Question

What is the sum of the measures of the angles of a trapezoid?

Answer 1

The sum of the interior angles of a trapezoid is `360^@`.

Explanation:

There's a general theorem that holds for each quadrilateral: the sum of interior angles of a convex quadrilateral is `360^@`. This can be generalized further to non-convex quadrilaterals. Since trapezoids are convex quadrilaterals, we prove the theorem using the convex assumption.

The proof is quite easy: let's choose a vertex of the convex quadrilateral and connect it to the opposite vertex. We divided the quadrilateral into two triangles and we are left with the following situation:

• each of the two angles at the opposite vertices that we connected is divided in two parts: one for each of the two triangles obtained.
• the angles at the two remaining opposite vertices are also angles of the two triangles.

So the sum of the interior angles of the two triangles equals the sum of the interior angles of the quadrilateral. Since the sum of the interior angles of a triangle is always `180^@`, the sum of the interior angles of two triangles is `360^@` and this ends the proof.