### Video Transcript

Given that line segment ๐ด๐ต and line segment ๐ท๐ถ are parallel, are triangles ๐ธ๐ถ๐ท and ๐ธ๐ด๐ต similar? If yes, why?

Letโs 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 weโre asked if triangle ๐ธ๐ถ๐ท, thatโs this smaller triangle, is similar to triangle ๐ธ๐ด๐ต, thatโs the larger triangle that ๐ธ๐ถ๐ท is part of.

Notice that weโre given this really important piece of information that the line segment ๐ด๐ต and the line segment ๐ท๐ถ are parallel. So letโs 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 arenโt given any information about the lengths of the sides, we could potentially use this grid paper to help us. However, letโs 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 ๐ธ appears in both triangles. If we wanted to write that formally, we could say that angle ๐ท๐ธ๐ถ is equal to angle ๐ต๐ธ๐ด. And thatโs because that angle is common to both triangles.

Next, letโs have a look at this angle ๐ธ๐ถ๐ท. Using the fact that these two line segments, ๐ด๐ต and ๐ถ๐ท, are parallel, then we can say that this angle at ๐ธ๐ด๐ต is equal to angle ๐ธ๐ถ๐ท. Thatโs because these two angles will be corresponding. The parallel lines are important because if we just had this triangle ๐ธ๐ด๐ต and then the line segments ๐ด๐ต and ๐ถ๐ท were not parallel, then we couldnโt say that angle ๐ธ๐ถ๐ท was equal to angle ๐ธ๐ด๐ต. Thatโs only true because we have a set of parallel lines.

Now weโve proved that we have two pairs of corresponding angles congruent. Therefore, weโve demonstrated that the AA or angle-angle similarity rule applies. And weโve shown that these two triangles, ๐ธ๐ถ๐ท and ๐ธ๐ด๐ต, are similar. We could also prove that angle ๐ถ๐ท๐ธ is equal to angle ๐ด๐ต๐ธ. 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, โYes, 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, โYes, as there are two pairs of corresponding angles in the triangles of equal measureโ would also be correct.