonal bisects angles and diagonals are congruent Do You UNDERSTAND? Do You KNOW HOW? 1. ESSENTIAL QUESTION Which For Exercises 5–8, is the parallelogram a properties of the diagonals of a rhombus, a square, or a rectangle? parallelogram help you to classify 5. 6. a parallelogram? 5 45 5 45 E 5 2. Error Analysis Sage 5 45 was asked to classify 45 DEFG. What was Sage’s error? D F 7. 8. DF EG G 9. What value of x will make the Since DF = EG, DEFG is a rectangle. parallelogram a rhombus? Since EG DF, DEFG is also a rhombus. Therefore, DEFG is a square. ✗ x 36 10. If m∠1 = 36 and m∠2 = 54, is PQRS a 3. Construct Arguments Write a biconditional rhombus, a square, a rectangle, or none of statement about the diagonals of rectangles. these? Explain. What theorems justify your statement? 4. Use Appropriate Tools Make a concept Q R map showing the relationships among U quadrilaterals, parallelograms, trapezoids, 1 2 isosceles trapezoids, kites, rectangles, P S squares, and rhombuses. LESSON 8-6 Conditions of Special Parallelograms 403 Scan for Practice Tutorial PRACTICE & PROBLEM SOLVING Multimedia Additional Exercises Available Online UNDERSTAND PRACTICE 11. Construct Arguments Write a proof for For Exercises 17 and 18, determine whether each Theorem 8-19 using the following figure is a rhombus. Explain your answer. diagram. SEE EXAMPLES 1 AND 2 B 17. 18. 100 A C 40 E 40 D P 12. Error Analysis Becky is 50 Q 19. What is the perimeter of parallelogram asked to classify PQRS. 50 WXYZ? SEE EXAMPLE 3 What is her error? 48 X Y S 48 R 13 13 13 13 W Z PR bisects opposite angles 24 SPQ and QRS, so For Exercises 20 and 21, determine the name PQRS must be a rhombus. ✗ that best describes each figure: parallelogram, rectangle, square, or rhombus. SEE EXAMPLE 4 20. 21. 13. Construct Arguments Write a proof for Theorem 8-21 using the following diagram. G H For Exercises 22–24, give the condition required for each figure to be the specified shape.
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