Question Video: Finding the Measure of an Arc Using the Relationship Between a Parallel Chord and Tangent Mathematics • Third Year of Preparatory School

Video Transcript

𝑚 is a circle, where line segment 𝐴𝐵 is a chord and line 𝐶𝐷 is a tangent. If 𝐴𝐵 is parallel to 𝐶𝐷 and the measure of arc 𝐴𝐵 is 72 degrees, find the measure of arc 𝐵𝐶.

Since 𝐴𝐵 is parallel to 𝐶𝐷, where 𝐴𝐵 is a chord and 𝐶𝐷 is a tangent, there’s a theorem we can use. We’re going to use the theorem that says that the measure of the arcs between a parallel chord and tangent of a circle are equal. So we can say that the measure of arc 𝐴𝐶 must be equal to the measure of arc 𝐵𝐶. In fact, the question tells us that the measure of arc 𝐴𝐵 is 72 degrees. And we can use the fact that the sum of all the measures of all arcs which make up the circle is 360 degrees. This means that the measure of arc 𝐴𝐶 plus the measure of arc 𝐴𝐵, which is 72 degrees, plus the measure of arc 𝐵𝐶 is 360. Then, we subtract 72 degrees from both sides. And we find that the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐶 is 288 degrees.

But earlier, we stated that the measure of arc 𝐴𝐶 is equal to the measure of arc 𝐵𝐶. So we can say that two times the measure of arc 𝐵𝐶 is 288 degrees. And then we could divide both sides of this equation by two. So the measure of arc 𝐵𝐶 is 288 divided by two, which is in fact 144 degrees. The measure of arc 𝐵𝐶 is 144 degrees.