Five-Minute Check (over Lesson 4–5)

Then/Now

New Vocabulary

Theorems:Isosceles Triangle

Example 1:Congruent Segments and Angles

Corollaries:Equilateral Triangle

Example 2:Find Missing Measures

Example 3:Find Missing Values

Example 4:Real-World Example: Apply Triangle Congruence

Over Lesson 4–5

A. A

B. B

C. C

D. D

A. ΔVXY

B. ΔVZY

C. ΔWYX

D. ΔZYW

Refer to the figure. Complete the congruence statement.ΔWXY Δ_____ by ASA. ?

Over Lesson 4–5

A. A

B. B

C. C

D. D

A. ΔVYX

B. ΔZYW

C. ΔZYV

D. ΔWYZ

Refer to the figure. Complete the congruence statement. ΔWYZ Δ_____ by AAS. ?

Over Lesson 4–5

A. A

B. B

C. C

D. D

A. ΔWXZ

B. ΔVWX

C. ΔWVX

D. ΔYVX

Refer to the figure. Complete the congruence statement. ΔVWZ Δ_____ by SSS. ?

Over Lesson 4–5

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. C D

B. A O

C. A G

D. T G

What congruence statement is needed to use AAS to prove ΔCAT ΔDOG?

You identified isosceles and equilateral triangles. (Lesson 4–1)

• Use properties of isosceles triangles.

• Use properties of equilateral triangles.

• legs of an isosceles triangle

• vertex angle

• base angles

Congruent 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.

___

___

Congruent Segments and Angles

B. Name two unmarked congruent segments.

Answer: BC BD

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

___

______ ___

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. PJM PMJ

B. JMK JKM

C. KJP JKP

D. PML PLK

A. Which statement correctly names two congruent angles?

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

B. Which statement correctly names two congruent segments?

A. JP PL

B. PM PJ

C. JK MK

D. PM PK

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.

Divide each side by 2.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.

Find Missing Measures

B. Find PR.

Answer: PR = 5 cm

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 30°

B. 45°

C. 60°

D. 65°

A. Find mT.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 1.5

B. 3.5

C. 4

D. 7

B. Find TS.

Find 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°.

Find Missing Values

mDFE = 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.

Find Missing Values

4 = y Divide each side by 2.

Answer: x = 17, y = 4

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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.

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.

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

and EX || OG.

Prove: ΔENX is equilateral.

___

Apply 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

Apply 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

Proof:ReasonsStatements

1. 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___ ___

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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

?