Question Video: Using the Right Angle-Hypotenuse-Side Triangle Congruence Criterion to Establish Congruence Mathematics

Video Transcript

In the given figure, points 𝐿 and 𝑁 are on a circle with center 𝑂. Which congruence criterion can be used to prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent?

In this problem, we need to determine how we can prove that these two triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent, which means that corresponding sides are congruent and corresponding angles are congruent. We might notice that both of these triangles have a right angle. And there is a congruency criterion which applies in right triangles. It is the RHS criterion, which tells us that two triangles are congruent if they have a right angle and the hypotenuse and one other side are equal.

Let’s see if we can apply this criterion here. And even though we aren’t given any length measurements, we can apply our knowledge of geometry to help. Because we’re given that 𝐿 and 𝑁 are on the circle and the center is 𝑂, then the line segments 𝑂𝐿 and 𝑂𝑁 are both radii of the circle. And importantly, that means that these are congruent. We also have this line segment 𝑂𝑀, which is a shared side between the two triangles. So this length will be equal in both triangles. Very helpfully, we can also recognize that the line segment 𝑂𝑀 is the hypotenuse in both of these triangles.

Now, if we look at the RHS criterion, we know that both triangles do have a right angle. We also know that the hypotenuse is congruent because this is a common side. And we have another pair of sides which are congruent. Therefore, it is by applying the RHS congruence criterion that we can prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent.