Question

in the circle below, \\(\overline{ik}\\) is a diameter. suppose \\(m\overarc{jk} = 122^\circ\\) and \\(m\angle kjl = 53^\circ\\). find the following.
(a) \\(m\angle ijl = \square^\circ\\)
(b) \\(m\angle ikj = \square^\circ\\)

Explanation:

Step1: Find \( m\angle IJL \)

Since \( \overline{IK} \) is a diameter, the measure of arc \( IK \) is \( 180^\circ \). We know \( m\overarc{JK} = 122^\circ \), so the measure of arc \( IJ \) is \( 180^\circ - 122^\circ = 58^\circ \). The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. But \( \angle IJL \) intercepts arc \( IL \)? Wait, no, actually, \( \angle KJL = 53^\circ \), and we can find \( \angle IJL \) by using the fact that \( \angle IJK \) is an inscribed angle over the diameter, so \( \angle IJK = 90^\circ \) (since an angle inscribed in a semicircle is a right angle). So \( \angle IJL + \angle KJL = \angle IJK = 90^\circ \). Given \( \angle KJL = 53^\circ \), then \( \angle IJL = 90^\circ - 53^\circ = 37^\circ \)? Wait, no, wait. Wait, arc \( JK \) is \( 122^\circ \), so the inscribed angle over arc \( JK \) would be \( \frac{122^\circ}{2} = 61^\circ \), but maybe I messed up. Wait, let's start over.

First, \( \overline{IK} \) is a diameter, so arc \( IK \) is \( 180^\circ \). Arc \( JK \) is \( 122^\circ \), so arc \( IJ \) is \( 180^\circ - 122^\circ = 58^\circ \). Then, \( \angle IJL \): wait, maybe \( \angle IJL \) intercepts arc \( IL \), but maybe another approach. Wait, the angle \( \angle KJL = 53^\circ \), which is an inscribed angle. The arc \( KL \) that it intercepts would be \( 2 \times 53^\circ = 106^\circ \)? No, that doesn't make sense. Wait, no, \( \angle KJL \) is an inscribed angle, so it intercepts arc \( KL \). So \( m\overarc{KL} = 2 \times m\angle KJL = 2 \times 53^\circ = 106^\circ \). Then, arc \( JK \) is \( 122^\circ \), arc \( KL \) is \( 106^\circ \), that can't be, since the total circle is \( 360^\circ \), but \( IK \) is a diameter, so the semicircle is \( 180^\circ \). Wait, I think I confused the semicircle. \( IK \) is a diameter, so the semicircle \( IJK \) is \( 180^\circ \). So arc \( IJ + arc JK = 180^\circ \). Arc \( JK = 122^\circ \), so arc \( IJ = 180 - 122 = 58^\circ \). Then, \( \angle IJL \): wait, maybe \( \angle IJL \) is equal to half the measure of arc \( IL \), but maybe not. Wait, the problem is part (a) \( m\angle IJL \). Let's look at the diagram: points \( I, J, K, L \) on the circle, \( IK \) diameter, \( JL \) a chord, \( JK \) a chord. So \( \angle IJK \) is an inscribed angle over diameter \( IK \), so \( \angle IJK = 90^\circ \) (since angle in semicircle is right angle). So \( \angle IJL + \angle KJL = \angle IJK = 90^\circ \). Given \( \angle KJL = 53^\circ \), then \( \angle IJL = 90^\circ - 53^\circ = 37^\circ \)? Wait, but that contradicts the arc measure. Wait, no, maybe \( \angle IJK \) is not 90^\circ? Wait, no, any angle inscribed in a semicircle is a right angle. So triangle \( IJK \) is right-angled at \( J \), so \( \angle IJK = 90^\circ \). Therefore, \( \angle IJL + \angle KJL = 90^\circ \), so \( \angle IJL = 90 - 53 = 37^\circ \). Wait, but let's check part (b): \( m\angle IKJ \). \( \angle IKJ \) is an inscribed angle intercepting arc \( IJ \). Arc \( IJ \) is \( 180 - 122 = 58^\circ \), so \( \angle IKJ = \frac{58^\circ}{2} = 29^\circ \)? Wait, no, arc \( IJ \) is \( 58^\circ \), so inscribed angle over arc \( IJ \) is \( 29^\circ \). Wait, but let's verify.

Wait, arc \( JK = 122^\circ \), so the central angle for arc \( JK \) is \( 122^\circ \), so the inscribed angle \( \angle IKJ \) intercepts arc \( IJ \), which is \( 180 - 122 = 58^\circ \), so \( \angle IKJ = \frac{58}{2} = 29^\circ \). And for part (a), \( \angle IJL \): since \( \angle IJK = 90^\circ \) (right angle), and \( \angle KJL = 53^\circ \), then \( \angle IJL = 90 - 53 = 37^\circ \). Wait, but let's check with arc measures. Arc \( IL \): if \( \angle IJL = 37^\circ \), then arc \( IL \) would be \( 74^\circ \). Then arc \( IJ = 58^\circ \), arc \( IL = 74^\circ \), arc \( LK \): wait, maybe not. Alternatively, maybe I made a mistake in \( \angle IJK \). Wait, \( \overline{IK} \) is a diameter, so \( \angle IJK \) is inscribed in the semicircle \( IJK \), so yes, \( \angle IJK = 90^\circ \). So \( \angle IJL + \angle KJL = 90^\circ \), so \( \angle IJL = 90 - 53 = 37^\circ \). Then for part (b), \( \angle IKJ \): in triangle \( IJK \), which is right-angled at \( J \), we have \( \angle IKJ + \angle JIK = 90^\circ \). \( \angle JIK \) is an inscribed angle over arc \( JK \), so \( \angle JIK = \frac{122^\circ}{2} = 61^\circ \). Therefore, \( \angle IKJ = 90^\circ - 61^\circ = 29^\circ \). Yes, that matches. So:

(a) \( \angle IJL = 90^\circ - 53^\circ = 37^\circ \)

(b) \( \angle IKJ = 90^\circ - \frac{122^\circ}{2} = 90^\circ - 61^\circ = 29^\circ \)

Step1 (for part a):

Since \( \overline{IK} \) is a diameter, \( \angle IJK = 90^\circ \) (angle in a semicircle).
Given \( \angle KJL = 53^\circ \), and \( \angle IJL + \angle KJL = \angle IJK \),
then \( \angle IJL = 90^\circ - 53^\circ = 37^\circ \).

Step2 (for part b):

The inscribed angle \( \angle JIK \) intercepts arc \( JK \), so \( m\angle JIK = \frac{1}{2}m\overarc{JK} = \frac{122^\circ}{2} = 61^\circ \).
In right triangle \( IJK \), \( \angle IKJ + \angle JIK = 90^\circ \),
so \( \angle IKJ = 90^\circ - 61^\circ = 29^\circ \).

Answer:

(a) \( \boxed{37} \)
(b) \( \boxed{29} \)