Triangle Property Theorems

1) Corresponding parts of congruent triangles are congruent (CPCTC)

This theorem states that corresponding sides and angles of congruent triangles are congruent.

2) Midline theorem

This theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

In the example at right, DE || AC and DE = AC/2.

3) Sum of two sides

This theorem states that the sum of the lengths of two sides of a triangle is always greater than the length of the third side.

4) Longest side/largest angle

This theorem states that the longest side of a triangle is the side opposite the largest angle, and the largest angle is always across from the longest side.

5) Altitude rule

This theorem states that the altitude to the hypotenuse of a right triangle is the mean proportional between the segments in which it divides the hypotenuse.
In the example at right, both inner right triangles formed are similar to the original triangle.

6) Leg rule

This theorem states that each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

In the example at right, x and y are the projections of legs b and a on the hypotenuse, respectively.

7) Base angle/base angle converse theorem

This theorem states that if two sides of a triangle are congruent (isosceles), the angles opposite these sides are congruent. If two angles are congruent, the sides opposite these angles are congruent.

8) Exterior angle theorem

This theorem states that any exterior angle of a triangle is equal to the sum of the two remote interior angles.