Question

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

Explanation:

Step1: Set angles equal (inscribed arcs)

Angles subtended by the same arc (arc AD) on the circumference are congruent, so:
$$4x + 10 = 5x - 2$$

Step2: Solve for x

Rearrange to isolate x:
$$10 + 2 = 5x - 4x$$
$$x = 12$$

Step3: Find inscribed angle measure

Substitute x into one angle formula:
$$\angle ABD = 4(12) + 10 = 58^\circ$$

Step4: Calculate arc AD measure

The measure of an inscribed angle is half the measure of its subtended arc. Let arc AD = $m$:
$$58^\circ = \frac{1}{2}m$$
$$m = 58^\circ \times 2 = 116^\circ$$

Answer:

116°