Question Video: Using the Congruency of Triangles to Identify Congruent Sides | Nagwa Question Video: Using the Congruency of Triangles to Identify Congruent Sides | Nagwa

# Question Video: Using the Congruency of Triangles to Identify Congruent Sides Mathematics • First Year of Preparatory School

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In the figure, if πβ π΄π·πΆ = πβ π΄πΈπ΅, then which of the following is true? [A] π·πΆ = π΄πΈ [B] π·πΆ = π΅πΈ [C] π·πΆ = πΈπΆ [D] π΄π· = π΅πΈ [E] π·π΅ = π΄πΈ

03:37

### Video Transcript

In the given figure, if the measure of angle π΄π·πΆ equals the measure of angle π΄πΈπ΅, then which of the following is true? (A) π·πΆ equals π΄πΈ. (B) π·πΆ equals π΅πΈ. (C) π·πΆ equals πΈπΆ. (D) π΄π· equals π΅πΈ. Or (E) π·π΅ equals π΄πΈ.

Letβs begin by noting that we are given that the measures of angles π΄π·πΆ and π΄πΈπ΅ are equal. And these are colored on the diagram. We also have a pair of congruent sides marked on the diagram. So, we can write that line segments π΄π· and π΄πΈ are congruent.

Now letβs consider two triangles within this larger triangle: triangle π΄π΅πΈ and triangle π΄π·πΆ. And sometimes, a separate sketch can be useful if we find it difficult to see these smaller triangles within a larger shape. In each triangle, we can still see the pair of congruent sides and the given congruent angle measures. And we may also notice that the angle at vertex π΄ appears in both triangles. We can write that the measure of angle π΅π΄πΈ equals the measure of angle πΆπ΄π·.

Now, we can see that in each triangle, we have two pairs of congruent angles and the pair of included sides between them are congruent. Therefore, this proves that triangles π΄πΆπ· and π΄π΅πΈ are congruent, using the ASA or angle-side-angle congruency criterion. We can note that the options that we were given concern the congruency of sides. So letβs think about what sides we can say are congruent by using the congruent triangles.

We already know that line segments π΄π· and π΄πΈ are congruent. And now we know that line segments π·πΆ and π΅πΈ are congruent because these are corresponding. The third side in each triangle, thatβs line segments π΄π΅ and π΄πΆ, are also congruent. The only one of these statements that appears in the answer options is that π·πΆ equals π΅πΈ. But letβs see if any of the others are also true.

In option (A), we can find line segments π·πΆ and π΄πΈ here on the figure. And we can observe that we have no information to prove that these pair of line segments are congruent. The same is true for option (C). We cannot prove that line segments π·πΆ and πΈπΆ are congruent. Line segments π΄π· and π΅πΈ also cannot be proved to be congruent. And neither can the line segments π·π΅ and π΄πΈ. Therefore, the only statement which is true is that π·πΆ equals π΅πΈ.

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