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Concept

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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

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BC 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.

A

B

C

D

A. Which statement correctly names two congruent angles?

A.PJM PMJ

B.JMK JKM

C.KJP JKP

D.PML PLK

A

B

C

D

A.JP PL

B.PM PJ

C.JK MK

D.PM PK

B. Which statement correctly names two congruent segments?

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.

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.

Find Missing Measures

B. 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

A

B

C

D

A. Find mT.

A.30°

B.45°

C.60°

D.65°

A

B

C

D

B. Find TS.

A.1.5

B.3.5

C.4

D.7

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°.

Find Missing Values

ALGEBRA Find the value of each variable.

Find Missing Values

mDFE= 60Definition of equilateral triangle

4x – 8 = 60Substitution

4x= 68Add 8 to each side.

x= 17Divide 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= FEDefinition of equilateral triangle

6y + 3= 8y – 5Substitution

3= 2y – 5Subtract 6y from each side.

8= 2yAdd 5 to each side.

Find Missing Values

4= yDivide each side by 2.

Answer:x = 17, y = 4

A

B

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

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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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.

Statements

Reasons

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:

Statements

Reasons

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:

Given: 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

A

B

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