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Isosceles Triangles
Geometry D – Chapter 4.6
Definitions - Review
Define an isosceles triangle.
A triangle with two congruent sides.
Name the parts of an isosceles triangle.
Legs are the congruent sides.
Vertex angle is the included angle of the legs.
Base is the side opposite the vertex angle.
Base angle is the included angle of the base and leg.
Definitions - Review
VABC is an isosceles triangle.
A
B
C
Name each item(s):
Vertex Angle B
Base
AC
Legs
AB, CB
Base Angles
A, C
Side opposite  C
AB
Angle opposite BC
A
Definitions - Review
Define an equilateral triangle.
A triangle with three congruent sides.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
A
B
C
Converse:
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
Proof - Isosceles Triangle Theorem
W
Given: VW  XW
Prove: V X
V
 Since every angle has a bisector, construct the
angle bisector of angle W.
X
Proof - Isosceles Triangle Theorem
W
Given: VW  XW
Prove: V X
V
Z
 Since every angle has a bisector, construct the
angle bisector of angle W.
 Given VW  XW
X
Proof - Isosceles Triangle Theorem
W
Given: VW  XW
Prove: V X
V
Z
X
 Since every angle has a bisector, construct the
angle bisector of angle W.
 Given VW  XW
 By the definition of angle bisectors VWZ XWZ
 WZ is congruent with itself by the reflexive property.
 By SAS, VVWZ VXWZ
 By CPCTC, V X
Corollary 1
A triangle is equilateral if and only if it is equiangular.
Corollary 2
Each angle of an equiangular triangle has a measure
of 60o.
Example 1
Find the measure of each angle.
mM  30o
M
L
mL  mN
N
x + x + 30o = 180o
2x + 30o = 180o
2x = 150o
x = 75o
mL mN 75o
Example 2
Find the length of each side.
E
3x – 6 = 6
3x – 6
2x
G
F
3x = 12
x=4
6
EF = 6
EG = 8
Example 3
Find the measure of each angle.
mP  2x  4
mQ  x  2
P
(2x – 4) + (x + 2) + (x + 2) = 180o
O
Q
4x = 180o
x = 45o
mP  2(45)  4  86o
mQ  45 2  47o
mO  mQ  47o
Example 4
Given: VABE VEDA , B is the midpoint of AC and
D is the midpoint of CE.
VACE
Prove:
C
 AED EAB
F
 AC is congruent to EC since the
sides opposite of congruent angles
are congruent.
D
B
A
is isosceles.
E
by CPCTC
 Triangle ACE is isosceles by the
definition of isosceles triangles.