### Theorems about Isosceles
Triangles

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1. If two sides in a triangle are congruent,
then the angles opposite the congruent sides are congruent angles.

2. If two angles in a triangle are
congruent, then the sides opposite the congruent angles are congruent
sides.

3. In an isosceles triangle, if a line
segment goes from the vertex angle to the base, the following
conditions are equivalent:

- The line segment meets the base at its midpoint,
- The line segment is perpendicular to the base.
- The line segment bisects the vertex angle.

4. If the bisector of an angle in a triangle
is perpendicular to the opposite side, the triangle is isosceles.

5. If the line from an angle of a triangle
which is perpendicular to the opposite side meets the opposite side
at its midpoint, then the triangle is isosceles.

6. (Extra Credit): If the bisector of an
angle in a triangle meets the opposite side at its midpoint, then the
triangle is isosceles.

7. A point is on the perpendicular bisector
of a line segment if and only if it lies the same distance from the
two endpoints.