Given that the measure of angle
𝐶𝐴𝐵 is 31 degrees, find 𝑦 and 𝑥.
The angle 𝐶𝐴𝐵 is the angle
formed when we move from 𝐶 to 𝐴 to 𝐵. So that’s the third angle in this
triangle. We notice the 𝑥 degrees and 𝑦
degrees are the measures of the two other angles in this triangle. Notice also that the line 𝐴𝐵 is a
diameter of this circle as it passes through the center 𝑀.
This means that the line 𝐴𝐵
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 𝐴𝐶𝐵, 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 𝑥, 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 𝑥 plus 𝑦
plus 31 equals 180 for the three angles in the triangle 𝐴𝐵𝐶.
Remember 𝑦 we have found to be
90. So we have 𝑥 plus 90 plus 31
equals 180. And we can solve this equation for
𝑥. 90 plus 31 is 121. So we have 𝑥 plus 121 equals
180. To find the value of 𝑥, we need to
subtract 121 from each side, giving 𝑥 equals 59. Again, there is no degree symbol
with this value as angle 𝐴𝐵𝐶 was 𝑥 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 𝑥 are 𝑦
equals 90; 𝑥 equals 59.