Question Video: Finding the Sum of Two Inscribed Arcs between Two Intersecting Chords given the Angle between the Chords Mathematics

Video Transcript

In the given figure, find the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷.

Let’s look carefully at the diagram we’ve been given. We have two chords, 𝐴𝐵 and 𝐶𝐷, which intersect inside a circle. We’re given the measure of one of the angles formed at the point of intersection and asked to find the sum of the measures of the arcs 𝐴𝐶 and 𝐵𝐷, which are the arcs now highlighted in orange and pink on our diagram.

To answer this question, we need to recall a theorem about the measures of angles formed by two chords that intersect inside a circle. This tells us that the measure of an angle between two chords is one-half the sum of the measures of its intercepted arc and its opposite arc. In this figure, the arc intercepted by the angle of 112 degrees is the arc 𝐵𝐷. And its opposite arc or the arc intercepted by its opposite angle is the arc 𝐴𝐶.

We can therefore form an equation. 112 degrees is equal to one-half the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷. And that’s great, because it is the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷 that we’re looking to find. We can work this out by multiplying both sides of the equation by two. This gives 224 degrees is equal to the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷.

So by recalling the angles of intersecting chords theorem, we found that the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷 is 224 degrees.