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

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Congruent Segments and Angles

A. Name two unmarked congruent angles.

Answer: BCAand A

Example 1BC is opposite D and BD is opposite BCD, so BC BD.

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Answer: BC BD

Congruent Segments and Angles

B. Name two unmarked congruent segments.

Example 1B

C

D

A. Which statement correctly names two congruent angles?

A.PJM PMJ

B.JMK JKM

C.KJP JKP

D.PML PLK

Example 1aB

C

D

A.JP PL

B.PM PJ

C.JK MK

D.PM PK

B. Which statement correctly names two congruent segments?

Example 1bSince 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.

Find Missing Measures

A. Find mR.

Triangle Sum Theorem

mQ = 60, mP = mR

Simplify.

Subtract 60 from each side.

Answer:mR = 60

Divide each side by 2.

Example 2B. Find PR.

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.

Answer:PR = 5 cm

Example 2Since 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°.

Find Missing Values

ALGEBRA Find the value of each variable.

Example 3mDFE = 60 Definition of equilateral triangle

4x – 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 triangle

6y + 3 = 8y – 5 Substitution

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

8 = 2y Add 5 to each side.

Example 3B

C

D

Find the value of each variable.

A.x = 20, y = 8

B.x = 20, y = 7

C.x = 30, y = 8

D.x = 30, y = 7

Example 3Given: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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Apply Triangle Congruence

NATURE Many geometric figures can be found in nature. Some honeycombs are shaped like a regular hexagon. That is, each of the six sides and interior angle measures are the same.

Example 4Reasons

1. HEXAGO is a regular polygon.

1. Given

2. Given

2. ΔONG is equilateral.

3. Definition of a regular hexagon

3. EX XA AG GO OH HE

4. N is the midpoint of GE

4. Given

6. Given

6. EX || OG

5. NG NE

5. Midpoint Theorem

Apply Triangle Congruence

Proof:

Example 4Reasons

7. Alternate Exterior Angles Theorem

7. NEX NGO

8. SAS

8. ΔONG ΔENX

9. Definition of Equilateral Triangle

9. OG NO GN

10. CPCTC

10. NO NX, GN EN

11. Substitution

11. XE NX EN

12. Definition of Equilateral Triangle

12.ΔENX is equilateral.

Apply Triangle Congruence

Proof:

Example 4Given: HEXAGO is a regular hexagon.NHE HEN NAG AGN

Prove: HN EN AN GN

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Proof:

Statements

Reasons

1. HEXAGO is a regular hexagon.

1. Given

2. NHEHENNAGAGN

2. Given

3. HE EX XA AG GO OH

3. Definition of regular hexagon

4. ΔHNE ΔANG

4. ASA

Example 4B

C

D

Proof:

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

Prove: HN EN AN GN

Statements

Reasons

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5. HN AN, EN NG

5. CPCTE

6. HN EN, AN GN

6. Converse of Isosceles Triangle Theorem

7. HN EN AN GN

7. Substitution

Example 4
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