Question

Use the above figure to answer the question. What's the angle between the tangents at T in the figure?

Answer 1

I believe that the answer is B) 60°.

Because the angle inside the circle equals 120°, and since these are 30°, 60°, 90° triangles, angles A and B are both 90°.

Which means that the remaining angles outside the circle each equal 30°.

Therefore, when both of these angles are added together, ∠T = 60°

Answer 2

Answer: 60 degrees (choice B)

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Explanation:

Triangle OTA is a 30-60-90 right triangle. This means that the three angles are 30 degrees, 60 degrees, 90 degrees for angles T, O and A in that order. We know that angle A is 90 degrees because the tangent segment AT is perpendicular to the radius at the point of tangency. This leaves angle T to be 30 degrees. Note how the three angles (30,60,90) add up to 180 degrees. Put another way: angle T and angle O are complementary angles, so they must add to 90 degrees.

Triangle OTB is a mirror reflection of triangle OTA. We can prove this using the hyponteuse leg (HL) congruence theorem. Specifically, OA = OB is the congruent pair of legs and OT = OT is the congruent pair of hypotenuses.

Because of this triangle congruence, all of the corresponding angles are congruent as well.

Check out the attached image below.

At this point, we simply add angle OTA and angle OTB to get the answer we want: 30+30 = 60. The angle ATB is the angle between the two tangents segment AT and segment BT.