Question

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

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of a central angle is equal to the measure of the arc it intercepts. So, $\angle ADC$ (central angle) and arc $AC$ have the same measure. Set up the equation $7x + 2=8x - 8$.

Step2: Solve the equation for $x$

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

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 the arc it intercepts. $\angle ABC$ is an inscribed angle intercepting arc $AC$. So, $m\angle ABC=\frac{1}{2}\times m\overset{\frown}{AC}$.

Step5: Calculate the measure of $\angle ABC$

$m\angle ABC=\frac{1}{2}\times72^{\circ}=36^{\circ}$.

Answer:

$36^{\circ}$