3150.11 – Calculating an Angle

What is the measure of ADB? It will be the same regardless of the placement of points A and B, as the figure below suggests, as long as the indicated angles are equal.

Solution

Solution. By the exterior angle theorem (twice):

\begin{aligned} 2x &=& \angle 2 + 40° \\ 2y &=& \angle 1 + 40°. \end{aligned}

Therefore,

\begin{aligned} \angle 2 &=& 2x - 40 \\ \angle 1 &=& 2y - 40. \end{aligned}

Adding gives

$\angle 1 + \angle 2 = 2x + 2y -80,$

but

$2x + 2y - 80 = \angle 1 + \angle 2 = 180 - 40 = 140,$

$2x + 2y = 220,$

$x + y = 110.$

Finally,

$z = 180 - (x + y) = 180 - 110 = 70°.$