Video: Finding the Measures of Two Inscribed Angles Where One Is Inscribed in a Semicircle

Given that π‘šβˆ πΆπ΄π΅ = 31Β°, find 𝑦 and π‘₯.

02:03

Video Transcript

Given that the measure of angle 𝐢𝐴𝐡 is 31 degrees, find 𝑦 and π‘₯.

The angle 𝐢𝐴𝐡 is the angle formed when we move from 𝐢 to 𝐴 to 𝐡. So that’s the third angle in this triangle. We notice the π‘₯ degrees and 𝑦 degrees are the measures of the two other angles in this triangle. Notice also that the line 𝐴𝐡 is a diameter of this circle as it passes through the center 𝑀.

This means that the line 𝐴𝐡 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 𝐴𝐢𝐡, which is given as 𝑦 degrees. So we have that 𝑦 degrees is equal to 90 degrees. And therefore the value of 𝑦 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 𝑦 plus 31 equals 180 for the three angles in the triangle 𝐴𝐡𝐢.

Remember 𝑦 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 𝐴𝐡𝐢 was π‘₯ degrees, so 59 degrees.

If we already had a degree symbol, it would be 59 degrees degrees which wouldn’t make sense. So the values of 𝑦 and π‘₯ are 𝑦 equals 90; π‘₯ equals 59.

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