# 4.5: Isosceles and Equilateral Triangles

```4.5: Isosceles and Equilateral Triangles
does not
not do
do to
to dwell
dwell on
on dreams…
dreams… and
and forget
forget to
to live.
live.
ItIt does
-Dumbledore
Isosceles Triangles
The congruent sides are the
LEGS.
The third side is the BASE.
The two congruent sides form the
VERTEX ANGLE.
The other two angles are the
BASE ANGLES.
Theorems 
Theorem 4.3: Isosceles Triangle Theorem – If two sides of a
triangle are congruent, then the angles opposite those sides are
congruent.
A  B
Proof of the Isosceles Triangle Theorem
Given : XY  XZ , XB bisects YXZ
Prove : Y  Z
Statements
1. XY  XZ ,
Reasons
1.
XB bisects YXZ
1  2
2.
3. XB  XB
3.
4. XYB  XZB
4.
5. Y  Z
5.
2.
Theorems 
Theorem 4.4: CONVERSE of Isosceles Triangle Theorem – If two
angles of a triangle are congruent, then the sides opposite the angles
are congruent.
AC  BC
Theorems 
Theorem 4.5: The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
CD  AB and CD bisects AB
Applications
Explain why each statement is true.
a. WVS  S
b. TR  TS
Applications
Find the value of y.
Corollaries
corollary – a statement that follows immediately from a theorem
Corollary to Theorem 4.3: If a triangle is
equilateral, then the triangle is equiangular.
X  Y  Z
Corollary to Theorem 4.4: If a triangle is
equiangular, then the triangle is equilateral.
XY  YZ  ZX
Applications
Find the value of the variables.
1)
2)
4.5: Isosceles and Equilateral Triangles
Homework 4.5: #1-13,
15, 20, 22, 28, 30
I don't know half of you half as well as I should like, and I like less
I don't know half of you half as well as I should like, and I like less
than half of you half as well as you deserve.
than half of you half as well as you deserve.
-Bilbo
```