Proof of Angle Sum of Quadrilaterals

Introduction

We have learned that the angle sum of a triangle is 180^\circ. What about the quadrilaterals? The square and the rectangles have four right angles, so clearly, the sum of their interior angles is 360^{\circ}. But what about other quadrilaterals?

Quadrilaterals

image via Wikipedia

Before proceeding with the proof, to those who want to explore first, draw different types of quadrilaterals and use the protractor to measure their interior angles. What do you think is the angle sum of the quadrilaterals? Are the sums equal?

Theorem

The sum of the interior angles of a quadrilateral is 360^\circ.

Proof

Given quadrilateral ABCD, we have to show that the angle sum of the interior angles is 360^\circ.

Draw diagonal AC to divide the quadrilateral into two triangles. Let p, q, r, s, t, and u be the measures of the interior angles as shown on the figure below.

quadrilateral angle sum

Using the triangle angle sum theorem, we know that p + q + r = 180^\circ and s + t + u = 180^\circ. But the sum of these six angle measures is the sum of the four interior angles of the quadrilateral (Can you see why?). Therefore, the sum of the interior angles of a quadrilateral is 360^{\circ}.

Note to the Student

In most countries, it is a convention (a standard) that angles should be named with Greek letters. I will not adopt that convention in this blog for convenience. I will be using letters in the English alphabet or even numbers to name angles. I believe that it is easier for you to learn if you are familiar with the symbols. On the other hand, if your teacher requires you to follow the convention, you should do so.

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