Question Video: Using the Properties of Congruence between Two Triangles to Solve a Given Problem Mathematics • First Year of Preparatory School

Video Transcript

If triangle 𝑅𝑆𝑇 is congruent to triangle 𝑀𝑁𝑂, what is line segment 𝑅𝑇 congruent to?

In this problem, we are given that there are two congruent triangles: triangle 𝑅𝑆𝑇 and triangle 𝑀𝑁𝑂. We can recall that congruent triangles, like any congruent polygons, have all pairs of corresponding angles congruent and all pairs of corresponding sides congruent. And even if we don’t know exactly what the triangles look like, we can use the given relationship to help us work out the congruent vertices and sides.

Since the letters represent the vertices, then by considering the first vertex in each triangle in the congruency relationship, we know that vertices 𝑅 and 𝑀 are corresponding. Then the second vertex 𝑆 in triangle 𝑅𝑆𝑇 corresponds to the second vertex 𝑁 in triangle 𝑀𝑁𝑂. And vertex 𝑇 is corresponding to vertex 𝑂.

And we could even sketch a diagram of what these triangles might look like. If we do this, we must make sure that the triangles are labeled in the same pattern. For example, triangle 𝑅𝑆𝑇 is labeled in a clockwise direction from the lower-left corner. And triangle 𝑀𝑁𝑂 is also labeled in a clockwise direction, starting from the lower-left corner.

The corresponding vertices can be shown like this. Now we want to consider the corresponding sides, and in particular, the line segment 𝑅𝑇, which joins vertices 𝑅 and 𝑇, which would be here on the diagram. Therefore, the corresponding line segment in triangle 𝑀𝑁𝑂 is that of the line segment 𝑀𝑂.

Now we could also work this out without drawing a diagram. If we return to the congruency relationship, we are considering the line segment joining vertices 𝑅 and 𝑇, which are the first and final letters in this congruency relationship for triangle 𝑅𝑆𝑇. The congruent line segment in triangle 𝑀𝑁𝑂 can be read in the same way. The first and final vertices are 𝑀 and 𝑂.

And so, we can give the answer that line segment 𝑀𝑂 is congruent to line segment 𝑅𝑇 when triangles 𝑅𝑆𝑇 and 𝑀𝑁𝑂 are congruent.