Given that line segment 𝐴𝐵 and line segment 𝐴𝐶 are two tangents to the circle 𝑀 and the measure of the angles 𝑀𝐴𝐶 is equal to 21 degrees, determine the measure of angle 𝐶𝐴𝐵.
Let’s simply begin by looking at what we do know about our diagram. We have a circle with two tangents 𝐴𝐵 and 𝐴𝐶. We can see that the point 𝑀 must be the center of the circle. And that means the lines 𝐵𝑀 and 𝐶𝑀 must be the radii of the circle. So what does it mean for triangles 𝐴𝐵𝑀 and 𝐴𝐶𝑀?
Well, we know by definition that the radii must be of equal length. So the length of the line 𝐵𝑀 must be equal to the length of the line 𝐶𝑀. We also know that a tangent and a radius meet at 90 degrees. So the measure of angle 𝐴𝐵𝑀 must be equal to the measure of angle 𝐴𝐶𝑀, which is equal to 90 degrees. We know that tangents from the same point are equal in length. So lines 𝐴𝐵 and 𝐴𝐶 must be equal. But there’s also a shared line here. The line 𝐴𝑀 is common to both triangles.
So now, we have two triangles which both have a right angle. They have a hypotenuse which is shared and they have two other equal sides. So we can pick and choose which condition for congruency we wish to quote. We could choose SSS. All three sides are equal in the triangles. So the two triangles must be congruent. We could choose RHS. The triangles have a right angle, an equal hypotenuse, and one other side of equal length. It doesn’t really matter which condition we choose though. What’s important is that we have identified that the triangles are congruent.
This means we can now assume the other angles must be equal. And therefore, angle 𝑀𝐴𝐵 must be equal to angle 𝑀𝐴𝐶, which we were told was 21 degrees. So the measure of angle 𝐶𝐴𝐵 must be equal to 21 plus 21, which is equal to 42. And the measure of angle 𝐶𝐴𝐵 is 42 degrees.