Question

exterior/interior angles what is the value of x? what is the value of x? this is a regular pentagon. (20x + 8)

Explanation:

Step1: Recall sum of exterior angles of polygon

The sum of exterior angles of any polygon is $360^{\circ}$.

Step2: Solve for left - hand problem

For the left - hand polygon, we have the equation $2x + 45+40 + x+77=360$. Combine like terms: $3x+162 = 360$. Then subtract 162 from both sides: $3x=360 - 162=198$. Divide both sides by 3: $x=\frac{198}{3}=66$.

Step3: Recall interior angle formula for regular polygon

The formula for the measure of an interior angle of a regular $n$ - sided polygon is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$. For a pentagon, $n = 5$, so $\theta=\frac{(5 - 2)\times180^{\circ}}{5}=\frac{3\times180^{\circ}}{5}=108^{\circ}$.

Step4: Solve for right - hand problem

The interior angle of the regular pentagon is $108^{\circ}$, and the exterior angle is $180 - 108 = 72^{\circ}$. If the exterior angle is represented as $20x+8$, then we set up the equation $20x+8 = 72$. Subtract 8 from both sides: $20x=72 - 8 = 64$. Divide both sides by 20: $x=\frac{64}{20}=3.2$.

Answer:

Problem on Left: $x = 66$
Problem on Right: $x = 3.2$