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Rule

Interior Angles Theorem

The sum of the measures of the interior angles of a triangle is
The triangle ABC with movable vertices A, B, and C and the angle measures written
Based on this diagram, the following relation holds true.

This theorem is also known as the Triangle Angle Sum Theorem.

Proof

Consider a triangle with vertices and and the parallel line to through Let and be the angles outside formed by this line and the sides and

Triangle ABC with the line parallel to BC

By the Alternate Interior Angles Theorem, is congruent to and is congruent to

Two pairs of alternate interior angles
By the definition of congruent angles, and have the same measure. For the same reason, and also have the same measure.
Furthermore, in the diagram it can be seen that and form a straight angle. Therefore, by the Angle Addition Postulate their measures add to
By the Substitution Property of Equality, the sum of the measures of and is equal to
Finally, in can be named