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In the circle below, bar(IK) is a diameter. Suppose mJK^(⏜)=102^(@) and m/_KJL=54^(@). Find the following. (a) m/_IJL= (b) m/_IKJ=

In the circle below, IK‾ \overline{I K} is a diameter. Suppose m \overparen{J K}=102^{\circ} and m∠KJL=54∘ m \angle K J L=54^{\circ} . Find the following.\newline(a) m∠IJL= m \angle I J L= \newline(b) m∠IKJ= m \angle I K J=

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Full solution

Q. In the circle below, IK‾ \overline{I K} is a diameter. Suppose m \overparen{J K}=102^{\circ} and m∠KJL=54∘ m \angle K J L=54^{\circ} . Find the following.\newline(a) m∠IJL= m \angle I J L= \newline(b) m∠IKJ= m \angle I K J=
  1. Identify Relationship: Identify the relationship between the diameter and angles in a circle.\newlineSince IKIK is a diameter, angle IKJIKJ is a right angle (9090 degrees) because an angle inscribed in a semicircle is a right angle.
  2. Calculate Angle: Calculate m/IKJm/_{\text{IKJ}} using the property of the right angle.m/IKJ=90m/_{\text{IKJ}} = 90 degrees.
  3. Use Exterior Angle Theorem: Use the given measure of the arc and the Exterior Angle Theorem.\newlinemJK◯=102∘mJK^{\bigcirc} = 102^\circ, which is the measure of the arc formed by points J, K, and L. The exterior angle, m∠KJLm\angle KJL, is given as 54∘54^\circ.
  4. Apply Theorem: Apply the Exterior Angle Theorem to find m∠IJLm\angle IJL. \newline$m\angle IJL = m\overset{\frown}{JK} - m\angle KJL = \(102\)^\circ - \(54\)^\circ = \(48\)^\circ.

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