in the given figure, what is the measure of angle cpe? figure not drawn to scale a. 64° b. 68° c. 70° d. 80°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle CPE=\angle FPD$. Also, the sum of the measures of the arcs intercepted by vertical angles in a circle is equal. So, $(2x + 28)+(4x + 12)=(x + 60)+ \text{arc}(CF)$. Since $\angle CPE$ and $\angle FPD$ are vertical angles, we can set up an equation based on the angle - arc relationship. The measure of an angle formed by two intersecting chords in a circle is half the sum of the measures of the intercepted arcs. But we can also use the fact that vertical angles are equal. So, $2x+28 = x + 60$.

Step2: Solve for x

Subtract x from both sides of the equation $2x+28 = x + 60$. We get $2x - x=60 - 28$, so $x = 32$.

Step3: Find the measure of $\angle CPE$

Substitute $x = 32$ into the expression for $\angle CPE$ (we can use the expression for the angle opposite to it which is $x + 60$). $\angle CPE=x + 60$. Plugging in $x = 32$, we have $\angle CPE=32+60=80^{\circ}$.

Answer:

D. $80^{\circ}$