Question Video: Finding Unknown Angles given Equal Chords in Congruent Circles Mathematics

Video Transcript

Consider that circles 𝑀 and 𝑁 are congruent and 𝐴𝐵 equals 𝐶𝐷. Find the value of 𝑥. Find the value of 𝑦.

Let’s start by seeing if we can work out the value of 𝑥, which appears in the circle 𝑀. The line segments 𝑀𝐶 and 𝑀𝐷 will both be radii of the circle. And that means that they will be the same length. This means that the triangle 𝑀𝐶𝐷 is an isosceles triangle. An isosceles triangle has two sides of equal measure and two angles of equal measure. And so the measure of angle 𝑀𝐷𝐶 will equal the measure of angle 𝑀𝐶𝐷. They will both be 25 degrees.

We can then use the fact that the interior angles in a triangle add up to 180 degrees to help us work out the value of 𝑥. So the three angles in the triangle, 25 degrees, 25 degrees, and 𝑥 degrees, will add to give 180 degrees. We can then simplify and subtract 50 degrees from both sides of the equation, which gives us that 𝑥 degrees is equal to 130 degrees. And so we’ve answered the first part of the question. 𝑥 is equal to 130.

Now, let’s see how we calculate the value of 𝑦 which appears in this circle 𝑁. The fact that these two circles are congruent is really very important. Because the circles are congruent, then the radii are also congruent. Therefore, we can say that the line segments 𝐴𝑁 and 𝐵𝑁 are congruent, but these are also congruent to the line segments 𝐶𝑀 and 𝐷𝑀. We are also told that 𝐴𝐵 is equal to 𝐶𝐷. Now we know that there are two equal chords in the circle which form part of these two triangles. Comparing these two triangles then, we have three pairs of corresponding sides congruent. And so we can say that triangle 𝑀𝐶𝐷 is congruent to triangle 𝑁𝐵𝐴 by the SSS or side-side-side congruency criterion.

Notice that because these are isosceles triangles, we could also have written that triangle 𝑀𝐶𝐷 is congruent to triangle 𝑁𝐴𝐵. And because we want to work out this angle measure of 𝑁𝐵𝐴, which is 𝑦 degrees, we can say that that’s equal to the measure of angle 𝑀𝐶𝐷. And so 𝑦 degrees is equal to 25 degrees. And hence 𝑦 is equal to 25.

In this question, we proved that two triangles are congruent. These triangles were each formed by chords which are congruent and two radii. But in fact, this is a standard property. It is worthwhile remembering the fact that if chords of the same length connect radii in the same circle or in congruent circles, then the two isosceles triangles formed are congruent. In this problem, we were given that two chords were the same length which connected radii in congruent circles.