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Isosceles and
Equilateral Triangles
Isosceles Triangle – Triangle with two
congruent sides.
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The congruent sides are the legs.
The third side is the base.
The two legs form the vertex angle.
The other two angles are the base
angles.
Legs of an isosceles triangle are congruent.
Base angles of an isosceles triangle are congruent.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then
the angles opposite those sides are
congruent.
Proof of Isosceles Triangle Theorem
Given: Isosceles XYZ with XY  XZ
Prove: Y  Z
This proof requires an auxiliary line.
Statements
1.
2.
3.
Isosceles XYZ with XY  XZ
XB bisects YXZ
1  2
XB  XB
5. YXB  ZXB
6. Y  Z
4.
Reasons
1. Given
2. Given
3. A bisector divides an angle into two
congruent angles. (2)
4. Reflexive Property
5. SAS (1, 3, 4)
6. CPCTC (5)
Converse of
Isosceles Triangle Theorem
If two angles of a triangle are congruent,
then the sides opposite those angles are
congruent.
Theorem
If a line bisects the vertex angle of an
isosceles triangle, then the line is also the
perpendicular bisector of the base.
Equilateral Triangle – Triangle with three
congruent sides.
Corollary – A theorem that can be proved
using another theorem.
Corollary
If a triangle is equilateral, then the triangle is
equiangular.
Corollary
If a triangle is equiangular, then the triangle
is equilateral.