Question

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

Explanation:

Step1: Set arc KL and arc ML equal

Since \(\overline{KL} \cong \overline{ML}\), their corresponding arcs \(\overset{\frown}{KL}\) and \(\overset{\frown}{ML}\) are equal. So, \(7x + 7 = 8x - 3\).

Step2: Solve for x

Subtract \(7x\) from both sides: \(7 = x - 3\). Then add 3 to both sides: \(x = 10\).

Step3: Find measure of arc JM

First, find the measure of arc JK: \(5x + 24 = 5(10) + 24 = 74^\circ\). Arc KL: \(7x + 7 = 7(10) + 7 = 77^\circ\). Arc ML: \(8x - 3 = 8(10) - 3 = 77^\circ\). The total circumference of a circle is \(360^\circ\), but we can also use the fact that the sum of arcs around a circle is \(360^\circ\), and arc JM is \(13x + 2\). Substitute \(x = 10\): \(13(10) + 2 = 132^\circ\)? Wait, no, wait. Wait, actually, the central angles correspond to the arcs. Wait, let's check again. Wait, the arcs: arc JK is \(5x + 24\), arc KL is \(7x + 7\), arc LM is \(8x - 3\), and arc MJ is \(13x + 2\). The sum of all arcs in a circle is \(360^\circ\). So, \((5x + 24) + (7x + 7) + (8x - 3) + (13x + 2) = 360\). Let's compute: \(5x + 7x + 8x + 13x + 24 + 7 - 3 + 2 = 360\). \(33x + 30 = 360\). \(33x = 330\), \(x = 10\). Then arc JM is \(13x + 2 = 13(10) + 2 = 132^\circ\)? Wait, no, wait the options have 132. Wait, but let's check the arcs. Wait, maybe I made a mistake. Wait, the arcs: arc JK is \(5x +24 = 74\), arc KL is \(77\), arc LM is \(77\), so arc JM is \(360 - 74 - 77 - 77 = 132\). Yes, that's correct.

Answer:

132° (the option with 132°)