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 π΄π·πΈ and π΄π΅πΆ in the given figure are similar. What, if anything, must be true of the lines π·πΈ and π΅πΆ?

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### Video Transcript

Triangles π΄π·πΈ and π΄π΅πΆ in the given figure are similar. What, if anything, must be true of the lines π·πΈ and π΅πΆ? 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 π΄π·πΈ, which is the smaller triangle, and triangle π΄π΅πΆ, 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 π·πΈ and π΅πΆ. But we are arenβt given any information about the lengths of any sides in this figure. So, letβ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 π΄π·πΈ is equal to the measure of angle π΄π΅πΆ. And, in the same way, the corresponding angles π΄πΈπ· and π΄πΆπ΅ must be of equal measure.

The third pair of angles in each triangle is the common angle at vertex π΄, which we could refer to as angle π·π΄πΈ in triangle π΄π·πΈ and angle π΅π΄πΆ in triangle π΄π΅πΆ. But we can consider the information from the first pair of angles. These angles are constructed between the lines π·πΈ and π΅πΆ and the line π΄π΅. 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 π·πΈ and π΅πΆ. The transversal is the line π΄π΅. And corresponding angles are congruent. Therefore, the lines π·πΈ and π΅πΆ are parallel. So, the answer to the question is that given in option (A). We can say that the lines π·πΈ and π΅πΆ are parallel.

Itβ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 π΄πΈπ· and π΄πΆπ΅ would be that the transversal would instead be the line π΄πΆ. But this would still prove that lines π·πΈ and π΅πΆ are parallel.

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