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Concept. ___.  BCA is opposite BA and  A is opposite BC , so  BCA   A. ___. Congruent Segments and Angles. A. Name two unmarked congruent angles. Answer:  BCA and  A. Example 1. ___. BC is opposite  D and BD is opposite  BCD , so BC  BD. ___. ___. ___. ___.

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Concept

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Concept

Concept


Example 1

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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 1


Example 11

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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.

Example 1


Example 1a

A

B

C

D

A. Which statement correctly names two congruent angles?

A.PJM PMJ

B.JMK JKM

C.KJP JKP

D.PML PLK

Example 1a


Example 1b

A

B

C

D

A.JP PL

B.PM PJ

C.JK MK

D.PM PK

B. Which statement correctly names two congruent segments?

Example 1b


Concept1

Concept


Example 2

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

Find Missing Measures

A. Find mR.

Triangle Sum Theorem

mQ = 60, mP = mR

Simplify.

Subtract 60 from each side.

Answer:mR = 60

Divide each side by 2.

Example 2


Example 21

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

Example 2


Example 2a

A

B

C

D

A. Find mT.

A.30°

B.45°

C.60°

D.65°

Example 2a


Example 2b

A

B

C

D

B. Find TS.

A.1.5

B.3.5

C.4

D.7

Example 2b


Example 3

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.

Example 3


Example 31

Find Missing Values

mDFE= 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.

Example 3


Example 32

Find Missing Values

4= yDivide each side by 2.

Answer:x = 17, y = 4

Example 3


Example 33

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

Example 3


Example 4

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.

Example 4


Example 41

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:

Example 4


Example 42

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:

Example 4


Example 43

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.NHEHENNAGAGN

2.Given

3.HE  EX  XA  AG  GO  OH

3.Definition of regular hexagon

4.ΔHNE ΔANG

4.ASA

Example 4


Example 44

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

Example 4


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