Question

for the figure shown, find m∠1 and m∠2. (the figure is not drawn to scale.) m∠1 = 118° m∠2 = □°

Explanation:

Step1: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the larger triangle, the exterior angle is \(134^{\circ}\), and one of the non - adjacent interior angles is \(84^{\circ}\). Let's find the other non - adjacent interior angle which is \(\angle1\).
We know that \(134^{\circ}=84^{\circ}+\angle1\).
So, \(\angle1 = 134^{\circ}-84^{\circ}=50^{\circ}\). But we are given \(m\angle1 = 118^{\circ}\), so we use the angle - sum property of a triangle for the smaller triangle containing \(\angle1\) and \(\angle2\).

Step2: Apply angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). In the triangle with angles \(16^{\circ}\), \(\angle1 = 118^{\circ}\), and \(\angle2\), we have \(16^{\circ}+118^{\circ}+\angle2=180^{\circ}\).
We can solve for \(\angle2\) as follows:
\[

$$\begin{align*} \angle2&=180^{\circ}-(16^{\circ} + 118^{\circ})\\ &=180^{\circ}-134^{\circ}\\ & = 46^{\circ} \end{align*}$$

\]

Answer:

\(m\angle2 = 46^{\circ}\)