Page 95 - Math Course 3 (Book 2)

Properties of Isosceles and Equilateral Triangles

Let’s Begin Solve the Test Item

Step 1 The base angles of ΔCDE are congruent.

Proof of Theorem Let x = m∠DEC = m∠DCE.
Angle Sum
m∠DEC + m∠DCE + m∠CDE = 180
Example Theorem
x + x + 120 = 180 Substitution

2x + 120 = 180 Add.
Write a two-column proof.
Subtract 120 from
Given: AB = CB = BD; ∠ACB ≅ ∠BCD D 2x = 60 each side.
Prove: ∠A ≅ ∠D. Divide each side
x = 30
by 2.
C B
So, m∠DEC = m∠DCE = 30.
Proof: A
Statements Reasons Step 2 ∠DCE and ∠BCA are vertical angles so
they have equal measures.
1. AB = CB = DB 1. Given
2. AB ≅ CB ≅ DB 2. Def. of ≅ segments m∠DEC = m∠BCA Def. of vertical angles
3. ΔABC and ΔBCD 3. Def. of isosceles Δ 30 = m∠BCA Substitution
are isosceles
4. ∠A ≅ ∠ACB, 4. Isosceles Δ Theorem Step 3 The base angles of ΔCBA are congruent.
∠BCD ≅ ∠D
5. ∠ACB ≅ ∠BCD 5. Given Let y = m∠CBA = m∠BAC.
6. ∠A ≅ ∠D 6. Transitive Property Angle Sum
m∠CBA + m∠BAC + m∠BCA = 180
Theorem
y + y + 30 = 180 Substitution
Find the Measure of a Missing Angle

2y + 30 = 180 Add.
Example Subtract 30 from
2y = 150
each side.
If DE ≅ CD, BC ≅ AC, and m∠CDE = 120, what is the y = 75 Divide each side
measure of ∠BAC? by 2.
The measure of ∠BAC is 75
D

Answer D
C
B E

A

A. 45.5 B. 57.5 C. 68.5 D. 75

Read the Test Item
ΔCDE is isosceles with base CE. Likewise, ΔCBA is
isosceles with BA.

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