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

Given that ๐‘šโˆ ๐ถ๐ด๐ต = 31ยฐ, find ๐‘ฆ and ๐‘ฅ.

02:03

Video Transcript

Given that the measure of angle ๐ถ๐ด๐ต is 31 degrees, find ๐‘ฆ and ๐‘ฅ.

The angle ๐ถ๐ด๐ต is the angle formed when we move from ๐ถ to ๐ด to ๐ต. So thatโ€™s the third angle in this triangle. We notice the ๐‘ฅ degrees and ๐‘ฆ degrees are the measures of the two other angles in this triangle. Notice also that the line ๐ด๐ต is a diameter of this circle as it passes through the center ๐‘€.

This means that the line ๐ด๐ต 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 ๐ด๐ถ๐ต, which is given as ๐‘ฆ degrees. So we have that ๐‘ฆ degrees is equal to 90 degrees. And therefore the value of ๐‘ฆ without the degree symbol is 90.

To find the value of ๐‘ฅ, 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 ๐‘ฅ plus ๐‘ฆ plus 31 equals 180 for the three angles in the triangle ๐ด๐ต๐ถ.

Remember ๐‘ฆ we have found to be 90. So we have ๐‘ฅ plus 90 plus 31 equals 180. And we can solve this equation for ๐‘ฅ. 90 plus 31 is 121. So we have ๐‘ฅ plus 121 equals 180. To find the value of ๐‘ฅ, we need to subtract 121 from each side, giving ๐‘ฅ equals 59. Again, there is no degree symbol with this value as angle ๐ด๐ต๐ถ was ๐‘ฅ degrees, so 59 degrees.

If we already had a degree symbol, it would be 59 degrees degrees which wouldnโ€™t make sense. So the values of ๐‘ฆ and ๐‘ฅ are ๐‘ฆ equals 90; ๐‘ฅ equals 59.

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