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