Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.

Question

Homi · Accepted Answer

We can divide any quadrilateral into two triangles by drawing a diagonal. The sum of angles in a triangle is 180 degrees.

Step 1 — Dividing the quadrilateral Let us draw a line segment connecting vertices B and D. This line segment BD divides the quadrilateral ABCD into two triangles. These triangles are $\triangle ABD$ and $\triangle CBD$ . Let us label the angles inside these triangles. In $\triangle ABD$ , we have $\angle A$ , $\angle ABD$ , and $\angle ADB$ . In $\triangle CBD$ , we have $\angle C$ , $\angle CBD$ , and $\angle CDB$ .

Step 2 — Sum of angles in triangles The sum of interior angles of any triangle is 180 degrees. For $\triangle ABD$ : $\angle A + \angle ABD + \angle ADB = 180^\circ$ For $\triangle CBD$ : $\angle C + \angle CBD + \angle CDB = 180^\circ$

Step 3 — Sum of angles in the quadrilateral Now, let us add the sums of angles from both triangles. $(\angle A + \angle ABD + \angle ADB) + (\angle C + \angle CBD + \angle CDB) = 180^\circ + 180^\circ$ We can rearrange the terms on the left side. $\angle A + (\angle ABD + \angle CBD) + \angle C + (\angle ADB + \angle CDB) = 360^\circ$ From the diagram, we observe the following: The angle at vertex B of the quadrilateral is $\angle B$ . This angle $\angle B$ is the sum of $\angle ABD$ and $\angle CBD$ . So, $\angle B = \angle ABD + \angle CBD$ . The angle at vertex D of the quadrilateral is $\angle D$ . This angle $\angle D$ is the sum of $\angle ADB$ and $\angle CDB$ . So, $\angle D = \angle ADB + \angle CDB$ . Let us substitute these into our equation. $\angle A + \angle B + \angle C + \angle D = 360^\circ$ Therefore, the sum of angles of quadrilateral ABCD is 360 degrees.

$\boxed{\text{Sum of angles in quadrilateral ABCD} = 360^\circ}$

Diagram 1

Step 4 — Verification by measurement We can draw this quadrilateral on a piece of paper. Then, we use a protractor to measure each interior angle. These angles are $\angle A$ , $\angle B$ , $\angle C$ , and $\angle D$ . Adding these four angles, their sum will be close to 360 degrees. Small differences might happen due to measurement errors.

Answer

(i) Yes, the sum of the angles in a quadrilateral such as the given one is 360°. (ii) Geometric reasoning shows that by dividing the quadrilateral into two triangles, the sum of its interior angles is 360°. (iii) By constructing the figure and measuring its angles with a protractor, the sum of the angles is found to be approximately 360°.

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