Question

if $overline{gh}$ is not parallel to $overline{ij}$, what is $mangle h$? 115° $mangle h=square^{circ}$

Explanation:

Step1: Recall polygon - angle sum formula

The sum of interior angles of a pentagon is given by $(n - 2)\times180^{\circ}$, where $n = 5$. So, $(5 - 2)\times180^{\circ}=540^{\circ}$.

Step2: Assume equal - length sides and some angle relationships (no parallel - side info given)

Since no other angle information is given except $\angle I = 115^{\circ}$ and no parallel - side relationships can be used to find angle measures, and assuming the polygon is a regular - like pentagon (as no other conditions), we can't directly use parallel - line angle rules. But if we assume the pentagon is equilateral (sides are equal as indicated by the red marks), we still lack enough information. However, if we assume the pentagon is a regular pentagon (a special case when sides are equal and angles are equal), the measure of each interior angle of a regular pentagon is $\frac{(5 - 2)\times180^{\circ}}{5}=108^{\circ}$. But this is a wrong assumption as we are given non - parallel sides. So, we need to use the fact that we have one angle $\angle I=115^{\circ}$. Let's assume the other four angles are equal (a non - standard but only way with given info). Let $m\angle H=x$. Then $115^{\circ}+4x = 540^{\circ}$.

Step3: Solve for $x$

First, subtract $115^{\circ}$ from both sides of the equation $115^{\circ}+4x = 540^{\circ}$. We get $4x=540^{\circ}-115^{\circ}=425^{\circ}$. Then $x=\frac{425^{\circ}}{4}=106.25^{\circ}$.

Answer:

$106.25$