Question

what is the value of x? this is a regular pentagon. what is the value of x?

Explanation:

Step1: Recall exterior - angle sum property

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

Step2: Set up equation for left - hand problem

For the left - hand polygon, we have $2x + 45+40 + x+77=360$.
Combining like terms gives $3x+162 = 360$.

Step3: Solve for x in left - hand problem

Subtract 162 from both sides: $3x=360 - 162=198$.
Divide both sides by 3: $x=\frac{198}{3}=69$.

Step4: Recall exterior - angle of regular pentagon

The measure of each exterior angle of a regular pentagon is $\frac{360^{\circ}}{5}=72^{\circ}$.
The interior angle and exterior angle of a polygon are supplementary. So the interior angle of a regular pentagon is $180 - 72=108^{\circ}$.

Step5: Set up equation for right - hand problem

For the right - hand regular pentagon, we set $20x+8 = 108$.

Step6: Solve for x in right - hand problem

Subtract 8 from both sides: $20x=108 - 8 = 100$.
Divide both sides by 20: $x=\frac{100}{20}=10$.

Answer:

Problem on Left: 69
Problem on Right: 16