Video: Finding the Measure of an Angle Using the Properties of Tangents to the Circle

Given that line segment 𝐴𝐡 and line segment 𝐴𝐢 are two tangents to the circle 𝑀, and π‘šβˆ π‘€π΄πΆ = 21Β°, determine π‘šβˆ πΆπ΄π΅.

01:55

Video Transcript

Given that line segment 𝐴𝐡 and line segment 𝐴𝐢 are two tangents to the circle 𝑀 and the measure of the angles 𝑀𝐴𝐢 is equal to 21 degrees, determine the measure of angle 𝐢𝐴𝐡.

Let’s simply begin by looking at what we do know about our diagram. We have a circle with two tangents 𝐴𝐡 and 𝐴𝐢. We can see that the point 𝑀 must be the center of the circle. And that means the lines 𝐡𝑀 and 𝐢𝑀 must be the radii of the circle. So what does it mean for triangles 𝐴𝐡𝑀 and 𝐴𝐢𝑀?

Well, we know by definition that the radii must be of equal length. So the length of the line 𝐡𝑀 must be equal to the length of the line 𝐢𝑀. We also know that a tangent and a radius meet at 90 degrees. So the measure of angle 𝐴𝐡𝑀 must be equal to the measure of angle 𝐴𝐢𝑀, which is equal to 90 degrees. We know that tangents from the same point are equal in length. So lines 𝐴𝐡 and 𝐴𝐢 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 𝑀𝐴𝐡 must be equal to angle 𝑀𝐴𝐢, which we were told was 21 degrees. So the measure of angle 𝐢𝐴𝐡 must be equal to 21 plus 21, which is equal to 42. And the measure of angle 𝐢𝐴𝐡 is 42 degrees.

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