1. Isosceles and Equilateral Triangles
Concept 25

2. Has exactly three congruent sides
Vertex Angle
the angle formed by the legs.
Leg
Leg
the 2 congruent sides of an isosceles triangle.
Base Angle
2 angles adjacent to the base.
the 3rd side of an isosceles triangle
Base

4. 1. 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

5. 2. Name two unmarked congruent segments.
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BC is opposite D and BD is opposite BCD, so BC  BD.
Answer: BC  BD

6. 3. Which statement correctly names two congruent angles?
A. PJM  PMJ
B. JMK  JKM
C. KJP  JKP
D. PML  PLK

7. 4. Which statement correctly names two congruent segments?
A. JP  PL
B. PM  PJ
C. JK  MK
D. PM  PK

9. 5. 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.

10. 6. 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

11. A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°

12. B. Find TS.
A. 1.5
B. 3.5
C. 4
D. 7

13. 7. Find the value of each variable.
mDFE = 60
4x – 8 = 60
4x = 68
x = 17
The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal.
DF = FE
6y + 3 = 8y – 5
3 = 2y – 5
8 = 2y
4 = y

14. 8. 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

15. CPCTC
Concept 26

16. Corresponding
Parts of
Congruent
Triangles are
Congruent

17. Use the diagram to answer the following.
What triangle appears to be congruent
to PAS?
to PAR?
∆𝐏𝐋𝐒
∆RLP
If 𝑃𝑆 ≅ 𝑅𝑆 and 𝑃𝐿 ≅ 𝐴𝑅 , what additional information would you need to prove PSL  RSA?
If LPA  ARL and PL≅AR, what additional information would you need to prove LPA  ARL?
𝑺𝑳 ≅ 𝑺𝑨
SSS
∠𝑺𝑷𝑳≅∠𝑺𝑨𝑹
SAS
∠𝑷𝑨𝑳≅∠𝑹𝑳𝑨
AAS
∠𝑷𝑳𝑨≅∠𝑹𝑨𝑳
ASA
𝑷𝑨 ≅ 𝑳𝑹
SAS

18. Use the diagram to answer the following questions.
To prove PSL  PSA, which triangles must you prove to be congruent?
To prove SL ≅ 𝑆𝐴 , which triangles must you prove to be congruent?
 LPS  APS
 LPS  APS
 LRS  ARS
 LPS  ARS
 APS  LRS

19. Use the marked diagrams to state the method used to prove the triangles are congruent. Give the congruence statement, then name the additional corresponding parts that could then be concluded to be congruent.
Missing Info/Why:
𝐷𝐵 ≅ 𝐵𝐷
Symmetric Prop.
Triangle Congruence/Why:
∆𝐴𝐵𝐷≅
∆𝐶𝐷𝐵
SSS
CPCTC:
∠𝐴≅ ∠𝐶
∠𝐴𝐵𝐷≅
∠𝐶𝐷𝐵
∠𝐴𝐷𝐵≅
∠𝐶𝐵𝐷

20. Use the marked diagrams to state the method used to prove the triangles are congruent. Give the congruence statement, then name the additional corresponding parts that could then be concluded to be congruent.
Missing Info/Why:
∠𝐴𝐶𝐵≅
∠𝐷𝐶𝐸
Vertical Angles
Triangle Congruence/Why:
∆𝐴𝐵𝐶≅
∆𝐷𝐸𝐶
ASA
CPCTC:
∠𝐵≅ ∠𝐸
𝐴𝐵 ≅ 𝐷𝐸
𝐵𝐶 ≅ 𝐸𝐶

21. Given: 𝑁𝑆 ≅ 𝑃𝑅 and  S  R
Prove: 𝑁𝑂 ≅ 𝑃𝑂
Statements
Reasons 1.
1. Given 2.
2. Given 3.
4. ∆NOS≅ 4. 5.
𝑁𝑆 ≅ 𝑃𝑅
 S  R
∠𝑆𝑂𝑁≅
∠𝑅𝑂𝑃
Vertical Angles
∆POR
ASA
𝑁𝑂 ≅ 𝑃𝑂
CPCTC