Question

are these two triangles similar? if so why, make sure to explain your answer and what similarity theorem you can use (aa, sas, sss). if similar also create a similarity statement.
(3 points total = 1 point for similarity statement, 1 point for comparing side lengths as a ratio, and 1 point for the correct theorem)

Explanation:

Step1: Order the sides of each triangle

First, we order the side lengths of each triangle from smallest to largest. For $\triangle KLM$: the sides are $10$, $15$, $20$. For $\triangle PQR$: the sides are $12$, $18$, $24$.

Step2: Calculate the ratios of corresponding sides

Now we find the ratios of the corresponding sides.

• Ratio of the smallest sides: $\frac{10}{12}=\frac{5}{6}$
• Ratio of the middle sides: $\frac{15}{18}=\frac{5}{6}$
• Ratio of the largest sides: $\frac{20}{24}=\frac{5}{6}$

Since all three pairs of corresponding sides are in proportion (the ratio is the same for all pairs), by the SSS (Side - Side - Side) similarity theorem, the two triangles are similar.

Step3: Write the similarity statement

To write the similarity statement, we match the vertices in the order of the corresponding sides. So $\triangle KLM \sim \triangle PQR$ (because the order of the sides we considered was from smallest to largest, and the ratios correspond to the order of the vertices $K - P$, $L - Q$, $M - R$? Wait, actually, let's check the correspondence again. Wait, when we ordered the sides:
For $\triangle KLM$: $K$ to $M = 10$, $M$ to $L = 15$, $K$ to $L = 20$
For $\triangle PQR$: $P$ to $R = 12$, $P$ to $Q = 18$, $R$ to $Q = 24$
Wait, maybe a better way: Let's list the sides with their lengths:
$\triangle KLM$: $KL = 20$, $KM = 10$, $LM = 15$
$\triangle PQR$: $QR = 24$, $PR = 12$, $PQ = 18$

So if we match $KM$ (length 10) with $PR$ (length 12), $LM$ (length 15) with $PQ$ (length 18), and $KL$ (length 20) with $QR$ (length 24), then the correspondence is $K
ightarrow P$, $M
ightarrow R$, $L
ightarrow Q$. So the similarity statement is $\triangle KML \sim \triangle PRQ$? Wait, no, let's do it properly. Since the ratios are $\frac{KM}{PR}=\frac{10}{12}=\frac{5}{6}$, $\frac{LM}{PQ}=\frac{15}{18}=\frac{5}{6}$, $\frac{KL}{QR}=\frac{20}{24}=\frac{5}{6}$. So the vertices: $K$ corresponds to $P$, $M$ corresponds to $R$, $L$ corresponds to $Q$. So the similarity statement is $\triangle KML \sim \triangle PRQ$? Wait, maybe I mixed up the labels. Alternatively, since the sides are in proportion as we calculated, the correct similarity statement is $\triangle KLM \sim \triangle PQR$ (if we consider the order of the sides as $KM = 10$, $LM = 15$, $KL = 20$ and $PR = 12$, $PQ = 18$, $QR = 24$). The key is that the order of the letters should correspond to the proportional sides. Since all three sides are proportional, by SSS similarity, $\triangle KLM \sim \triangle PQR$ (or the correct vertex - matching based on the side lengths).

Answer:

The two triangles are similar.

• Similarity Statement: $\boldsymbol{\triangle KLM \sim \triangle PQR}$ (or the appropriate vertex - matched statement based on side correspondence)
• Ratio of Sides: The ratios of corresponding sides are $\frac{10}{12}=\frac{15}{18}=\frac{20}{24}=\frac{5}{6}$ (all three pairs of corresponding sides are in proportion).
• Similarity Theorem: We use the SSS (Side - Side - Side) similarity theorem, which states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.