7.2 Isosceles and Equilateral Triangles

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© Houghton Mifflin Harcourt Publishing Company Module 7 B A characteristics/properties triangles. of special these In activity, this and youtriangles construct isosceles investigate will other potential The angles that have as asideare base the the The sideopposite vertex the angle is the The angle formed by legsis the the The congruent sidesare the called An Essential Question: 7.2 Name isosceles triangle isosceles

Explore sides of angle. the points the Label Using acompass, place point the on vertex the and draw an arc that intersects the yourLabel angle yourDo work spaceprovided. inthe Use astraightedge to draw an angle.

Triangles Isosceles andEquilateral

is a triangle with is at atriangle least two congruent sides.

and equilateral triangles? What are thespecialrelationships amonganglesandsidesinisosceles ∠

Investigating Isosceles Triangles Isosceles Investigating A , as shown figure. inthe legs legs vertex angle vertex of triangle. the base B B B and and . base angles base . C A A A . Class 327

. C C Date Vertex angle

Base angles

Base Legs Resource Locker Lesson 2

∠

Triangle 4 Triangle Triangle 3 Triangle C 328

. _ BC A Triangle 2 Triangle B Looking at your results, what conjecture can be made about the base angles, be can about made conjecture what results, your Looking at

Triangle 1 Triangle Proving the Isosceles Triangle Theorem Theorem Proving the Triangle Isosceles and Its Converse

? B A C C ∠ ∠ ∠ ∠ m m m and

B Use a protractor to measure each angle. Record the measures in the table under the column the column under in the table the measures Record each angle. measure to a protractor Use 1. Triangle for Use the straightedge to draw line segment segment line draw to the straightedge Use Repeat steps A–D at least two more times and record the results in the table. Make sure sure Make in the table. the results record times and more two least at A–D steps Repeat size each time. a different is Explain 1

How do you know the triangles you constructed are isosceles triangles? are constructed the triangles you know do you How a Conjecture Make ∠

Isosceles Triangle Theorem Triangle Isosceles Reflect If two sides of a triangle are congruent, then the two angles opposite the sides are the sides are opposite angles then the two congruent, a triangle are sides of two If congruent.

D Module 7 Module C This conjecture can be proven so it can be stated as a theorem. as be can stated so it be can proven conjecture This In the Explore, you made a conjecture that the base angles of an isosceles triangle are congruent. congruent. isosceles an triangle are the base of angles that a conjecture made you the Explore, In

2. 1.

E This theorem is sometimes called the Base Angles Theorem and can also be stated as “Base angles angles “Base also as can be called and stated sometimes is the Base theorem Theorem This Angles congruent.” isosceles an triangle are of

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© Houghton Mifflin Harcourt Publishing Company Prove: oue 7 Module 3. Reflect Given: Step 3 those two If Given: Step 1 Step 2 Prove: Example 1 so thatso vertex might it help to sketch of areflection given the next triangle to triangle, original the Discussion 5. 3. 2. 4. 4. 3. 2. 5. 1. 1. Complete proof the of Converse the of Isosceles the Triangle Theorem. Complete proof the of Isosceles the Triangle Theorem. Complete statement the of Converse the of Isosceles the Triangle Theorem.

∠ ∠ AB _ △ ∠ △ CA _ BA BA _ Prove Triangle theIsosceles Theorem and its converse.

A ABC

AB _ ABC BAC ∠

≅ ≅ ≅ ∠ ≅ B

∠ AB ≅ _ BA BA _ AC AC _ ≅ ≅ CA CA _ ≅ ≅ Statements Statements B ≅ In proofs the of Isosceles the Triangle Theorem and its converse, how A

AC AC _

∠

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of a are are

B B A A . are congruent, two the then 5. 4. 2. 1. 3. 5. 3. 2. 1. 4. C C CPCTC CPCTC Reflexive Property ofCongruenceReflexive Property Given Symmetric Property ofEquality Symmetric Property

Symmetric Property ofEquality Symmetric Property Given 329 Reasons Reasons

opposite esn 2 Lesson © Houghton Mifflin Harcourt Publishing Company C Lesson 2 A . C .

B _ BC ≅

A _ AC ≅

_ AB

B . , and therefore, therefore, , and

. Property of Congruence. of Property by the

330

. ≅ ∠ B ∠

the Isosceles Triangle Theorem, since Theorem, Triangle the Isosceles .

by by the B ∠

by the by ≅

A C ≅ C by

we know that know we ∠ ∠

_ BC

∠ C _ BC

_ AB ≅

, ≅ ≅ ≅ ∠ _ AC

≅

B C is a triangle with three congruent angles. angles. congruent a triangle three is with B B and Its Converse is a triangle with three congruent sides. congruent a triangle three is with Proving the Equilateral Triangle Theorem Theorem Proving the Triangle Equilateral ≅ ∠

≅ ∠ ∠

∠ _ AC

_ AC A

≅ ≅ ≅ _ AB . ∠ ≅ ∠ ≅ ≅

B B A A A by the Converse of the Isosceles Triangle Theorem because Theorem Triangle the Isosceles of the Converse by

∠ ∠ ∠

_ AB ∠ _ AB ∠ _ BC ≅ converse. its and Theorem the Equilateral Triangle Prove ≅

that also is known It Given that Because _ AC

Complete the proof of the Equilateral Triangle Theorem. Triangle the Equilateral of the proof Complete Complete the proof of the Converse of the Equilateral Triangle Theorem Triangle the Equilateral of the Converse of theproof Complete To prove the Equilateral Triangle Theorem, you applied the theorems of isosceles of the theorems applied you Theorem, Triangle the Equilateral prove To triangles between equilateral the relationship about be can concluded triangles. What isosceles triangles?and

equiangular triangle equilateral triangle Converse of the Equilateral Triangle Theorem Triangle of the Equilateral Converse Equilateral Triangle Theorem Triangle Equilateral If a triangle is equiangular, then it is equilateral. is then it a triangle equiangular, is If If a triangle is equilateral, then it is equiangular. is then it a triangle equilateral, is If Example 2 Finally,

Prove: An Therefore, Module Module 7 Reflect 4. Given: The converse of the Equilateral Triangle Theorem is also is true. Theorem Triangle the Equilateral of converse The Given: Step 1 Step An 2 Explain

Prove: Congruence, of Property the Transitive by Thus, Step 2 Step

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© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Nelvin C. Cepeda/ZUMA Press/Corbis 6 Substitute 6for 6 oue 7 Module To measure the find of vertex the angle of value the triangle, the firstof find B To length the of find eachsideof value the triangle, the firstof find A You properties can the use of and isosceles equilateral to triangles solve problems involving theorems. these Explain3 So, length the of eachsideof is 31cm. triangle the Example 3 Katie is stitching center the inlay onto abanner that m So, So, centimeters. What is length the of eachsideof triangle? the an equilateral with following the triangle dimensions in she created to represent It hertutorial new service. is x ( ∠ 6

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. esn 2 Lesson © Houghton Mifflin Harcourt Publishing Company 1

Lesson 2 - y C 4 5 A

B + y 3 10 332 Discuss how the sides of an isosceles triangle relate isosceles an triangle relate the sides of Discuss how

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- x Q (5 R 3)°

Consider the vertex and base angles of an isosceles Can they an triangle. be base of angles and the vertex Consider +

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QuestionEssential Check-In

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Your Turn Your 8. Module 7 Module 7.

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© Houghton Mifflin Harcourt Publishing Company Prove: oue 7 Module 1. Prove: ie: Given: 2. ie: Given: 3. about about formed. What about is true angles two the base of intersection the Label of angle the sides as point along line the and draw copied the angle that so angle the angle. original facesthe acompassThen use to copy angle. the Placecompass the point at another point Use astraightedge. Draw aline. Draw an acute angle with vertex Prove Isosceles the Triangle Theorem as aparagraph proof. Complete flowproof the of Equilateral the Triangle Theorem. Evaluate: Homework andPractice CA _ ∠ ∠ AB AB _ _ A B

and and

AC ≅ ≅ ≅ ≅ Giv

≅ ∠ AC AC AC ∠ _ _ en CB _

BC C B

≅ ≅ ? What kindof didyou triangle form? Explain your reasoning.

BC BC ∠ _ C

Isosceles Triangle Theorem C 333 . Look at. Look you triangle the have △ ABC ? What doyou know ∠ A ≅

A ∠ B along line. the ≅

∠ C B

• Extra Practice Extra • Help and Hints • Homework Online • esn 2 Lesson © Houghton Mifflin Harcourt Publishing Company Lesson 2 F 1)° + R 4)°

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2 Lesson 2 © Houghton Mifflin Harcourt Publishing Company Lesson 2 C T B 40º 80º 2.4 mi ge07se_c04l08003aa ABeckmann A All angles angles are congruent. are congruent. 336

. , is the triangle an acute, isosceles, acute, the triangle, is an . After . After → ‾

° ° AC 107° 54 = ° x L Isosceles Triangles Triangles ∠ , the angle to the to , the angle B ? BT and m ° 63 = , an air-traffic controller in controller air-traffic , an sides J

sides A ∠ are congruent. are congruent. effect the theorems and of the cause show to the diagram Complete with m . How can you find you can . How ° JKL measures the angle to the plane as 40 as the plane to the angle measures △ . Explain your reasoning. The horizontal lines are parallel. are lines horizontal The reasoning. your . Explain T x

Summarize Find Find Given triangle? right or obtuse, A plane is flying parallel to the ground along along flying the ground is parallel to A plane covered in the lesson. Explain why the arrows show the direction going both ways. going the direction show the arrows why Explain in the lesson. covered When the plane is at at is the plane When tower 2.4 miles to traveled has the plane 80 is plane

14. 13. 12. Module Module 7 15.

© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©De Klerk/ Alamy oue 7 Module 18. 17. 16. or false. Select the correct the answeror Select false. for eachlettered Explain part. your reasoning. isosceles triangle? triangle? isosceles shown. If m John is building adoghouse. He roof to the decides use design truss Justify Reasoning measure of angle. abase sum the Determine of measures the ofangles. base the The measure of vertex the angle of is 12more antriangle isosceles than the 5times a. e. e. d. c. b. All isosceles triangles have triangles isosceles All at least triangle is obtuse.triangle The exterior angle of an equilateral can determine its perimeter. legsofthe an triangle, you isosceles If you know length the of one of equilateral triangles. are triangles isosceles All each of its sidesis is triangle If of perimeter the an equilateral two acute angles. F ∠ DBF P , then the length the of, then D =

35 Determine whether eachof whether Determine following the statements is true B P __ 3

. ° , what is measure the of vertex the angle of the (2 x + A

15)° C E 337 G True True True True True False False False False False esn 2 Lesson © Houghton Mifflin Harcourt Publishing Company C Lesson 2 . B D CBD 40° and A ABD

and B

. _ BC Reasons

, describe triangles _ AC ⟘ C

are angle bisectors, bisectors, angle are _ BD has a right angle at at angle a right has _ CD

is the midpoint of of the midpoint is . If . If ABC M 338

and _ AC A M _ AD is on on is D , given point point , given B C ∠

≅ , and point point , and B B Isosceles right triangle right Isosceles ∠

Prove

Statements is an isosceles triangle and isosceles an triangle is and ?

bisects angle bisects angle ABC ADC _ △ BD

∠ . _ CB Focus on Higher Order Thinking on Higher Order Focus ≅

_ Analyze Relationships Analyze AB Critical Thinking Critical Given that that Given what iswhat m Explain. HINT: Draw a diagram. Draw HINT: Explain.

H.O.T. H.O.T. Module 7 Module 21. 19. 20.

Construct antriangle. Explain isosceles howConstruct you know that two sidesare congruent. congruent. an equilateralConstruct triangle. Explain how you know three the sidesare I know two sidesare congruent because I know three the sidesare congruent because • • • • • • • and to intersection the point. Use astraightedge to draw line segments connecting centers both to eachother Mark one point where circles the intersect. circle. should have (Both same the radius length.) Draw another circle of same the sizethat through goes center the of first the Use acompass to draw acircle. Draw aline segment (chord) two the points between on circle. the of two the points on circle the (radii). Use astraightedge to draw aline segment from center the of circle the to each Use acompass to draw acircle. Mark two different points on circle. the

Follow the method to atriangle.Follow construct themethod

339

Lesson 2

, when it is is , when it P , 4 miles from the airport. , 4 miles from Q 340 P T 5 mi 40° Q 80° at constant speed, how fast is it traveling? traveling? it is fast speed, how constant at Q A is in contact with an airplane flying at point point flying at airplane an with in contact is 4 mi to A P

Module 7 Module 5 miles from the airport, and 30 seconds later when it is at point point at is when it later 30 seconds the airport, and 5 miles from flies the plane both times. If at theground with makes the plane the angles shows diagram The from the ground parallel to The control tower at airport at tower control The Lesson Lesson Task Performance

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