Math / Geometry

QuestionDetermine which pairs of triangles are similar. Use a sketch to help explain how you know. \begin{tabular}{|l|l|l|} \hline \multicolumn{1}{|c|}{ Triangle } & \multicolumn{1}{|c|}{ Angles } & \multicolumn{1}{c|}{ Sides } \\ \hline $\triangle \mathrm{ABC}$ & $\angle \mathrm{A}=90^{\circ}$ & $\mathrm{AB}=6$ \\ & $\angle \mathrm{~B}=45^{\circ}$ & $\mathrm{BC}=8.4$ \\ & $\angle \mathrm{C}=45^{\circ}$ & $\mathrm{AC}=6$ \\ \hline $\triangle \mathrm{EFG}$ & $\angle \mathrm{E}=90^{\circ}$ & $\mathrm{EF}=3$ \\ & $\angle \mathrm{~F}=45^{\circ}$ & $\mathrm{FG}=4.2$ \\ & $\angle \mathrm{G}=45^{\circ}$ & $\mathrm{EG}=3$ \\ \hline $\triangle \mathrm{HIJ}$ & $\angle \mathrm{H}=90^{\circ}$ & $\mathrm{HI}=9.2$ \\ & $\angle \mathrm{I}=60^{\circ}$ & $\mathrm{IJ}=18.4$ \\ & $\angle \mathrm{~J}=30^{\circ}$ & $\mathrm{HJ}=15.9$ \\ \hline $\triangle \mathrm{KLM}$ & $\angle \mathrm{K}=90^{\circ}$ & $\mathrm{KL}=9$ \\ & $\angle \mathrm{~L}=45^{\circ}$ & $\mathrm{LM}=12.6$ \\ & $\angle \mathrm{M}=45^{\circ}$ & $\mathrm{KM}=9$ \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? Which of these triangles are like twins, but maybe different sizes? Watch out! Just because two triangles look similar, doesn't mean they are!
We need to check if their angles match and their sides are proportional.

STEP 2

1. Check for Similar Triangles using Angle-Angle Similarity
2. Check for Similar Triangles using Side-Side-Side Similarity

STEP 3

Let's look at the angles!
If two triangles have two matching angles, they're similar!
This is called Angle-Angle similarity.

STEP 4

Triangles $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ all have angles of $90^\circ$ , $45^\circ$ , and $45^\circ$ .
Boom! They're similar! $\triangle HIJ$ has different angles, so it's not similar to the others.

STEP 5

Even though we know $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are similar because of their angles, let's check if their sides are proportional too!
This is called Side-Side-Side similarity.

STEP 6

For $\triangle ABC$ and $\triangle EFG$ , let's see.
We have: $\frac{AB}{EF} = \frac{6}{3} = \textbf{2}$ $\frac{BC}{FG} = \frac{8.4}{4.2} = \textbf{2}$ $\frac{AC}{EG} = \frac{6}{3} = \textbf{2}$ All the ratios are 2!
That means the sides are proportional!

STEP 7

Now, let's compare $\triangle ABC$ and $\triangle KLM$ : $\frac{AB}{KL} = \frac{6}{9} = \frac{2}{3}$ $\frac{BC}{LM} = \frac{8.4}{12.6} = \frac{2}{3}$ $\frac{AC}{KM} = \frac{6}{9} = \frac{2}{3}$ Again, all the ratios are the same, this time $\frac{\textbf{2}}{\textbf{3}}$ !
So, these triangles are also similar!

STEP 8

Finally, let's compare $\triangle EFG$ and $\triangle KLM$ : $\frac{EF}{KL} = \frac{3}{9} = \frac{1}{3}$ $\frac{FG}{LM} = \frac{4.2}{12.6} = \frac{1}{3}$ $\frac{EG}{KM} = \frac{3}{9} = \frac{1}{3}$ All ratios are $\frac{\textbf{1}}{\textbf{3}}$ , confirming they're similar!

STEP 9

$\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are all similar to each other. $\triangle HIJ$ is not similar to any of the other triangles.

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Studdy Solution

STEP 1

What is this asking? Which of these triangles are like twins, but maybe different sizes? Watch out! Just because two triangles look similar, doesn't mean they are!
We need to check if their angles match and their sides are proportional.

STEP 2

1. Check for Similar Triangles using Angle-Angle Similarity
2. Check for Similar Triangles using Side-Side-Side Similarity

STEP 3

Let's look at the angles!
If two triangles have two matching angles, they're similar!
This is called Angle-Angle similarity.

STEP 4

Triangles $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ all have angles of $90^\circ$ , $45^\circ$ , and $45^\circ$ .
Boom! They're similar! $\triangle HIJ$ has different angles, so it's not similar to the others.

STEP 5

Even though we know $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are similar because of their angles, let's check if their sides are proportional too!
This is called Side-Side-Side similarity.

STEP 6

For $\triangle ABC$ and $\triangle EFG$ , let's see.
We have: $\frac{AB}{EF} = \frac{6}{3} = \textbf{2}$ $\frac{BC}{FG} = \frac{8.4}{4.2} = \textbf{2}$ $\frac{AC}{EG} = \frac{6}{3} = \textbf{2}$ All the ratios are 2!
That means the sides are proportional!

STEP 7

STEP 8

Finally, let's compare $\triangle EFG$ and $\triangle KLM$ : $\frac{EF}{KL} = \frac{3}{9} = \frac{1}{3}$ $\frac{FG}{LM} = \frac{4.2}{12.6} = \frac{1}{3}$ $\frac{EG}{KM} = \frac{3}{9} = \frac{1}{3}$ All ratios are $\frac{\textbf{1}}{\textbf{3}}$ , confirming they're similar!

STEP 9

$\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are all similar to each other. $\triangle HIJ$ is not similar to any of the other triangles.

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Studdy Solution

STEP 1

What is this asking? Which of these triangles are like twins, but maybe different sizes? Watch out! Just because two triangles look similar, doesn't mean they are!We need to check if their angles match and their sides are proportional.

STEP 2

1. Check for Similar Triangles using Angle-Angle Similarity2. Check for Similar Triangles using Side-Side-Side Similarity

STEP 3

Let's **look** at the angles!If two triangles have two matching angles, they're similar!This is called Angle-Angle similarity.

STEP 4

Triangles △ABC\triangle ABC△ABC, △EFG\triangle EFG△EFG, and △KLM\triangle KLM△KLM all have angles of 90∘90^\circ90∘, 45∘45^\circ45∘, and 45∘45^\circ45∘.Boom! They're similar! △HIJ\triangle HIJ△HIJ has different angles, so it's not similar to the others.

STEP 5

Even though we know △ABC\triangle ABC△ABC, △EFG\triangle EFG△EFG, and △KLM\triangle KLM△KLM are similar because of their angles, let's check if their sides are proportional too!This is called Side-Side-Side similarity.

STEP 6

STEP 7

STEP 8

STEP 9

△ABC\triangle ABC△ABC, △EFG\triangle EFG△EFG, and △KLM\triangle KLM△KLM are all similar to each other. △HIJ\triangle HIJ△HIJ is not similar to any of the other triangles.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Which of these triangles are like twins, but maybe different sizes? Watch out! Just because two triangles look similar, doesn't mean they are!
We need to check if their angles match and their sides are proportional.

1. Check for Similar Triangles using Angle-Angle Similarity
2. Check for Similar Triangles using Side-Side-Side Similarity

Let's look at the angles!
If two triangles have two matching angles, they're similar!
This is called Angle-Angle similarity.

Triangles $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ all have angles of $90^\circ$ , $45^\circ$ , and $45^\circ$ .
Boom! They're similar! $\triangle HIJ$ has different angles, so it's not similar to the others.

Even though we know $\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are similar because of their angles, let's check if their sides are proportional too!
This is called Side-Side-Side similarity.

$\triangle ABC$ , $\triangle EFG$ , and $\triangle KLM$ are all similar to each other. $\triangle HIJ$ is not similar to any of the other triangles.