Question

in the circle below, suppose ( moverarc{kli} = 156^circ ) and ( mangle lkj = 58^circ ). find the following.
(a) ( mangle kli = square^circ )
(b) ( mangle lij = square^circ )

Explanation:

Step1: Analyze ∠KLI (Part a)

∠KLI is an inscribed angle? Wait, no, wait. Wait, the arc \( \overarc{KLI} \) is 156°, but ∠KLI: Wait, no, maybe I misread. Wait, the problem says "m \( \overarc{KLI} = 156^\circ \)" and "m ∠LKJ = 58°". Wait, for part (a), m ∠KLI: Wait, maybe ∠KLI is related to the cyclic quadrilateral? Wait, no, maybe ∠KLI is an inscribed angle? Wait, no, let's check the diagram. Points K, L, I, J are on the circle, so KLIJ is a cyclic quadrilateral? Wait, ∠KLI: Wait, maybe ∠KLI is a right angle? No, wait, maybe I made a mistake. Wait, the problem says "m \( \overarc{KLI} = 156^\circ \)" – no, wait, the arc KLI is 156°, but ∠KLI: Wait, no, maybe the question (a) is m ∠KLI, but wait, maybe it's a typo? Wait, no, the original problem: "In the circle below, suppose m \( \overarc{KLI} = 156^\circ \) and m ∠LKJ = 58°. Find the following. (a) m ∠KLI = [ ]° (b) m ∠LIJ = [ ]°" Wait, maybe ∠KLI is an inscribed angle subtended by arc KJ? Wait, no, let's think again. Wait, ∠LKJ is 58°, which is an inscribed angle. The measure of an inscribed angle is half the measure of its subtended arc. So ∠LKJ subtends arc LJ. So m arc LJ = 2 * 58° = 116°. Then, the total circumference is 360°, so arc KLI is 156°, arc LJ is 116°? Wait, no, that can't be. Wait, maybe the circle is divided into arcs. Wait, maybe KLI is an arc, and then ∠KLI: Wait, no, maybe ∠KLI is a right angle? No, that doesn't make sense. Wait, maybe I misread the problem. Wait, the problem says "m \( \overarc{KLI} = 156^\circ \)" – no, wait, maybe it's arc KJI? No, the problem says KLI. Wait, maybe ∠KLI is an inscribed angle subtended by arc KI? No, this is confusing. Wait, maybe the first part (a) is a typo, and it's supposed to be m ∠KJI or something else. Wait, no, the user's problem: (a) m ∠KLI = [ ]°. Wait, maybe ∠KLI is equal to 90°? No, that's not right. Wait, maybe the diagram shows that KLIJ is a cyclic quadrilateral, and ∠KLI is opposite to ∠KJ I? Wait, no, let's check the second part. For part (b), m ∠LIJ. Let's think about cyclic quadrilaterals: opposite angles in a cyclic quadrilateral sum to 180°. Wait, ∠LKJ is 58°, which is an inscribed angle. Let's try again.

Wait, ∠LKJ is 58°, so arc LJ (the arc opposite to ∠LKJ) is 2*58° = 116°. Then, arc KLI is 156°, so arc KI would be arc KLI - arc KL? No, maybe not. Wait, maybe the total circle is 360°, so arc KJ: wait, no. Wait, maybe the arc KLI is 156°, so the remaining arc KJ I would be 360° - 156° = 204°? No, that doesn't help. Wait, maybe ∠KLI is an inscribed angle subtended by arc KJ. Wait, no, let's look at part (b): m ∠LIJ. ∠LIJ is an inscribed angle? Or maybe ∠LIJ is related to ∠LKJ. Wait, maybe KLIJ is a cyclic quadrilateral, so ∠LIJ + ∠LKJ = 180°? No, that would be if they are opposite angles. Wait, ∠LKJ is 58°, so ∠LIJ would be 180° - 58° = 122°? But that's for part (b). Wait, maybe part (a) is a mistake, but the problem says "m \( \overarc{KLI} = 156^\circ \)" – no, wait, maybe the arc KLI is 156°, so the inscribed angle subtended by arc KI? No, I'm confused. Wait, maybe the first part (a) is actually m ∠KJI, but no. Wait, maybe the user made a typo, but assuming that ∠KLI is a right angle? No, that's not. Wait, wait, maybe the problem is that ∠KLI is an inscribed angle subtended by arc KJ, but arc KJ: wait, ∠LKJ is 58°, so arc LJ is 116°, so arc KJ would be 360° - 156° - 116° = 88°? No, that's not. Wait, I think I made a mistake. Let's start over.

In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. ∠LKJ is 58°, so it intercepts arc LJ. Therefore, m(arc LJ) = 2 * 58° = 116°. Now, the arc KLI is given as 156°. Wait, arc KLI: points K, L, I. So arc KLI is from K to L to I. Then, the remaining arc from I to K would be 360° - 156° = 204°? No, that can't be. Wait, maybe the circle is divided into arc KLI (156°) and arc KJI (360° - 156° = 204°). Now, ∠KLI: let's see, ∠KLI is an inscribed angle intercepting arc KJ. Wait, no, ∠KLI is at point L, between K and I. So the arc intercepted by ∠KLI is arc KJ. Wait, but arc KJ: if arc LJ is 116°, and arc KL is part of arc KLI (156°), then arc KJ would be arc KL + arc LJ? No, that's not. Wait, maybe I'm overcomplicating. Let's look at part (b): m ∠LIJ. ∠LIJ is an inscribed angle intercepting arc LK? No, wait, ∠LIJ is at point I, between L and J. So it intercepts arc LK. Wait, no, ∠LIJ: points L, I, J. So the angle at I between L and J. So it intercepts arc LJ? No, ∠LIJ intercepts arc LK? Wait, no, the inscribed angle at I, between L and J, intercepts arc LJ. Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. So ∠LIJ intercepts arc LK? Wait, maybe ∠LIJ is supplementary to ∠LKJ because they are in a cyclic quadrilateral. Wait, KLIJ is a cyclic quadrilateral, so opposite angles sum to 180°. ∠LKJ and ∠LIJ: are they opposite? ∠LKJ is at K, ∠LIJ is at I. So if KLIJ is cyclic, then ∠LKJ + ∠LIJ = 180°? Wait, 58° + ∠LIJ = 180°, so ∠LIJ = 122°. That makes sense for part (b). Now, part (a): m ∠KLI. Wait, maybe ∠KLI is a right angle? No, that's not. Wait, maybe the problem has a typo, and part (a) is m ∠KJI, but no. Wait, the arc KLI is 156°, so the inscribed angle subtended by arc KLI would be half of 156°, which is 78°. Wait, maybe ∠KLI is an inscribed angle subtended by arc KI? No, arc KLI is 156°, so the angle subtended by arc KLI at the center is 156°, so the inscribed angle would be 78°. So m ∠KLI = 78°? Let's check: if arc KLI is 156°, then the inscribed angle ∠KJI (at J) would be half of 156°, which is 78°, but ∠KLI: maybe ∠KLI is equal to ∠KJI because they subtend the same arc? Wait, yes! In a circle, angles subtended by the same arc are equal. So arc KI is subtended by ∠KLI (at L) and ∠KJI (at J). So if arc KLI is 156°, wait, no, arc KI: wait, arc KLI is K to L to I, so arc KI is part of that? No, arc KLI is K-L-I, so the arc from K to I through L is 156°, so the inscribed angle subtended by arc KI (the minor arc) would be different, but if we take the major arc KI, it's 360° - 156° = 204°, but that's too big. Wait, maybe I'm wrong. Let's try again.

Wait, the problem says "m \( \overarc{KLI} = 156^\circ \)" – so arc KLI is 156°, so the inscribed angle ∠KJI (at J) subtended by arc KLI would be half of 156°, which is 78°. Then, ∠KLI (at L) subtended by arc KJ: wait, no. Wait, ∠LKJ is 58°, which is an inscribed angle subtended by arc LJ, so arc LJ is 116°, as before. Then, arc KLI is 156°, so arc KI (the arc from K to I through L) is 156°, so the arc from K to I through J would be 360° - 156° = 204°. Then, the inscribed angle subtended by arc KI (minor arc) would be half of (360° - 156°) = 102°, but that's not. Wait, I'm getting confused. Let's look at part (b): m ∠LIJ. ∠LIJ is at I, between L and J. So it's an inscribed angle subtended by arc LK. Wait, arc LK: if arc LJ is 116°, and arc KLI is 156°, then arc LK is part of arc KLI. Wait, arc KLI is K-L-I, so arc LK is from L to K, which is part of arc KLI. Wait, no, arc KLI is K to L to I, so arc LK is from L to K, which is the reverse of arc KL. Wait, maybe the total arc KLI is 156°, so arc KL + arc LI = 156°. Then, arc LJ is 116°, so arc KJ = arc KL + arc LJ? No, that's not. Wait, maybe the key is that ∠LIJ is supplementary to ∠LKJ because KLIJ is cyclic. So ∠LIJ = 180° - 58° = 122°, which is part (b). Then, part (a): m ∠KLI. If arc KLI is 156°, then the inscribed angle ∠KLI subtended by arc KI? No, maybe ∠KLI is 78°, as half of 156°? Wait, 156° / 2 = 78°, so m ∠KLI = 78°. Let's check:

Part (a): m ∠KLI = 78° (since it's an inscribed angle subtended by arc KI, which is 156°? Wait, no, inscribed angle is half the arc. So if arc KLI is 156°, then the inscribed angle subtended by arc KLI is 78°, so ∠KLI = 78°.

Part (b): ∠LIJ is supplementary to ∠LKJ (because KLIJ is cyclic, opposite angles sum to 180°), so 180° - 58° = 122°.

Yes, that makes sense. So:

(a) m ∠KLI = 78° (half of arc KLI, 156° / 2 = 78°)

(b) m ∠LIJ = 180° - 58° = 122° (cyclic quadrilateral opposite angles sum to 180°)

Step1: Solve for m ∠KLI (Part a)

The measure of an inscribed angle is half the measure of its intercepted arc. ∠KLI intercepts arc \( \overarc{KLI} \), which has a measure of \( 156^\circ \). So:
\( m \angle KLI = \frac{1}{2} \times m \overarc{KLI} = \frac{1}{2} \times 156^\circ = 78^\circ \).

Step2: Solve for m ∠LIJ (Part b)

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary (sum to \( 180^\circ \)). \( \angle LKJ \) and \( \angle LIJ \) are opposite angles in cyclic quadrilateral \( KLIJ \). Given \( m \angle LKJ = 58^\circ \):
\( m \angle LIJ = 180^\circ - m \angle LKJ = 180^\circ - 58^\circ = 122^\circ \).

Answer:

(a) \( \boldsymbol{78} \)
(b) \( \boldsymbol{122} \)