Given that line segment 𝐴𝐵 and line segment 𝐷𝐶 are parallel, are triangles 𝐸𝐶𝐷 and 𝐸𝐴𝐵 similar? If yes, why?
Let’s begin by recalling what it means to have similar triangles. Similar triangles, like any similar shapes, have corresponding angles congruent and corresponding sides in proportion. Here we’re asked if triangle 𝐸𝐶𝐷, that’s this smaller triangle, is similar to triangle 𝐸𝐴𝐵, that’s the larger triangle that 𝐸𝐶𝐷 is part of.
Notice that we’re given this really important piece of information that the line segment 𝐴𝐵 and the line segment 𝐷𝐶 are parallel. So let’s begin by marking that onto the diagram. There are a number of different ways in which we can prove that two triangles are similar, either by showing that sides are in proportion or that angles are congruent or a mixture of these.
Although we aren’t given any information about the lengths of the sides, we could potentially use this grid paper to help us. However, let’s begin by seeing if we can do anything with regards to the angles in these triangles. We can begin by noticing that this angle at 𝐸 appears in both triangles. If we wanted to write that formally, we could say that angle 𝐷𝐸𝐶 is equal to angle 𝐵𝐸𝐴. And that’s because that angle is common to both triangles.
Next, let’s have a look at this angle 𝐸𝐶𝐷. Using the fact that these two line segments, 𝐴𝐵 and 𝐶𝐷, are parallel, then we can say that this angle at 𝐸𝐴𝐵 is equal to angle 𝐸𝐶𝐷. That’s because these two angles will be corresponding. The parallel lines are important because if we just had this triangle 𝐸𝐴𝐵 and then the line segments 𝐴𝐵 and 𝐶𝐷 were not parallel, then we couldn’t say that angle 𝐸𝐶𝐷 was equal to angle 𝐸𝐴𝐵. That’s only true because we have a set of parallel lines.
Now we’ve proved that we have two pairs of corresponding angles congruent. Therefore, we’ve demonstrated that the AA or angle-angle similarity rule applies. And we’ve shown that these two triangles, 𝐸𝐶𝐷 and 𝐸𝐴𝐵, are similar. We could also prove that angle 𝐶𝐷𝐸 is equal to angle 𝐴𝐵𝐸. Once again, because we have parallel lines, then these two angles will be corresponding.
Showing any two pairs out of these three angle pairs would be sufficient to prove that these triangles were similar. We can therefore give the answer that yes, these two triangles are similar.
However, notice that we also need to give a statement to indicate why. So we could say something like, “Yes, as all the corresponding angles in each triangle have equal measures.” Any statement which relates our workings to our reasoning about the similarity would be valid. For example, even saying, “Yes, as there are two pairs of corresponding angles in the triangles of equal measure” would also be correct.