Question Video: Finding the Measure of an Angle given Its Arcβs Measure Using Another Inscribed Angle Mathematics

In the figure, line segment π΅πΆ is a diameter of the circle. πβ π΄πΆπ΅ = πβ πΆπ·πΈ, πβ  arc π΅π· = 42Β°. And πΈ is the midpoint of arc πΆπ·. Find πβ  arc π΄πΆ.

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Video Transcript

In the following figure, line segment π΅πΆ is a diameter of the circle. The measure of angle a π΄πΆπ΅ is equal to the measure of angle πΆπ·πΈ. The measure of arc π΅π· is 42 degrees. And πΈ is the midpoint of arc πΆπ·. Find the measure of arc π΄πΆ.

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 π΄πΆ and π΅πΆ intersect at point πΆ. This means that angle π΄πΆπ΅ is subtended by the arc π΄π΅. And we conclude that the measure of the angle is equal to the measure of the arc.

We are told that the Angles, π΄πΆπ΅ and πΆπ·πΈ are equal. This means that the arcs π΄π΅ and πΆπΈ are also equal. We are told that the measure of arc π΅π· is 42 degrees. Since πΈ is the midpoint of Arc πΆπ·, then the triangle πΆπ·πΈ is isosceles, where πΆπΈ is equal to π·πΈ. We know that two angles in an isosceles triangle are equal. And this also means that the arc π·πΈ is equal to the arc πΆπΈ and the arc π΄π΅.

Our aim in this question is to find the measure of arc π΄πΆ. Since π΅πΆ 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 π΅π·, π·πΈ, and πΆπΈ 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 π·πΈ, πΆπΈ, and π΄π΅ 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 π΄πΆ. Adding three lots of 69 to 42 gives us 249. The measure of our arc aπΆ plus 249 degrees is equal to 360 degrees. We can then subtract 249 degrees from both sides such that the measure of arc π΄πΆ is equal to 111 degrees.