Question

in the diagram below, m∠cih = 100° and m∠bgd = 43°. find m∠ahf.

Explanation:

Step1: Identify vertical - angles

$\angle BGD$ and $\angle AGF$ are vertical - angles. Since vertical angles are equal, and $m\angle BGD = 43^{\circ}$, then $m\angle AGF=43^{\circ}$.

Step2: Identify linear - pair

$\angle CIH$ and $\angle HIG$ form a linear - pair. A linear - pair of angles is supplementary, so $m\angle HIG = 180^{\circ}-m\angle CIH$. Given $m\angle CIH = 100^{\circ}$, then $m\angle HIG=180 - 100=80^{\circ}$.

Step3: Use angle - sum property in triangle

In $\triangle HIG$, we want to find $\angle AHF$. Note that $\angle AHF$ and $\angle GHI$ are vertical - angles. In $\triangle HIG$, we know that the sum of interior angles of a triangle is $180^{\circ}$. But we can also use the fact that $\angle AHF$ can be found by considering the angles around point $H$.
$\angle AHF$ and $\angle AGF$ and $\angle HIG$ are related. Since $\angle AHF$ and $\angle GHI$ are vertical - angles, and considering the angles formed by the intersecting lines.
We know that $\angle AHF = 180^{\circ}-(43^{\circ}+80^{\circ})$.
$180-(43 + 80)=180 - 123=57^{\circ}$.

Answer:

$57^{\circ}$