Question

n below, ( mangle cih = 106^{circ} ) and ( mangle bgd = 42^{circ} ). find ( mangle ahf ).
you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Explanation:

Step1: Find the measure of $\angle HGI$

Since $\angle BGD$ and $\angle HGI$ are vertical angles, they are equal. So, $m\angle HGI = 42^{\circ}$.

Step2: Find the measure of $\angle GHI$

$\angle CIH$ and $\angle GHI$ are vertical angles. So, $m\angle GHI=106^{\circ}$.

Step3: Use the triangle - angle sum property

In $\triangle HGI$, we know that the sum of the interior angles of a triangle is $180^{\circ}$. Let $m\angle AHG$ be $x$. Then $x + 106^{\circ}+42^{\circ}=180^{\circ}$.

$$x=180^{\circ}-(106^{\circ} + 42^{\circ})=180^{\circ}-148^{\circ}=32^{\circ}$$

Step4: Find the measure of $\angle AHF$

Since $\angle AHG$ and $\angle AHF$ are supplementary (they form a linear pair), $m\angle AHF = 180^{\circ}-m\angle AHG$.

$$m\angle AHF=180^{\circ}- 32^{\circ}=148^{\circ}$$

Answer:

$148^{\circ}$