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Home » Quadrangles » Trapezoids » Trapezoids: Adjacent angles are Supplementary

Trapezoids: Adjacent angles are Supplementary

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In a trapezoid, the two angles that are on the same leg (one on the top base, one on the bottom base) are called 'adjacent angles'. These adjacent angles are supplementary, which means their measures sum up to 180°, as we will now show.

Problem

In a trapezoid ABCD, prove that the adjacent angles are supplementary.

trapezoid with angles

Strategy

We need to show that  m∠BAD + m∠CDA = 180°, and that m∠ABC + m∠DCB = 180°.

We have a trapezoid without any special features (that is, it is not an isosceles trapezoid and not a right trapezoid). So, all we know about this trapezoid is that the two bases are parallel. This is what we will need to use in our proof.

If this looks familiar, it is because we have already proven it for the general case of two parallel lines intersected by transversal line - here in the consecutive interior angles theorem.

So today, we just need to see that a trapezoid is no more than two parallel lines (the bases) intersected by two transversal lines (the legs) - and then apply the theorem, twice.

Proof

(1) ABCD is a trapezoid         //given

(2) AB||CD                             //definition of trapezoid

(3) m∠BAD + m∠CDA = 180°  //consecutive interior angles theorem

(4) m∠ABC + m∠DCB = 180°  //consecutive interior angles theorem

Alternatively, we could have applied the theorem to just the first set of angles. And since the sum of the angles in a simple convex quadrangle - and that includes trapezoids - is 360° - the other set of angles must be 360°-180°= 180°.

« Rhombus Diagonals Bisect the Angles
Circles: Finding the Area of a Ring »

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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