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Home » Quadrangles » The Sum of Interior Angles in a Simple Convex Quadrangle

The Sum of Interior Angles in a Simple Convex Quadrangle

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will prove that the sum of interior angles in a simple convex quadrangle is always 360° by showing that this quadrangle is composed of two triangles.

Problem

Show that the sum of interior angles in a simple convex quadrangle is always 360°.

Simple Convex Quadrangle

Strategy

Let’s think about the strategy to answer this geometry problem. 360 is a familiar number - it is the measure of angles in a circle, but there are no circles here, so let’s rule that out.

360 is also 2x180, and that is also a familiar number. It is the sum of the angles in a triangle.

So, if we can show that this quadrangle is composed of two triangles, we will have the proof- and that’s all we need.

Proof

In any simple convex polygon, a line connecting 2 points on the perimeter of the polygon is entirely within the polygon, per the definition of convex polygons.

So, let’s draw a line connecting two opposite corners of the quadrangle (such a line is a called a ‘diagonal’) – say from C to A:

Diagonal line in a simple convex polygon.

(1)    m∠A1 + m∠D+ m∠C1 = 180°                        //sum of the interior angles in triangle ΔADC

(2)    m∠A2 + m∠B+ m∠C2 = 180°                         //sum of the interior angles in triangle ΔABC

(3)    m∠A1 + m∠D+ m∠C1+m∠A2 + m∠B+ m∠C2 = 360°   //add both equations

(4)    m∠A1 + m∠A2 +m∠D+ m∠C1+ m∠C2 + m∠B= 360°   //re-arrange terms

(5)    m∠A1 + m∠A2 = m∠A                                      // angle addition postulate

(6)    m∠C1 + m∠C2 = m∠C                                       // angle addition postulate

(7)    m∠A+m∠D+ m∠C + m∠B= 360°   ∎         //substitution property of equality

« Triangle Exterior Angles
Pythagorean Theorem: Lengths of Edges in a Right Triangle »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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