Question

this is a regular decagon. what is the value of x? what are the measures of ∠a and ∠e?

Explanation:

Step1: Recall interior - angle formula for regular polygon

For a regular polygon with $n$ sides, the measure of each interior angle $\theta$ is given by $\theta=\frac{(n - 2)\times180^{\circ}}{n}$.

Step2: Solve for left - hand problem

A decagon has $n = 10$ sides. Substitute $n = 10$ into the formula: $\theta=\frac{(10 - 2)\times180^{\circ}}{10}=\frac{8\times180^{\circ}}{10}=144^{\circ}$, so $x = 144^{\circ}$.

Step3: Solve for right - hand problem

The sum of the interior angles of a pentagon ($n = 5$) is $(n - 2)\times180^{\circ}=(5 - 2)\times180^{\circ}=540^{\circ}$. Given angles are $164^{\circ}$, $102^{\circ}$, and $90^{\circ}$ (right - angle at $B$). Let $\angle A = 90^{\circ}$ (right - angle shown). Then $\angle E=540^{\circ}-(164^{\circ}+102^{\circ}+90^{\circ}+90^{\circ})=540^{\circ}-446^{\circ}=140^{\circ}$.

Answer:

Problem on Left:
$x = 144^{\circ}$
Problem on Right:
$\angle A=90^{\circ}$, $\angle E = 140^{\circ}$