Isosceles and
Equilateral
Triangles
Geometry (Holt 4-9)
K.Santos
Parts of an Isosceles Triangle
Isosceles triangle—is a triangle with at least two congruent sides
A
B
C
Legs—are the congruent sides
𝐴𝐵 and 𝐴𝐶
Vertex angle—angle formed by the legs
<A
Base—side opposite the vertex angle
𝐵𝐶
Base angles—two angles that have the base as a side
< B and < C
Isosceles Triangle
Theorem (4-9-1)
If two sides of a triangle are congruent, then the angles
opposite the sides are congruent.
A
B
C
Given: 𝐴𝐵 ≅ 𝐵𝐶
Then: <A ≅ <C
congruent sides
congruent angles
Converse of the Isosceles
Triangle Theorem (4-9-2)
If two angles of a triangle are congruent, then the sides
opposite those angles are congruent.
A
B
C
Given: <A ≅ <C
Then: 𝐴𝐵 ≅ 𝐵𝐶
congruent angles
congruent sides
Example—finding angle
measures
A
Find the measure of <C.
C
m< B = m< A = x
m < C + m < B + m < A = 180
x + 50 + 50 = 180
x + 100 = 180
x = 80 which means m< C = 80°
50°
B
Example—finding angle
measures
A
Find the measure of < C.
50°
x
C
m< C = m< B = x
m < C + m < B + m < A = 180
x + x + 50 = 180
2x + 50 = 180
2x = 130
x = 65 which means m < C = 65°
B
Example—finding angle
measures (algebraic)
S
Find x.
x + 38°
T
3x°
R
m<R=m<S
3x = x + 38
2x = 38
x = 19
Corollary(4-9-3)—
Equilateral Triangle
If a triangle is equilateral, then it is equiangular.
M
N
Given: 𝑀𝑁 ≅ 𝑁𝑂 ≅ 𝑀𝑂
Then: <M ≅ <N ≅ <O
equilateral
O
180/3 = 60°
equiangular
Corollary (4-9-4)—
Equiangular Triangle
If a triangle is equiangular, then it is equilateral.
M
N
Given: <M ≅ <N ≅ <O
O
Then: 𝑀𝑁 ≅ 𝑁𝑂 ≅ 𝑀𝑂
equiangular
equilateral
Example—Finding angles
Find x.
G
4x+12
H
I
triangle is equilateral-----equiangular each angle is 60°
4x + 12 = 60
4x = 48
x = 12°
Example—finding sides
J
Find t.
3t + 3
K
5t – 9
L
Triangle is equiangular---equilateral (all equal sides)
𝐾𝐿 ≅ 𝐽𝐿
5t – 9 = 3t + 3
2t – 9 = 3
2t = 12
t=6
Example—Multiple
Triangles
Find the measures of the numbered angles.
m< 3 = m< 4
x + x + 80 = 180
2x + 80 = 180
2x = 100
x = 50 so m <3 and m < 4 = 50
m< 1 and m< 4 are supplementary
m< 1 + m < 4 = 180
m< 1 + 50 = 180
m< 1 = 130
m< 1 + m< 2 + m< 5 = 180 with m< 2 = m< 5
130 + y + y = 180
130 + 2y = 180
2y = 50
y = 25 so m< 2 = 25, and m< 5 = 25
80 5
3
4
1
2