Question Video: Finding the Measures of Two Inscribed Angles Where One Is Inscribed in a Semicircle Mathematics • 11th Grade

Given that πβ πΆπ΄π΅ = 31Β°, find π¦ and π₯.

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Video Transcript

Given that the measure of angle πΆπ΄π΅ is 31 degrees, find π¦ and π₯.

The angle πΆπ΄π΅ is the angle formed when we move from πΆ to π΄ to π΅. So thatβs the third angle in this triangle. We notice the π₯ degrees and π¦ degrees are the measures of the two other angles in this triangle. Notice also that the line π΄π΅ is a diameter of this circle as it passes through the center π.

This means that the line π΄π΅ divides the circle into two semicircles. So we can apply one of our circle theorems, which is that the angle inscribed in a semicircle is 90 degrees, a right angle. The inscribed angle is the angle at the circumference, angle π΄πΆπ΅, which is given as π¦ degrees. So we have that π¦ degrees is equal to 90 degrees. And therefore the value of π¦ without the degree symbol is 90.

To find the value of π₯, we need to apply one of our more basic angle facts, which is that the angle sum in a triangle is 180 degrees. This gives the equation π₯ plus π¦ plus 31 equals 180 for the three angles in the triangle π΄π΅πΆ.

Remember π¦ we have found to be 90. So we have π₯ plus 90 plus 31 equals 180. And we can solve this equation for π₯. 90 plus 31 is 121. So we have π₯ plus 121 equals 180. To find the value of π₯, we need to subtract 121 from each side, giving π₯ equals 59. Again, there is no degree symbol with this value as angle π΄π΅πΆ was π₯ degrees, so 59 degrees.

If we already had a degree symbol, it would be 59 degrees degrees which wouldnβt make sense. So the values of π¦ and π₯ are π¦ equals 90; π₯ equals 59.