In the diagram, the ratios of two pairs of corresponding sides are equal.To prove that △LMN ~ △XYZ by the SAS similarity theorem, it also needs to be shown that
∠N ≅ ∠Z
∠N ≅ ∠X
∠L ≅ ∠Z
∠L ≅ ∠Y
No diagram. Cannot copy and paste here.
To prove that △LMN ~ △XYZ by the SAS similarity theorem, we need to show that two pairs of corresponding sides are proportional and one pair of corresponding angles is congruent. In this case, the given information states that the ratios of two pairs of corresponding sides are equal, but it does not directly mention any angle measures.
To show that △LMN ~ △XYZ by the SAS similarity theorem, we need to prove that one pair of corresponding angles is congruent. Let's go through the steps to do so:
Step 1: Given the diagram with the statement that the ratios of two pairs of corresponding sides are equal, we can label the corresponding sides and angles:
In △LMN:
LM/NX = MN/YZ
LN = XY
Step 2: We can also label the corresponding angles:
∠L and ∠X
∠M and ∠Y
∠N and ∠Z
Step 3: Now, we need to show that one pair of corresponding angles is congruent. We can start by considering ∠N and ∠Z:
Assuming that ∠N ≅ ∠Z, we can argue as follows:
From the given information, we have LM/NX = MN/YZ
By the SAS similarity theorem, if the ratios of two pairs of corresponding sides are equal and one pair of corresponding angles is congruent, then the triangles are similar.
Therefore, △LMN ~ △XYZ by the SAS similarity theorem can be proved.
Note: In this scenario, to prove that △LMN ~ △XYZ, it is sufficient to show that any one pair of corresponding angles is congruent, such as ∠N ≅ ∠Z. The other angle pairs mentioned (∠N ≅ ∠X, ∠L ≅ ∠Z, ∠L ≅ ∠Y) are not necessary to establish the similarity.