Determine Similar Figures

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Question 1 of 3

Are the two triangles similar?

1. No
2. Yes

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Similar triangles have corresponding side lengths which are proportional (equal). This rule is called Side-Side-Side

$(SSS)$ Similarity.

First, let’s label the values of each side length

For triangle

$ABC$ :

$AB=6,$ $BC=12,$ $AC=9$

For triangle

$DEF$ :

$DE=4,$ $EF=8,$ $DF=6$

To prove that

$ABC$ is similar to

$DEF$ the following equation must be true

$\frac{\color\green{AB}}{\color\green{DE}}=\frac{\color{#004ec4}{BC}}{\color{#004ec4}{EF}}=\frac{\color{#e85e00}{AC}}{\color{#e85e00}{DF}}$			Definition of Similarity

$\frac{\color\green{6}}{\color\green{4}}=\frac{\color{#004ec4}{12}}{\color{#004ec4}{8}}=\frac{\color{#e85e00}{9}}{\color{#e85e00}{6}}$			Plug in the side lengths

$\frac{\color\green{3}}{\color\green{2}}=\frac{\color{#004ec4}{3}}{\color{#004ec4}{2}}=\frac{\color{#e85e00}{3}}{\color{#e85e00}{2}}$			Simplify each fraction

Since corresponding side lengths are proportional,

$ABC$ is similar to

$DEF$

Yes,

$ABC$ is similar to

$DEF$

Question 2 of 3

2. Question
Are the two triangles similar?
- 1. No
- 2. Yes
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Similar triangles have corresponding angles which are proportional (equal). This rule is called Angle-Angle-Angle $(A A A)$ Similarity.

First, take note of equal angles from the diagram.

To prove that $ABC$ is similar to $DEF$ , check for equal angles.

$angle A$ $=$ $angle D$ as shown by the dot

$angle B$ $=$ $angle E$ as shown by the purple mark

$angle C$ $=$ $angle F$ Angle Sum of Triangle

Since all angles are equal, $ABC$ and $DEF$ are equiangular and therefore similar.

Yes, $ABC$ is similar to $DEF$

Question 3 of 3

Are the two triangles similar?

1. No
2. Yes

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If two triangles have two corresponding side lengths and an included angle to be proportional, then the triangles are similar. This rule is called Side-Angle-Side

$(SAS)$ Similarity.

First, lets label the values of each side length

For triangle

$ABC$ :

$AB=8,$ $AC=17.6$

For triangle

$DEF$ :

$DE=5,$ $DF=11$

$angle A = angle D$ as shown by the dots

To prove that

$ABC$ is similar to

$DEF$ the following equation must be true

$\frac{\color\green{AB}}{\color\green{DE}}=\frac{\color{#e85e00}{AC}}{\color{#e85e00}{DF}}$			Definition of Similarity

$\frac{\color\green{8}}{\color\green{5}}=\frac{\color{#e85e00}{17.6}}{\color{#e85e00}{11}}$			Plug in the side lengths

$\frac{\color\green{8}}{\color\green{5}}=\frac{\color{#e85e00}{8}}{\color{#e85e00}{5}}$			Simplify each fraction

Since corresponding side lengths are proportional, and included angles are equal,

$ABC$ is similar to

$DEF$

Yes,

$ABC$ is similar to

$DEF$

Determine Similar Figures

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2. Question

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3. Question

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Quizzes

$angle A$	$=$	$angle D$	as shown by the dot
$angle B$	$=$	$angle E$	as shown by the purple mark
$angle C$	$=$	$angle F$	Angle Sum of Triangle