Similarity Theorems: Identifying and defining elements

The triangles are similar according to the angle-angle theorem.

Having two pairs of equal angles is sufficient to conclude that the triangles are similar.

Given that the data shows that there are two pairs with equal angles:

$B=E=40$

$C=F=60$

The triangles are similar according to the angle-angle theorem, therefore triangle ABC is similar to triangle DEF.

To determine whether the triangles $\triangle ABC$ and $\triangle DEF$ are similar, we shall apply the Side-Side-Side (SSS) similarity theorem, which requires that the ratios of corresponding sides of the triangles be equal.

Let's compute the ratios:

• Ratio of corresponding sides $BC$ and $EF$: $\frac{BC}{EF} = \frac{8}{4} = 2$
• Ratio of corresponding sides $AB$ and $DE$: $\frac{AB}{DE} = \frac{4}{2} = 2$
• Ratio of corresponding sides $AC$ and $DF$: $\frac{AC}{DF} = \frac{6}{3} = 2$

Since all the corresponding side ratios are equal ($2$), the triangles $\triangle ABC$ and $\triangle DEF$ are similar by the SSS similarity theorem.

Therefore, the solution to the problem is Yes.

To determine if the triangles ABC and DEF are similar, we need to examine the ratios of corresponding sides.

• Side AC (6) corresponds to side DF (3).
• Side BC (4) corresponds to side EF (2).
• Side AB (2) corresponds to side DE (1).

We calculate the ratios of corresponding sides:

• $\frac{AC}{DF} = \frac{6}{3} = 2$
• $\frac{BC}{EF} = \frac{4}{2} = 2$
• $\frac{AB}{DE} = \frac{2}{1} = 2$

All the corresponding side ratios are equal to 2, indicating that the sides of triangle ABC are proportional to the sides of triangle DEF by a common ratio. According to the Side-Side-Side (SSS) similarity criterion, this means the triangles are similar.

Therefore, the triangles are indeed similar. The correct answer is Yes.

To solve this problem, we'll determine if the triangles $\triangle ABC$ and $\triangle DEF$ are similar using the Side-Side-Side (SSS) similarity criterion.

Step 1: Identify the sides of both triangles:
For $\triangle ABC$, the side lengths are $AB = 5$, $BC = 4$, and $CA = 4$.
For $\triangle DEF$, the side lengths are $DE = 5$, $EF = 4$, and $FD = 4$.

Step 2: Calculate the ratios of the corresponding sides:
$\frac{AB}{DE} = \frac{5}{5} = 1$
$\frac{BC}{EF} = \frac{4}{4} = 1$
$\frac{CA}{FD} = \frac{4}{4} = 1$

Step 3: Verify similarity:
All three ratios are equal, so by the SSS criterion, the triangles are similar.

Therefore, the triangles $\triangle ABC$ and $\triangle DEF$ are similar.

The sides of the triangles are not equal and, therefore, the triangles are not similar.

To determine whether the triangles are similar, we will use the Side-Side-Side (SSS) criterion for similarity. According to this criterion, triangles are similar if the ratios of their corresponding sides are equal.

We have two triangles: $\triangle ABC$ with sides 7, 5, and 4, and $\triangle DEF$ with sides 7, 5, and 3.

We will calculate the ratios of the corresponding sides:

• For sides $AB$ and $DE$: $\frac{AB}{DE} = \frac{7}{7} = 1$
• For sides $BC$ and $EF$: $\frac{BC}{EF} = \frac{5}{5} = 1$
• For sides $AC$ and $DF$: $\frac{AC}{DF} = \frac{4}{3}$

From the calculations, we observe that two of the side ratios are equal to 1, but the third ratio $\frac{4}{3}$ does not match the others. Thus, the side ratios are not all identical, meaning the triangles are not similar according to the SSS criterion.

Therefore, the triangles $\triangle ABC$ and $\triangle DEF$ are not similar.

To determine if the triangles are similar, we will use the Side-Side-Side (SSS) similarity criterion, which checks if the corresponding sides of both triangles are proportional.

Let's analyze the given side lengths:
Triangle $\triangle ABC$ has sides $AB = 6$, $BC = 2$, and $AC = 4$.
Triangle $\triangle DEF$ has sides $DE = 12$, $EF = 4$, and $DF = 8$.

Now, calculate the ratios of corresponding sides:

• Ratio for sides $AB$ and $DE$: $\frac{6}{12} = \frac{1}{2}$
• Ratio for sides $BC$ and $EF$: $\frac{2}{4} = \frac{1}{2}$
• Ratio for sides $AC$ and $DF$: $\frac{4}{8} = \frac{1}{2}$

Since all corresponding sides are in the same proportion $\frac{1}{2}$, the triangles satisfy the SSS criterion for similarity.

Therefore, the triangles $\triangle ABC$ and $\triangle DEF$ are similar.

Thus, the answer is Yes.

We will analyze the given triangles to establish which ones are similar:

• Triangle I: Sides are $8$, $6$, and $4$.
• Triangle II: Sides are $4$, $3$, and $2$.
• Triangle III: Sides are $6$, $4$, and $2$.

To check for similarity using the Side-Side-Side (SSS) criterion, we compare the ratios of the corresponding sides of each triangle:

• For Triangle I and II:
  $\frac{8}{4} = 2$, $\frac{6}{3} = 2$, $\frac{4}{2} = 2$
  All sides are in the ratio $2:1$.
• For Triangle I and III:
  The ratios of sides will be:
  $\frac{8}{6} \neq \frac{6}{4} \neq \frac{4}{2}$
  These do not confirm similarity as the ratios differ.
• For Triangle II and III:
  $\frac{6}{4} = 1.5$, $\frac{4}{2} = 2$, which are not equal in proportions resulting in no similarity.

The only pair of triangles meeting the similarity condition based on the SSS criterion is Triangle II and Triangle III, with a similarity ratio of $2:1$.

Therefore, Triangles II and III are similar with a similarity ratio of 2.

This matches with the correct given answer, choice 4: $II, III, 2$.