1. Section 4-4: The Isosceles Triangle Theorems

2. vertex
angle
leg
leg
base
angle
base
angle
base

3. Isosceles Triangle Definition: _______________________________________
A triangle with at least two congruent sides

4. Isosceles Triangle Theorem: __________________ _________________________________________ Converse of Isosceles Triangle Theorem: ________ __________________________________________________________________________________
If two sides of a
triangle are congruent, then the angles opposite those sides are congruent.
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

5. Corollary 1: An equilateral triangle is also ______________________.
Corollary 2: An equilateral triangle has three __________ angles. 
equiangular
60°

6. Corollary 3: The bisector of the vertex angle of an isosceles triangle is ___________________ to the base at its ____________.
Corollary 4: An equiangular triangle is also _______________________.
perpendicular
midpoint
equilateral

7. x = 70
x = 75
y = 40
y = 75

8. 2x – 4 = x + 5
2x + 7 = 5x – 8
x = 9
15 = 3x
x = 5

9. 62°
(2x)°
2x + 2x + x = 180
x = 42
5x = 180
x = 36

10. 40°
65°
70°
x = 50
x = 50

11. In Exs. 9 and 10, 𝐵𝐶 ≅ 𝐴𝐶 .
9. If the m∠1=140°,
m∠2 = _________.
m∠3 = _________,
m∠4 = _________.
10. If the m∠4 = 65°,
m∠3 = ________,
m∠2 = ________,
m∠1 = ________.
40°
70°
70°
65°
50°
130°

12. 11. Place the statements in an appropriate order for a proof and give the reasons.
Given: ∠1≅∠2
Prove: 𝑂𝐾 ≅ 𝑂𝐽
a) ∠3≅∠4
b) ∠2≅∠4; ∠3≅∠1
c) 𝑂𝐾 ≅ 𝑂𝐽
d) ∠1≅∠2 d;
Given b;
Vertical ∠ ′ 𝑠 𝑎𝑟𝑒 ≅ a;
Substitution Prop c;
Converse of Isosceles Triangle Thm.

13. 𝑂𝐾 ≅ 𝑂𝐽 Given ∠3 ≅ ∠4 Isoc. ∆ Thm. ∠1 ≅ ∠3; ∠2 ≅ ∠4;
12. Write a two-column proof. Given: 𝑂𝐾 ≅ 𝑂𝐽 Prove: ∠1 ≅ ∠2 STATEMENTS REASONS 1. ______________________________ 1. _____________________ 2. ______________________________ 2. _____________________ 3. ______________________________ 3. _____________________ 4. ______________________________ 4. _____________________
𝑂𝐾 ≅ 𝑂𝐽
Given
∠3 ≅ ∠4
Isoc. ∆ Thm.
∠1 ≅ ∠3;
∠2 ≅ ∠4;
Vertical ∠’s are ≅.
∠1 ≅ ∠2
Substitution Prop.

14. HOMEWORK: page 137 #1-10 all (#9, 10 give reasons)