4.6Isosceles and Equilateral

CCSS

Content StandardsG.CO.10 Prove theorems about triangles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Mathematical Practices2 Reason abstractly and quantitatively.3 Construct viable arguments and critique the reasoning of others.

Then/Now

You identified isosceles and equilateral triangles.

• Use properties of isosceles triangles.

• Use properties of equilateral triangles.

Vocabulary

• legs of an isosceles triangle

• vertex angle

• base angles

Concept

Example 1Congruent Segments and Angles

A. Name two unmarked congruent angles.

Answer: BCA and A

BCA is opposite BA and A is opposite BC, so BCA A.

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Example 1Congruent Segments and Angles

B. Name two unmarked congruent segments.

Answer: BC BD

___BC is opposite D and BD is opposite BCD, so BC BD.

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______ ___

Example 1a

A. PJM PMJ

B. JMK JKM

C. KJP JKP

D. PML PLK

A. Which statement correctly names two congruent angles?

Example 1bB. Which statement correctly names two congruent segments?

A. JP PL

B. PM PJ

C. JK MK

D. PM PK

Concept

Since QP = QR, QP QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR.

Example 2Find Missing Measures

A. Find mR.

.

.Answer: mR = 60

Since all three angles measure 60, the triangle is equiangular. Because an equiangular triangle is also equilateral, QP = QR = PR. Since QP = 5, PR = 5 by substitution.

Example 2Find Missing Measures

B. Find PR.

Answer: PR = 5 cm

Example 2a

A. 30°

B. 45°

C. 60°

D. 65°

A. Find mT.

Example 2b

A. 1.5

B. 3.5

C. 4

D. 7

B. Find TS.

Example 3Find Missing Values

ALGEBRA Find the value of each variable.

Since E = F, DE FE by the Converse of the Isosceles Triangle Theorem. DF FE, so all of the sides of the triangle are congruent. The triangle is equilateral. Each angle of an equilateral triangle measures 60°.

Example 3Find Missing Values

mDFE = 60 Definition of equilateral triangle4x – 8 = 60 Substitution

4x = 68 Add 8 to each side.x = 17 Divide each side by 4.

The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal.

DF = FE Definition of equilateral triangle6y + 3 = 8y – 5 Substitution

3 = 2y – 5 Subtract 6y from each side.8 = 2y Add 5 to each side.

Example 3Find Missing Values

4 = y Divide each side by 2.

Answer: x = 17, y = 4

Example 3

A. x = 20, y = 8

B. x = 20, y = 7

C. x = 30, y = 8

D. x = 30, y = 7

Find the value of each variable.

Example 4Apply Triangle Congruence

Given: HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX || OG.

Prove: ΔENX is equilateral.

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Example 4Apply Triangle Congruence

Proof:ReasonsStatements1. Given1. HEXAGO is a regular polygon.

5. Midpoint Theorem5. NG NE

6. Given6. EX || OG

2. Given2. ΔONG is equilateral.

3. Definition of a regular hexagon

3. EX XA AG GO OH HE

4. Given4. N is the midpoint of GE.

Example 4Apply Triangle Congruence

Proof:ReasonsStatements7. Alternate Exterior Angles

Theorem 7. NEX NGO

8. ΔONG ΔENX 8. SAS

9. OG NO GN 9. Definition of Equilateral Triangle

10. NO NX, GN EN 10. CPCTC

11. XE NX EN 11. Substitution

12. ΔENX is equilateral. 12. Definition of Equilateral Triangle

Example 4

Proof:ReasonsStatements1. Given1. HEXAGO is a regular hexagon.2. Given2. NHE HEN NAG AGN

3. Definition of regular hexagon

4. ASA

3. HE EX XA AG GO OH

4. ΔHNE ΔANG

___ ___

Given: HEXAGO is a regular hexagon.NHE HEN NAG AGN

Prove: HN EN AN GN___ ___

Example 4

A. Definition of isosceles triangle

B. Midpoint Theorem

C. CPCTC

D. Transitive Property

Proof:ReasonsStatements

5. ___________5. HN AN, EN NG

6. Converse of Isosceles Triangle Theorem

6. HN EN, AN GN

7. Substitution7. HN EN AN GN

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