Question Video: Identifying Similar Triangles | Nagwa Question Video: Identifying Similar Triangles | Nagwa

Question Video: Identifying Similar Triangles Mathematics • Second Year of Preparatory School

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Which two of these triangles are similar?

04:54

Video Transcript

Which two of these triangles are similar? And we’re given four different triangles numbered (1) through (4).

Let’s begin by recalling the meaning of similar. Similar triangles, like any similar shapes, will have corresponding angles equal and corresponding sides in proportion.

If we take a quick look at the triangles that we’re given, we can see that triangles (1) and (2) both have an angle of 40 degrees and triangle (4) has an angle of 100 degrees. There’s not enough information given in any of these triangles to prove that they’re similar. So let’s take a look and see if we can work out any of the other angles in each triangle.

To help us do this, we’ll need to remember that the angles in a triangle add up to 180 degrees. So let’s take a look at triangle number (1). Triangle (1) has this markation on the lines, indicating that there are two sides the same length. We remember that a triangle with two sides the same length is an isosceles triangle. An isosceles triangle also has a pair of equal angles. The two equal angles will be these ones marked in orange. And we’re given that the measure of one of these angles is 40 degrees. Therefore, they’re both 40 degrees. We can then find the third unknown angle in this triangle using the fact that these three angles must add up to 180 degrees. 40 plus 40 gives us 80. And subtracting that from 180 degrees leaves us with 100 degrees.

Now that we know the three angles in this triangle, we can see if any of the other triangles will have similar angle measurements. Let’s have a look then at triangle number (2). And we notice that once again this is also an isosceles triangle. It’s got three line markings this time to indicate that these signs are not necessarily the same length as those in triangle (1). When we consider which two angles will be the same size in this isosceles triangle number (2), it will be these two angles at the base of the triangle. Because we know that these angles are the same size and we’re given that the other angle is 40 degrees, then we can work out their value.

Let’s say that we call this unknown angle 𝑥 degrees. We know that the three angles must add up to give 180 degrees. So we could write the equation two 𝑥 degrees plus 40 degrees is equal to 180 degrees. We could then subtract 40 degrees from both sides. And dividing through by two gives us that 𝑥 is equal to 70 degrees. Therefore, in triangle (2), we have two angles of 70 degrees and an angle of 40 degrees. We now know that triangles (1) and (2) are not similar because we don’t have corresponding pairs of angles equal.

Let’s take a look then at triangle number (3). Once again, we can see that this is an isosceles triangle. And although we’re not given any angle measurement, we are given the indication that we have a right angle of 90 degrees. In this isosceles triangle, we’ll have two pairs of equal angles, which will be here in orange. So let’s see if we can work out the value of these. Using the same approach as before, we can take the value of the unknown angle as 𝑦. Therefore, 𝑦 plus 𝑦 plus 90 degrees is 180 degrees. Simplifying to two 𝑦 degrees plus 90 degrees is 180 degrees. Subtracting 90 degrees from both sides and then dividing through by two gives us that 𝑦 must be equal to 45 degrees.

Now that we have the three angle measurements in triangle number (3), we can also see that none of these three triangles would be similar to each other. So let’s have a look at the fourth triangle.

Triangle (4) is also an isosceles triangle with a pair of equal angles here in pink. As these two angles are the same and all three angles add up to 180 degrees, then the two unknown angles must be 40 degrees. The other triangle in this set, which has angles of 100 degrees, 40 degrees, and 40 degrees, is triangle number (1).

Therefore, we can give the answer for which two of these triangles are similar as triangles (1) and (4).

As we were working through this problem, we found all three angles in each triangle. However, in order to prove that two triangles are similar, we only need to prove that there are two pairs of corresponding angles equal. Proving that means that the AA rule is fulfilled and the triangles are similar. However, showing that all three corresponding angles are equal in triangles (1) and (4) as we did here is also valid.

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