### Dr. Wilson

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. (The Isosceles DecompositionTheorem) In an isosceles triangle, if a line segment goes from the vertex angle to the base, the following conditions are equivalent:

1. The line segment meets the base at its midpoint,
2. The line segment is perpendicular to the base.
3. 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.