Question Video: Finding the Measure of the Inscribed Angle between Two Secants given the Measure of the Two Inscribed Arcs

Video Transcript

Find the value of 𝑥.

In the given figure, line segment 𝐶𝐸 and line segment 𝐴𝐸 intersect at point 𝐸. The angle created at their intersection is the one labeled 𝑥. And as these lines have intersected the circle, they have created two intercepted arcs. If we sketch the center of the circle, we can show arc 𝐷𝐵 that measures 71 degrees and arc 𝐶𝐴 that measures 144 degrees.

Since the line segment 𝐶𝐸 and the line segment 𝐴𝐸 can both be described as secants of the circle, we can use the angles of intersecting secants theorem. Which tells us the angle created by the intersection of two secants, in this case, the measure of angle 𝐶𝐸𝐴, will be equal to one-half the difference of the two intercepted arcs. And that means we can say that 𝑥 is equal to 144 minus 71 divided by two. 𝑥 is equal to 36.5 degrees.