Finding the Measure of an Angle Outside a Circle

By: Shelly Kaufmann

Using a circle and a point placed outside the circle, discover how to find the measure of an angle outside the circle. What else did you discover?

Construct a circle using GSP.  Place a point A outside the circle. Construct two lines that pass through the circle and intersect at A. Label the points where the lines intersect the circle.

Next, use one of your two lines and Point A, add one line segment so that you form a triangle.

Locate the two inscribed angles inside the circle. We know that the angle arc of an inscribed angle is twice the measure of the inscribed angle. Measure one inscribed angle DCE. Test that the angle arc is twice the measure. To do this, you must add a third point on the circle between D and E to measure. Your measures will most likely not match mine exactly!

Next, measure the second inscribed angle CEB. Measure the arc angle. Check to make certain your inscribed angle is half the measure of your arc angle! Again, your measures will not necessarily match mine!

Label the angle in your first inscribed angle as x and your second inscribed angle as y.

Mathematically, we know that one half of the angle arc of X minus the angle arc of Y should give us the measure of angle A.

Angle A = 1/2 (57.63 - 11.65)  = 1/2 (45.98) = 22.99

Is the sum of the angles in the triangle 180º ?

Can you see another way to look at the problem?

Try looking at line segment AD as a straight angle. If you know the measure of X, you can subtract that measure from the straight angle measure of 180°.

180 – 28.81 = 151.19

Add 151.19 to Y which we know is 5.83. The sum of these two angles is 157.02.

180° - 157.02  = 22.98

Wow! Isn’t this great!