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

Given that line segment π΄π΅ and line segment π΄πΆ are two tangents to the circle π, and πβ ππ΄πΆ = 21Β°, determine πβ πΆπ΄π΅.

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### Video Transcript

Given that line segment π΄π΅ and line segment π΄πΆ are two tangents to the circle π and the measure of the angles ππ΄πΆ is equal to 21 degrees, determine the measure of angle πΆπ΄π΅.

Letβs simply begin by looking at what we do know about our diagram. We have a circle with two tangents π΄π΅ and π΄πΆ. We can see that the point π must be the center of the circle. And that means the lines π΅π and πΆπ must be the radii of the circle. So what does it mean for triangles π΄π΅π and π΄πΆπ?

Well, we know by definition that the radii must be of equal length. So the length of the line π΅π must be equal to the length of the line πΆπ. We also know that a tangent and a radius meet at 90 degrees. So the measure of angle π΄π΅π must be equal to the measure of angle π΄πΆπ, which is equal to 90 degrees. We know that tangents from the same point are equal in length. So lines π΄π΅ and π΄πΆ must be equal. But thereβs also a shared line here. The line π΄π 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 doesnβt really matter which condition we choose though. Whatβ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 ππ΄π΅ must be equal to angle ππ΄πΆ, which we were told was 21 degrees. So the measure of angle πΆπ΄π΅ must be equal to 21 plus 21, which is equal to 42. And the measure of angle πΆπ΄π΅ is 42 degrees.