Determine Similar Triangles

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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

(S S S)

$(SSS)$ Similarity.

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

For triangle

A B C

$ABC$ :

$A B = 6,$ $AB=6,$ $B C = 12,$ $BC=12,$ $A C = 9$ $AC=9$

For triangle

D E F

$DEF$ :

$D E = 4,$ $DE=4,$ $E F = 8,$ $EF=8,$ $D F = 6$ $DF=6$

To prove that

A B C

$ABC$ is similar to

D E F

$DEF$ 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,

A B C

$ABC$ is similar to

D E F

$DEF$

Yes,

A B C

$ABC$ is similar to

D E F

$DEF$

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 $(A A A)$ $(A A A)$ Similarity.

First, take note of equal angles from the diagram.

To prove that $A B C$ $ABC$ is similar to $D E F$ $DEF$ , check for equal angles.

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

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

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

Since all angles are equal, $A B C$ $ABC$ and $D E F$ $DEF$ are equiangular and therefore similar.

Yes, $A B C$ $ABC$ is similar to $D E F$ $DEF$

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

(S A S)

$(SAS)$ Similarity.

First, lets label the values of each side length

For triangle

A B C

$ABC$ :

$A B = 8,$ $AB=8,$ $A C = 17.6$ $AC=17.6$

For triangle

D E F

$DEF$ :

$D E = 5,$ $DE=5,$ $D F = 11$ $DF=11$

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

To prove that

A B C

$ABC$ is similar to

D E F

$DEF$ 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,

A B C

$ABC$ is similar to

D E F

$DEF$

Yes,

A B C

$ABC$ is similar to

D E F

$DEF$

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1. Question

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2. Question

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3. Question

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Quizzes

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