1. Isosceles and Equilateral Triangles
LESSON 4–6
Isosceles and Equilateral Triangles

2. Five-Minute Check (over Lesson 4–5) TEKS Then/Now New Vocabulary
Theorems: Isosceles Triangle
Example 1: Congruent Segments and Angles
Proof: Isosceles-Triangle Theorem
Corollaries: Equilateral Triangle
Example 2: Find Missing Measures
Example 3: Find Missing Values
Example 4: Real-World Example: Apply Triangle Congruence
Lesson Menu

3. Refer to the figure. Complete the congruence statement
Refer to the figure. Complete the congruence statement. ΔWXY  Δ_____ by ASA. ?
A. ΔVXY
B. ΔVZY
C. ΔWYX
D. ΔZYW
5-Minute Check 1

4. Refer to the figure. Complete the congruence statement
Refer to the figure. Complete the congruence statement. ΔWYZ  Δ_____ by AAS. ?
A. ΔVYX
B. ΔZYW
C. ΔZYV
D. ΔWYZ
5-Minute Check 2

5. Refer to the figure. Complete the congruence statement
Refer to the figure. Complete the congruence statement. ΔVWZ  Δ_____ by SSS. ?
A. ΔWXZ
B. ΔVWX
C. ΔWVX
D. ΔYVX
5-Minute Check 3

6. What congruence statement is needed to use AAS to prove ΔCAT  ΔDOG?
A. C  D
B. A  O
C. A  G
D. T  G
5-Minute Check 4

7. Mathematical Processes G.1(E), G.1(F)
Targeted TEKS
G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of
interior angles, base angles of isosceles triangles, mid-segments, and medians, and apply these relationships to
solve problems.
Mathematical Processes
G.1(E), G.1(F)
TEKS

8. You identified isosceles and equilateral triangles.
Use properties of isosceles triangles.
Use properties of equilateral triangles.
Then/Now

9. legs of an isosceles triangle vertex angle base angles
Vocabulary

10. Concept

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

12. B. Name two unmarked congruent segments.
Congruent Segments and Angles
B. Name two unmarked congruent segments.
___
BC is opposite D and BD is opposite BCD, so BC  BD.
Answer: BC  BD
Example 1

13. A. Which statement correctly names two congruent angles?
A. PJM  PMJ
B. JMK  JKM
C. KJP  JKP
D. PML  PLK
Example 1a

14. B. Which statement correctly names two congruent segments?
A. JP  PL
B. PM  PJ
C. JK  MK
D. PM  PK
Example 1b

15. Concept

16. Concept

17. Subtract 60 from each side. Answer: mR = 60 Divide each side by 2.
Find Missing Measures
A. Find mR.
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.
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

18. 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

19. A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°
Example 2a

20. B. Find TS.
A. 1.5
B. 3.5
C. 4
D. 7
Example 2b

21. ALGEBRA Find the value of each variable.
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°.
Example 3

22. mDFE = 60 Definition of equilateral triangle 4x – 8 = 60 Substitution
Find Missing Values
mDFE = 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 3

23. 4 = y Divide each side by 2. Answer: x = 17, y = 4 Find Missing Values
Example 3

24. 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

25. Prove: ΔENX is equilateral.
Apply Triangle Congruence
Given: HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX || OG.
Prove: ΔENX is equilateral.
___
Example 4

26. Proof: Reasons Statements 1. Given 1. HEXAGO is a regular polygon.
Apply Triangle Congruence
Proof:
Reasons
Statements
1. Given
1. HEXAGO is a regular polygon.
2. Given
2. ΔONG is equilateral.
3. Definition of a regular hexagon
3. EX  XA  AG  GO  OH  HE
4. Given
4. N is the midpoint of GE.
5. Midpoint Theorem
5. NG  NE
6. Given
6. EX || OG
Example 4

27. Proof: Reasons Statements 7. NEX  NGO
Apply Triangle Congruence
Proof:
Reasons
Statements
7. 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

28. Given: HEXAGO is a regular hexagon. NHE  HEN  NAG  AGN
___
Given: HEXAGO is a regular hexagon. NHE  HEN  NAG  AGN
Prove: HN  EN  AN  GN
Proof:
Reasons
Statements
1. Given
1. HEXAGO is a regular hexagon.
2. Given
2. NHE  HEN  NAG  AGN
3. Definition of regular hexagon
4. ASA
3. HE  EX  XA  AG  GO  OH
4. ΔHNE  ΔANG
Example 4

29. A. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC
Proof:
Reasons
Statements
5. ___________
5. HN  AN, EN  NG
6. Converse of Isosceles Triangle Theorem
6. HN  EN, AN  GN
7. Substitution
7. HN  EN  AN  GN ?
A. Definition of isosceles triangle
B. Midpoint Theorem
C. CPCTC
D. Transitive Property
Example 4

30. Isosceles and Equilateral Triangles
LESSON 4–6
Isosceles and Equilateral Triangles