Question

question in the diagram below, m∠cih = 107° and m∠bgd = 39°, find m∠fhg.

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. $\angle CIH$ and $\angle DIG$ are vertical - angles. So, $m\angle DIG=m\angle CIH = 107^{\circ}$.

Step2: Use the angle - sum property in a triangle

In the triangle formed by the intersection of the lines at point $G$, we know that the sum of angles in a triangle is $180^{\circ}$. Let's assume we want to find $\angle FHG$.
We know that $\angle DIG$ is an exterior angle of the triangle with an interior angle $\angle BGD = 39^{\circ}$.
By the exterior - angle property of a triangle (the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles), if we consider the triangle with one vertex at $G$ and the line intersecting at $H$, and the exterior angle at $G$ related to the angles at $H$ and the other non - adjacent interior angle.
Let $\angle FHG=x$.
We know that $\angle DIG$ (exterior angle) $=\angle BGD+\angle FHG$.
So, $x=\angle DIG-\angle BGD$.

Step3: Calculate the angle

Substitute $m\angle DIG = 107^{\circ}$ and $m\angle BGD = 39^{\circ}$ into the equation.
$m\angle FHG=107^{\circ}-39^{\circ}=68^{\circ}$.

Answer:

$68^{\circ}$