Question Video: Using Similarity to Recognize Geometric Properties | Nagwa Question Video: Using Similarity to Recognize Geometric Properties | Nagwa

Question Video: Using Similarity to Recognize Geometric Properties Mathematics • Second Year of Preparatory School

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Triangles 𝐴𝐷𝐸 and 𝐴𝐡𝐢 in the given figure are similar. What, if anything, must be true of the lines 𝐷𝐸 and 𝐡𝐢?

03:15

Video Transcript

Triangles 𝐴𝐷𝐸 and 𝐴𝐡𝐢 in the given figure are similar. What, if anything, must be true of the lines 𝐷𝐸 and 𝐡𝐢? Option (A) they are parallel, or option (B) they are perpendicular.

In this question, we are told that there are two similar triangles. They are triangle 𝐴𝐷𝐸, which is the smaller triangle, and triangle 𝐴𝐡𝐢, which is the larger triangle. We can recall that similar triangles have corresponding angles congruent and corresponding sides in proportion.

Now, we are asked what must be true of the two lines 𝐷𝐸 and 𝐡𝐢. But we are aren’t given any information about the lengths of any sides in this figure. So, let’s see what we can work out by using the angle properties of these similar triangles. Since the corresponding angles are congruent, we know that the measure of angle 𝐴𝐷𝐸 is equal to the measure of angle 𝐴𝐡𝐢. And, in the same way, the corresponding angles 𝐴𝐸𝐷 and 𝐴𝐢𝐡 must be of equal measure.

The third pair of angles in each triangle is the common angle at vertex 𝐴, which we could refer to as angle 𝐷𝐴𝐸 in triangle 𝐴𝐷𝐸 and angle 𝐡𝐴𝐢 in triangle 𝐴𝐡𝐢. But we can consider the information from the first pair of angles. These angles are constructed between the lines 𝐷𝐸 and 𝐡𝐢 and the line 𝐴𝐡. And we know that these angles are congruent.

We know that if we have a pair of parallel lines and a transversal, then the corresponding angles are congruent. And remember that the converse of this is also true. That is, if corresponding angles in a transversal of two lines are congruent, then the lines are parallel. And this is the situation that we have here. The two lines are 𝐷𝐸 and 𝐡𝐢. The transversal is the line 𝐴𝐡. And corresponding angles are congruent. Therefore, the lines 𝐷𝐸 and 𝐡𝐢 are parallel. So, the answer to the question is that given in option (A). We can say that the lines 𝐷𝐸 and 𝐡𝐢 are parallel.

It’s worth noting that we could have proved the same property using the second pair of angles that we found. The only difference in using the corresponding congruent angles 𝐴𝐸𝐷 and 𝐴𝐢𝐡 would be that the transversal would instead be the line 𝐴𝐢. But this would still prove that lines 𝐷𝐸 and 𝐡𝐢 are parallel.

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