Example 2: Verifying Triangle Similarity - Demonstrates how to determine if triangles are similar using side ratios and angle congruence.

Let’s solve each part step by step.

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Part A:

We are given two triangles: △STU and △XVW.

Side lengths:

△STU:
- ST = 6
- TU = 8
- SU = 10

△XVW:
- XV = 9
- VW = 12
- XW = 15

To check if they are similar, we compare the ratios of corresponding sides. We need to match the smallest side to smallest, middle to middle, largest to largest.

In △STU: sides are 6, 8, 10 → ordered: 6 < 8 < 10
In △XVW: sides are 9, 12, 15 → ordered: 9 < 12 < 15

So pair them as:
- 6 ↔ 9
- 8 ↔ 12
- 10 ↔ 15

Now compute ratios:

6/9 = 2/3
8/12 = 2/3
10/15 = 2/3

All three ratios are equal → so the triangles are similar!

Now write the similarity statement. Match vertices in order of corresponding sides.

Since 6 (ST) corresponds to 9 (XV),
8 (TU) corresponds to 12 (VW),
10 (SU) corresponds to 15 (XW)

That means:
S ↔ X
T ↔ V
U ↔ W

So the similarity statement is:
△STU ~ △XVW

*(Note: In the image it says △STU ~ △XVW — that matches our result.)*

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Part B:

We are given triangle KLP with points M on KM and N on KN? Wait — let’s look carefully.

Actually, from the diagram description:

Triangle KMN has point L on KM and P on KN? Or maybe it's triangle KLP inside triangle KMN?

Wait — looking at labels:

Triangle KLP:
KL = 8
LP = ? Not labeled directly — but wait, the diagram shows:

Actually, based on standard notation and the given numbers:

It seems we have triangle KMN, and inside it, triangle KLP, where L is on KM and P is on KN.

Given:

In △KLP:
KL = 8
KP = 12
LP = ? — not needed if we use SAS or SSS.

Wait — actually, the problem gives:

For △KLP: KL = 8, KP = 12
For △KMN: KM = 10, KN = 15

And angle K is common to both triangles.

So we can use SAS Similarity: If two sides are proportional and the included angle is congruent, then triangles are similar.

Check ratio of sides around angle K:

In △KLP and △KMN:

Side KL corresponds to KM → 8 / 10 = 4/5
Side KP corresponds to KN → 12 / 15 = 4/5

Same ratio! And angle K is shared → ∠K ≅ ∠K

Therefore, by SAS similarity, △KLP ~ △KMN

Similarity statement:
Vertices must correspond:
K ↔ K (common vertex)
L ↔ M (since KL corresponds to KM)
P ↔ N (since KP corresponds to KN)

So: △KLP ~ △KMN

*(Matches what’s written in green in the image.)*

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Final Answer:

A. Yes, the triangles are similar. Similarity statement: △STU ~ △XVW
B. Yes, the triangles are similar. Similarity statement: △KLP ~ △KMN