Question

in △ghi,

• m∠g = 80°, and
• the measure of the exterior angle at vertex h is 140°.

what is m∠i?
a. 140°
b. 60°
c. 40°
d. 120°

Explanation:

Step1: Recall exterior - angle property

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Let the interior angle at vertex $H$ be $\angle H$ and its exterior angle be $140^{\circ}$.

Step2: Find $\angle H$

Since the exterior angle at $H$ and $\angle H$ are supplementary (their sum is $180^{\circ}$), then $m\angle H=180^{\circ}- 140^{\circ}=40^{\circ}$.

Step3: Use angle - sum property of a triangle

In $\triangle GHI$, the sum of the interior angles of a triangle is $180^{\circ}$, i.e., $m\angle G + m\angle H+m\angle I=180^{\circ}$. We know $m\angle G = 80^{\circ}$ and $m\angle H = 40^{\circ}$. Substitute these values into the equation: $80^{\circ}+40^{\circ}+m\angle I=180^{\circ}$.

Step4: Solve for $m\angle I$

First, simplify the left - hand side: $120^{\circ}+m\angle I=180^{\circ}$. Then, subtract $120^{\circ}$ from both sides: $m\angle I=180^{\circ}-120^{\circ}=60^{\circ}$.

Answer:

B. $60^{\circ}$