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

Explanation:

Step1: Recall the central - inscribed angle relationship

The measure of an arc is equal to the measure of its central angle. So the central angle $\angle ADC$ and the arc $AC$ are related as $\angle ADC$ (central angle) and arc $AC$ have equal measures when the angle is subtended by the arc at the center of the circle. So we set up the equation $7x + 2=8x - 8$. $7x+2 = 8x - 8$

Step2: Solve the equation for $x$

Subtract $7x$ from both sides: $2=x - 8$. Then add 8 to both sides to get $x = 10$. $x=10$

Step3: Find the measure of arc $AC$

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

Step4: Recall the inscribed - angle formula

The measure of an inscribed angle is half the measure of the arc it subtends. $\angle ABC$ is an inscribed angle and arc $AC$ is the arc it subtends. So $m\angle ABC=\frac{1}{2}m\overset{\frown}{AC}$. $m\angle ABC=\frac{1}{2}\times72^{\circ}$

Answer:

$36^{\circ}$