Question

in circle n, kl ≅ ml. what is the measure of jm? (13x + 2)° (5x + 24)° (8x - 3)° (7x + 7)° 77° 154° 132° 90°

Explanation:

Step1: Use the property of congruent chords

Since $\overline{KL}\cong\overline{ML}$, the central - angles subtended by these chords are equal. Let's first find the value of $x$ using the fact that the sum of the measures of the angles around a point is $360^{\circ}$. The angles at the center of the circle formed by the radii to the points $J$, $K$, $L$, and $M$ are related to the inscribed - angles. The measure of an inscribed angle is half the measure of the central angle subtended by the same arc. But we can also use the fact that the sum of the inscribed angles in a cyclic quadrilateral $JKLM$ and the properties of equal - chord angles.
We know that in a circle, if two chords are equal, the arcs they subtend are equal. Also, the sum of the inscribed angles in a cyclic quadrilateral is $360^{\circ}$. But we can use the angle - sum property of a triangle formed by the radii. Since $\overline{KL}\cong\overline{ML}$, the angles subtended by these chords at the circumference are equal.
Let's assume we use the fact that the sum of the inscribed angles in the cyclic quadrilateral $JKLM$: $(5x + 24)+(13x+2)+(8x - 3)+(7x + 7)=360$.
Combine like terms: $(5x+13x + 8x+7x)+(24 + 2-3 + 7)=360$.
$33x+30 = 360$.
Subtract 30 from both sides: $33x=360 - 30=330$.
Divide both sides by 33: $x = 10$.

Step2: Find the measure of $\overset{\frown}{JM}$

The measure of an arc is twice the measure of the inscribed angle subtended by it. The inscribed angle $\angle{JKM}$ and $\angle{JLM}$ are related to the arc $\overset{\frown}{JM}$.
First, find the measure of $\angle{JKM}=5x + 24$. Substitute $x = 10$ into it, $\angle{JKM}=5\times10+24=74^{\circ}$.
The measure of $\angle{JLM}=8x - 3$. Substitute $x = 10$ into it, $\angle{JLM}=8\times10-3 = 77^{\circ}$.
The measure of the arc $\overset{\frown}{JM}$ is twice the measure of the inscribed angle $\angle{JKM}$ or $\angle{JLM}$ (angles in the same segment).
The measure of $\overset{\frown}{JM}=2\times77=154^{\circ}$.

Answer:

$154^{\circ}$