Question

angle kjl measures (7x - 8)°. angle kml measures (3x + 8)°. what is the measure of arc kl? 20° 40° 48° 96°

Explanation:

Step1: Recall the inscribed - angle theorem

In a circle, angles inscribed in the same arc are equal. So, $\angle KJL=\angle KML$.
$7x - 8=3x + 8$

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $7x-3x - 8=3x-3x + 8$, which simplifies to $4x-8 = 8$.
Add 8 to both sides: $4x-8 + 8=8 + 8$, so $4x=16$.
Divide both sides by 4: $x = 4$.

Step3: Find the measure of $\angle KML$

Substitute $x = 4$ into the expression for $\angle KML$: $\angle KML=(3x + 8)^{\circ}=(3\times4 + 8)^{\circ}=(12 + 8)^{\circ}=20^{\circ}$.

Step4: Use the inscribed - angle and arc relationship

The measure of an inscribed angle is half the measure of the intercepted arc. Let the measure of arc $KL$ be $y$. Then $\angle KML=\frac{1}{2}y$.
Since $\angle KML = 20^{\circ}$, we have $20^{\circ}=\frac{1}{2}y$.
Multiply both sides by 2: $y = 40^{\circ}$.

Answer:

$40^{\circ}$