KindiakMath

Leftover Angle Properties

We say that two angles \alpha, \beta are supplementary if \alpha + \beta = 180^\circ.

Question 1. In the diagram below, show that \alpha + \beta = 360^\circ. That is, angles at a point sum to 360^\circ.

(Click for Solution)

Solution. Extend the straight line as follows.

Then \beta = \gamma + 180^\circ. Since adjacent angles on a straight line are supplementary, \alpha + \gamma = 180^\circ. Therefore

\begin{aligned} \alpha + \beta &= \alpha + (\gamma + 180^\circ) \\ &= (\alpha + \gamma) + 180^\circ \\ &= 180^\circ + 180^\circ \\ &= 360^\circ. \end{aligned}

Question 2. In the diagram below, show that \alpha + \beta = \gamma. That is, the external angle equals the sum of opposite interior angles.

(Click for Solution)

Solution. Construct the angle \theta adjacent to \gamma.

Since angles in a triangle are supplementary,

\alpha + \beta + \theta = 180^\circ.

Since adjacent angles on a straight line are supplementary,

\gamma + \theta = 180^\circ.

Therefore,

\alpha + \beta + \theta = 180^\circ = \gamma + \theta.

Canceling \theta from both sides,

\alpha + \beta = \gamma,

as required.

—Joel Kindiak, 15 Jan 26, 1808H

,

Published by

joelkindiak

Leave a comment