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Radii to Tangents - Problem 2

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Two radii with two lines connecting each end form a quadrilateral, in which the measure of a central angle is given. Recall that a radius to a point of tangency forms a right angle. As a result, the quadrilateral formed has a pair of parallel lines, making this quadrilateral a trapezoid. Therefore, the other two angles of the trapezoid are supplementary, meaning that their sum is 180°. As a result, it is possible to find the value of the missing angle by subtracting the measure of the given angle from 180°.

In this problem we have two circles that are sharing a point of tangency and we have this blue line here which is tangent to both of the circles and I’m asking you to find this angle X. You are also given that this angle right here is 80 degrees.

Well, if I look at this, we have a quadrilateral where we only know two angles. So if I go back over here, I’m going to remind myself that whenever I have a radius to a point of tangency, it forms a right angle. So I’m going to go back to my problem here and I’m going to see that I have two right angles. I have a right angle right there and I have another right angle right there.

Now, an interesting point to make right here is that since these are both right angles, we can assume that these two lines are parallel. So if you’re ever asked what type of quadrilateral is formed here, you can assume that it is a trapezoid, since you have one pair of parallel sides.

Getting back to the problem, if we want to find X, we know that the sum of these four angles is 360 degrees. However two of those angles are 90 degrees. So I know that X and 80 must sum to 180 degrees. So X must be 100 degrees, which I found by subtracting 80 degrees on both sides of this equation. So the key to this problem is realizing that since we have a trapezoid, we have two radii to a point of tangency which create right angles.

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