Question

angle abd measures (4x + 10)°. angle acd measures (5x - 2)°. what is the measure of arc ad? 58° 116° 12° 96°

Explanation:

Step1: Recall inscribed - angle theorem

In a circle, inscribed angles that intercept the same arc are equal. So, $\angle ABD=\angle ACD$.

Step2: Set up the equation

Set $4x + 10=5x - 2$.

Step3: Solve the equation for $x$

Subtract $4x$ from both sides: $10=x - 2$. Then add 2 to both sides, we get $x = 12$.

Step4: Find the measure of an inscribed - angle

Substitute $x = 12$ into the measure of $\angle ABD$ (we could also use $\angle ACD$). $\angle ABD=4x + 10=4\times12+10=48 + 10=58^{\circ}$.

Step5: Use the inscribed - angle and arc relationship

The measure of an arc is twice the measure of the inscribed angle that intercepts it. If $\angle ABD$ intercepts arc $AD$, then the measure of arc $AD = 2\angle ABD$.

Step6: Calculate the measure of arc $AD$

$m\overset{\frown}{AD}=2\times58^{\circ}=116^{\circ}$.

Answer:

$116^{\circ}$