1. The Isosceles Triangle Theorems
Section 4-4

2. Recall: Parts of the Isosceles Triangle
Vertex
Leg
Leg
Base angle
Base angle
Base

3. Theorem 4-1
If 2 sides of a triangle are congruent then the angles opposite those sides are congruent.
“the angle opposite” refers to the angle that does not use the segment as one of its sides.
<1 and <2 are opposite the congruent sides. 2
<1  <2 1

4. Some corollaries Corollary 1 Corollary 2 Corollary 3
An equiangular triangle is also equilateral
Corollary 2
An equiangular triangle has three 60˚ angles
Corollary 3
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint

5. Did somebody say “slice?”
Corollary 3
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
Slice!
Did somebody say “slice?”
We can prove this by drawing the angle bisector of the vertex angle…
…it slices the triangle into 2 congruent triangles by SAS

6. Corollary 3
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
We can prove this by drawing the angle bisector of the vertex angle…
…it slices the triangle into 2 congruent triangles by SAS
…Then the bisector splits the base into 2 congruent segments and creates 2 congruent angles (right angles)

7. Theorem 4-2
If 2 angles of a triangle are congruent then the sides opposite those angles are congruent. 2
<1  <2
“the sides opposite <1 and <2 must be congruent 1

8. Examples: Find x: 1.) 2.) 30˚ 2x – 4 x˚ 2x + 2 x + 5 3.) 42 41 56˚ 62˚