Question

in circle n, $overline{kl}congoverline{ml}$. what is the measure of $widehat{jm}$? (13x + 2)° 77° 90° (5x + 24)° n m 132° (8x - 3)° k 154° (7x + 7)° l

Explanation:

Step1: Use the property of congruent chords

Since $\overline{KL}\cong\overline{ML}$, the central - angles subtended by these chords are equal. So, $m\angle KNL = m\angle MLN$. That is, $7x + 7=8x - 3$.
Solve for $x$:
\[

$$\begin{align*} 7x+7&=8x - 3\\ 7 + 3&=8x-7x\\ x&=10 \end{align*}$$

\]

Step2: Find the measure of $\angle JNM$

The sum of the central - angles in a circle is $360^{\circ}$.
First, find the measures of the known central - angles:
$m\angle JNK=(5x + 24)^{\circ}$, substituting $x = 10$, we get $m\angle JNK=(5\times10 + 24)^{\circ}=74^{\circ}$.
$m\angle KNL=(7x + 7)^{\circ}$, substituting $x = 10$, we get $m\angle KNL=(7\times10 + 7)^{\circ}=77^{\circ}$.
$m\angle MLN=(8x - 3)^{\circ}$, substituting $x = 10$, we get $m\angle MLN=(8\times10 - 3)^{\circ}=77^{\circ}$.
$m\angle JNM=360-(m\angle JNK + m\angle KNL+m\angle MLN)$.
$m\angle JNK + m\angle KNL+m\angle MLN=74 + 77+77 = 228^{\circ}$.
$m\angle JNM=360 - 228=132^{\circ}$.
The measure of an arc is equal to the measure of its central - angle. So, the measure of $\widehat{JM}$ is $132^{\circ}$.

Answer:

$132^{\circ}$