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

Given that πβ π΄π΅πΆ = 27Β°, and πβ π΅π·π΄ = 62Β°, find πβ π΅π·πΆ and πβ π΅πΆπ·.

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### Video Transcript

Given that the measure of angle π΄π΅πΆ equals 27 degrees and the measure of angle π΅π·π΄ equals 62 degrees, find the measure of angle π΅π·πΆ and the measure of angle π΅πΆπ·.

The first angle measure that weβre given is the measure of angle π΄π΅πΆ, which is 27 degrees. The second angle measure is that of angle π΅π·π΄, which weβre told is 62 degrees. The first angle weβre asked to calculate the measure of is that of angle π΅π·πΆ, which is the whole angle at the top of this figure. The second angle we need to calculate is that of angle π΅πΆπ·, which is on the right of the figure.

The first thing we might observe is that we have this bow tie shape within the circle, which may indicate that weβll be using inscribed angles theorems. One of these theorems tells us that inscribed angles subtended by the same arc are equal. This angle of πΆπ΅π΄ is inscribed by the arc πΆπ΄. And so it will be equal in measure to any other angle inscribed by the same arc. And so the measure of angle πΆπ·A will be the same size. It will be 27 degrees. The first angle we needed to calculate was the measure of angle π΅π·πΆ. And so it will be the sum of these two angles, 62 degrees plus 27 degrees, which gives us an answer of 89 degrees. And so thatβs the first angle measure found.

Now, letβs see how we work towards finding the other angle measure. Because we know that inscribed angles subtended by the same arc are equal, then the measure of angle π·π΄π΅ will be equal to the measure of angle π·πΆπ΅. However, we donβt know the measure of this angle either. Letβs see if we can calculate either of these angles. To do this, letβs observe that we have a pair of congruent line segments. The line segment π΄π· is equal to the line segment π΄π΅. And so when we consider these line segments as part of the triangle π΄π΅π·, then we know that π΄π΅π· will be an isosceles triangle.

Isosceles triangles have two equal sides and two equal angle measures. And so the two angle measures that are equal will be the measure of angle π΄π·π΅ and the measure of angle π΄π΅π·. These will both be 62 degrees. We can now use the triangle π΄π΅π· and the fact that the interior angles in a triangle sum to 180 degrees to find the measure of angle π·π΄π΅. We can write the equation that the three angles in the triangle, thatβs 62 degrees, 62 degrees, and the measure of angle π·π΄π΅, must sum to 180 degrees. 62 degrees plus 62 degrees simplifies to 124 degrees. Subtracting 124 degrees from both sides gives us that the measure of angle π·π΄π΅ is 56 degrees.

Now, we know that this angle subtended by the arc π΅π· will be equal to the measure of angle π΅πΆπ·, which is subtended by the same arc. And so we can give the answer for both angle measures. The measure of angle π΅π·πΆ is 89 degrees, and the measure of angle π΅πΆπ· is 56 degrees.