Question

in circle d, angle adc measures (7x + 2)°. arc ac measures (8x - 8)°. what is the measure of ∠abc? 36° 43° 72° 144°

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of a central angle is equal to the measure of its intercepted arc. So, $\angle ADC$ (central angle) and arc $AC$ are related as $7x + 2=8x - 8$.

Step2: Solve for $x$

Subtract $7x$ from both sides of the equation $7x + 2=8x - 8$:
$2=x - 8$.
Then add 8 to both sides: $x=10$.

Step3: Find the measure of arc $AC$

Substitute $x = 10$ into the expression for arc $AC$: $8x-8=8\times10 - 8=72^{\circ}$.

Step4: Recall the inscribed - arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc. $\angle ABC$ is an inscribed angle and arc $AC$ is its intercepted arc. So, $m\angle ABC=\frac{1}{2}m\overset{\frown}{AC}$.
Since $m\overset{\frown}{AC}=72^{\circ}$, then $m\angle ABC = 36^{\circ}$.

Answer:

A. $36^{\circ}$