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Question Video: Proving the Similarity of Two Triangles Mathematics • 8th Grade

Given that line segment ๐ด๐ต and line segment ๐ท๐ถ are parallel, are triangles ๐ธ๐ถ๐ท and ๐ธ๐ด๐ต similar? If yes, why?


Video Transcript

Given that line segment ๐ด๐ต and line segment ๐ท๐ถ are parallel, are triangles ๐ธ๐ถ๐ท and ๐ธ๐ด๐ต similar? If yes, why?

Letโ€™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 weโ€™re asked if triangle ๐ธ๐ถ๐ท, thatโ€™s this smaller triangle, is similar to triangle ๐ธ๐ด๐ต, thatโ€™s the larger triangle that ๐ธ๐ถ๐ท is part of.

Notice that weโ€™re given this really important piece of information that the line segment ๐ด๐ต and the line segment ๐ท๐ถ are parallel. So letโ€™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 arenโ€™t given any information about the lengths of the sides, we could potentially use this grid paper to help us. However, letโ€™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 ๐ธ appears in both triangles. If we wanted to write that formally, we could say that angle ๐ท๐ธ๐ถ is equal to angle ๐ต๐ธ๐ด. And thatโ€™s because that angle is common to both triangles.

Next, letโ€™s have a look at this angle ๐ธ๐ถ๐ท. Using the fact that these two line segments, ๐ด๐ต and ๐ถ๐ท, are parallel, then we can say that this angle at ๐ธ๐ด๐ต is equal to angle ๐ธ๐ถ๐ท. Thatโ€™s because these two angles will be corresponding. The parallel lines are important because if we just had this triangle ๐ธ๐ด๐ต and then the line segments ๐ด๐ต and ๐ถ๐ท were not parallel, then we couldnโ€™t say that angle ๐ธ๐ถ๐ท was equal to angle ๐ธ๐ด๐ต. Thatโ€™s only true because we have a set of parallel lines.

Now weโ€™ve proved that we have two pairs of corresponding angles congruent. Therefore, weโ€™ve demonstrated that the AA or angle-angle similarity rule applies. And weโ€™ve shown that these two triangles, ๐ธ๐ถ๐ท and ๐ธ๐ด๐ต, are similar. We could also prove that angle ๐ถ๐ท๐ธ is equal to angle ๐ด๐ต๐ธ. 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, โ€œYes, as all the corresponding angles in each triangle have equal measures.โ€ Any statement which relates our workings to our reasoning about the similarity would be valid. For example, even saying, โ€œYes, as there are two pairs of corresponding angles in the triangles of equal measureโ€ would also be correct.

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