1. Five-Minute Check (over Lesson 4–5) Mathematical Practices 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

2. A. yes; Leg-Leg Congruence B. yes; Hypotenuse-Angle Congruence
Determine whether each pair of triangles is congruent. If yes, include the theorem that can be used to prove congruence.
A. yes; Leg-Leg Congruence
B. yes; Hypotenuse-Angle Congruence
C. yes; Hypotenuse-Leg Congruence
D. not enough information
5-Minute Check 1

3. A. yes; Leg-Angle Congruence B. yes; Hypotenuse-Angle Congruence
Determine whether each pair of triangles is congruent. If yes, include the theorem that can be used to prove congruence.
A. yes; Leg-Angle Congruence
B. yes; Hypotenuse-Angle Congruence
C. yes; Hypotenuse-Leg Congruence
D. not enough information
5-Minute Check 2

4. A. yes; Leg-Leg Congruence B. yes; Leg-Angle Congruence
Determine whether each pair of triangles is congruent. If yes, include the theorem that can be used to prove congruence.
A. yes; Leg-Leg Congruence
B. yes; Leg-Angle Congruence
C. yes; Hypotenuse-Leg Congruence
D. not enough information
5-Minute Check 3

5. In JKL,  J is a right angle and in RST,  R is a right angle
In JKL,  J is a right angle and in RST,  R is a right angle. To prove JKL  RST by LL, what pieces of information do you need to know? A. B. C. D.
and
 K  S
5-Minute Check 4

6. Mathematical Practices 2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
Content Standards
G.CO.10 Prove theorems about triangles.
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). MP

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

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

9. Concept

10. 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.
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Answer: BCA and A
Example 1

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

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

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

14. Concept

15. Concept

16. 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

17. 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

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

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

20. 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

21. 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

22. 4 = y Divide each side by 2. Answer: x = 17, y = 4 Find Missing Values
Example 3

23. 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

24. 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.
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Example 4

25. 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

26. 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

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

28. 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