In the given figure, line 𝐴𝐵 is a tangent to a circle with center 𝑀 at 𝐵, line segment 𝐶𝐷 is a diameter, the measure of angle 𝐵𝐴𝑀 equals 𝑥, and the measure of angle 𝑀𝐷𝐵 equals two 𝑥 minus 55. Find the value of 𝑥 in degrees.
We are told in the question that the line 𝐴𝐵 is a tangent to the circle at point 𝐵. We know that a tangent meets the radius at the point of contact at right angles. The measure of angle 𝐴𝐵𝑀 is therefore equal to 90 degrees. As all radii are equal in length, we know that the line segments 𝑀𝐷 and 𝑀𝐵 have the same length. This means that triangle 𝑀𝐷𝐵 is isosceles. This in turn means that the measures of angles 𝑀𝐷𝐵 and 𝑀𝐵𝐷 are equal. And they are equal to the expression two 𝑥 minus 55. We are also told in the question that the measure of angle 𝐵𝐴𝑀 is equal to 𝑥.
We will now consider triangle 𝐴𝐵𝐷 and the fact that angles in a triangle sum to 180 degrees. This means that the measures of the angles 𝐴𝐷𝐵, 𝐴𝐵𝐷, and 𝐵𝐴𝐷 must equal 180 degrees. Angle 𝐴𝐷𝐵 is equal to two 𝑥 minus 55, angle 𝐴𝐵𝐷 is equal to two 𝑥 minus 55 plus 90, and angle 𝐵𝐴𝐷 is equal to 𝑥.
This gives us the following equation. Two 𝑥 minus 55 plus two 𝑥 minus 55 plus 90 plus 𝑥 is equal to 180. Collecting like terms on the left-hand side gives us five 𝑥 minus 20, and this is equal to 180. We can then add 20 to both sides. Finally, dividing through by five gives us 𝑥 is equal to 40. The value of 𝑥 in degrees is 40 degrees.