Question

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

Answer 1

Hint: For finding out the sum of the measures of the angles of a trapezoid, we need to consider a trapezoid ABCD, with the side AB parallel to the side CD. Then we need to draw a diagonal so as to divide the trapezoid ABCD into two triangles, namely ACD and ABC. By applying the angle sum property in the two triangles, we can find out the measure of the angles of them. Finally, on adding these two sums we will get the required sum of the measures of the angles of the trapezoid ABCD.

Complete step-by-step solution:
Let us consider a trapezoid ABCD. We know that in a trapezoid, one pair of opposite sides are parallel to each other. Therefore, we take the sides AB and CD to be parallel as shown in the figure below.

According to the questions, we have to find out the sum of the angles of the trapezoid. For this, we divide the above trapezoid into two triangles ACD and ABC by the diagonal AC as shown below.

Now, applying the angle sum property in the triangle ACD we get
$\Rightarrow \angle ADC+\angle ACD+\angle CAD={{180}^{\circ }}........\left( i \right)$
Similarly, for the triangle ACB we can write
$\Rightarrow \angle ABC+\angle ACB+\angle BAC={{180}^{\circ }}........\left( ii \right)$
Adding the equations (i) and (ii) we get
$\begin{align}
  & \Rightarrow \angle ADC+\angle ACD+\angle CAD+\angle ABC+\angle ACB+\angle BAC={{180}^{\circ }}+{{180}^{\circ }} \\
 & \Rightarrow \angle ADC+\left( \angle ACD+\angle ACB \right)+\left( \angle CAD+\angle BAC \right)+\angle ABC={{360}^{\circ }} \\
 & \Rightarrow \angle D+\angle C+\angle A+\angle B={{360}^{\circ }} \\
\end{align}$
Hence, the required sum of the angles is equal to ${{360}^{\circ }}$.

Note: We must not get confused between the terms “trapezium” and “trapezoid”. Both mean the same. Since the trapezoid is a four sided quadrilateral, we can also use the formula $S=\left( n-2 \right)\times {{180}^{\circ }}$ where we need to substitute $n=4$ to get the required sum of the angles.