Question Video: Finding the Measure of an Angle given Its Arc’s Measure Using Its Central Angle and Another Inscribed Angle | Nagwa Question Video: Finding the Measure of an Angle given Its Arc’s Measure Using Its Central Angle and Another Inscribed Angle | Nagwa

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

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Given that line segment π΄π΅ is a diameter in the circle π, and πβ π΅ππ· = 59Β°β, find the πβ π΄πΆπ· in degrees.

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### Video Transcript

Given that line segment π΄π΅ is a diameter in circle π and the measure of angle π΅ππ· equals 59 degrees, find the measure of angle π΄πΆπ· in degrees.

Letβs put what we know into the diagram. Angle π΅ππ· measures 59 degrees, and weβre trying to find the measure of angle π΄πΆπ·. If we start with what we know about angle π΅ππ·, since π΅ππ· has a vertex at the center of the circle, π΅ππ· is a central angle. And because angle π΅ππ· is a central angle, its subtended arc, arc π΅π·, also measures 59 degrees. Weβre also interested in angle π΄πΆπ·. But angle π΄πΆπ· is not a central angle. Itβs an inscribed angle because its vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle π΄πΆπ· would be arc π΄π·. We have a partial measurement for this arc, but weβre missing the distance from π΄ to π΅. But because we know that π΄π΅ is a diameter, it cuts the circle in half. And that means the measure of arc π΄π΅ is 180 degrees. If arc π΄π΅ equals 180 and arc π΅π· equals 59 degrees, we can say the measure of arc π΄π· is equal to the measure of arc π΄π΅ plus the measure of arc π΅π·.

If we plug in what we know, the measure of arc π΄π· is 239 degrees. Because angle π΄πΆπ· is an inscribed angle and it has a subtended arc measure of 239 degrees, we can find out the exact measure of angle π΄πΆπ·. The measure of the inscribed angle π΄πΆπ· will be one-half its subtended arc, arc π΄π·. Since that arc is 239 degrees, we take half of that and we get 119.5 degrees for the measure of angle π΄πΆπ·.

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