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Question Video: Finding the Measure of an Angle given Its Arc’s Measure Using Another Inscribed Angle Mathematics

In the figure, line segment 𝐡𝐢 is a diameter of the circle. π‘šβˆ π΄πΆπ΅ = π‘šβˆ πΆπ·πΈ, π‘šβˆ  arc 𝐡𝐷 = 42Β°. And 𝐸 is the midpoint of arc 𝐢𝐷. Find π‘šβˆ  arc 𝐴𝐢.

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Video Transcript

In the following figure, line segment 𝐡𝐢 is a diameter of the circle. The measure of angle a 𝐴𝐢𝐡 is equal to the measure of angle 𝐢𝐷𝐸. The measure of arc 𝐡𝐷 is 42 degrees. And 𝐸 is the midpoint of arc 𝐢𝐷. Find the measure of arc 𝐴𝐢.

There is a lot of information here. And before starting, it is worth recalling what we mean by an inscribed angle. An inscribed angle is formed by the intersection of two chords on the circumference of a circle. For example, the chords 𝐴𝐢 and 𝐡𝐢 intersect at point 𝐢. This means that angle 𝐴𝐢𝐡 is subtended by the arc 𝐴𝐡. And we conclude that the measure of the angle is equal to the measure of the arc.

We are told that the Angles, 𝐴𝐢𝐡 and 𝐢𝐷𝐸 are equal. This means that the arcs 𝐴𝐡 and 𝐢𝐸 are also equal. We are told that the measure of arc 𝐡𝐷 is 42 degrees. Since 𝐸 is the midpoint of Arc 𝐢𝐷, then the triangle 𝐢𝐷𝐸 is isosceles, where 𝐢𝐸 is equal to 𝐷𝐸. We know that two angles in an isosceles triangle are equal. And this also means that the arc 𝐷𝐸 is equal to the arc 𝐢𝐸 and the arc 𝐴𝐡.

Our aim in this question is to find the measure of arc 𝐴𝐢. Since 𝐡𝐢 is a diameter, we know it splits the circle in half. Since the angles in a circle sum to 360 degrees, in a semicircle, they must sum to 180 degrees. The measure of arcs 𝐡𝐷, 𝐷𝐸, and 𝐢𝐸 must sum to 180 degrees. This gives us an equation 42 degrees plus π‘₯ plus π‘₯ is equal to 180 degrees. π‘₯ plus π‘₯ is equal to two π‘₯. And subtracting 42 degrees from both sides, this is equal to 138 degrees. We can then divide through by two such that π‘₯ is equal to 69 degrees. This means that the measure of the arcs 𝐷𝐸, 𝐢𝐸, and 𝐴𝐡 are all equal to 69 degrees.

Since all five arcs on the circumference of our circle sum to 360 degrees, we can use this to calculate the measure of arc 𝐴𝐢. Adding three lots of 69 to 42 gives us 249. The measure of our arc a𝐢 plus 249 degrees is equal to 360 degrees. We can then subtract 249 degrees from both sides such that the measure of arc 𝐴𝐢 is equal to 111 degrees.

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