Question Video: Finding the Measure of an Angle Given Its Arc’s Measure Using an Inscribed Angle | Nagwa Question Video: Finding the Measure of an Angle Given Its Arc’s Measure Using an Inscribed Angle | Nagwa

# Question Video: Finding the Measure of an Angle Given Its Arcβs Measure Using an Inscribed Angle Mathematics • Third Year of Preparatory School

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Suppose that the line ππ is a tangent to the circle with center π at π΅, line segment π΅π΄ β₯ line segment ππΆ, and πβ π΄π΅π = 51.8Β°. Find πβ πΆπ΅π.

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### Video Transcript

Suppose that the line ππ is a tangent to the circle with center π at π΅, line segment π΅π΄ is parallel to line segment ππΆ, and the measure of angle π΄π΅π is 51.8 degrees. Find the measure of angle πΆπ΅π.

Letβs begin by marking the angle weβve been given and the angle we wish to calculate on the diagram. Angle π΄π΅π is 51.8 degrees, and weβre looking for the measure of angle πΆπ΅π. We can now see that the angle whose measure weβve been asked to calculate is an angle of tangency, as it is the angle between the tangent ππ and the chord π΅πΆ. We can therefore recall a key theorem, which states that the measure of an angle of tangency is equal to half the measure of the central angle subtended by the same arc. The arc connecting the endpoints of the chord π΅πΆ is the minor arc π΅πΆ, and the central angle subtended by this arc is angle π΅ππΆ. So we have that the measure of angle πΆπ΅π is half the measure of angle π΅ππΆ.

Now we need to consider how to find the measure of this angle. So letβs look at the other information given in the figure. We know that the measure of angle π΄π΅π is 51.8 degrees, and weβre told that the line segments π΅π΄ and ππΆ are parallel. Angles π΅ππΆ and ππ΅π΄ are therefore alternate angles in parallel lines and so are of equal measure.

The final detail to observe is that the angle ππ΅π is the angle formed between the radius ππ΅ and the tangent ππ at the point of tangency. We can then recall a second key theorem, which is that a tangent to a circle is perpendicular to the radius at the point of contact. This means that the measure of angle ππ΅π is 90 degrees. We can therefore calculate the measure of angle ππ΅π΄ by subtracting 51.8 degrees from 90 degrees, giving 38.2 degrees. The measure of angle π΅ππΆ is also 38.2 degrees, due to alternate angles in parallel lines being equal.

Weβre finally able to calculate the measure of angle πΆπ΅π. Itβs one-half of the measure of angle π΅ππΆ, so itβs one-half of 38.2 degrees. Thatβs 19.1 degrees. So, by identifying that angle πΆπ΅π is an angle of tangency and then recalling that the measure of an angle of tangency is half the measure of the central angle subtended by the same arc, along with using other key angle properties, weβve found that the measure of angle πΆπ΅π is 19.1 degrees.

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