# Similar Triangles

See how to find the side of similar triangles.

12 examples and their solutions.

## Definition

their shapes are the same

but their sizes are different.

So the interior angles are congruent.

And their sides are proportional.

aa' = bb' = cc'

[~] is the similar symbol.

Congruent Triangles

## AA Similarity

### Postulate

if 2 angles are congruent

(Angle-Angle),

then those two triangles are similar.

### Example

Given: AB // CD

Prove: △PAB ≌ △PDC

Solution Prove: △PAB ≌ △PDC

Statement | Reason |
---|---|

1. AB // CD | Given |

2. ∠PAB ≌ ∠PDC | Alternate interior angles in parallel lines are congruent. |

3. ∠APB ≌ ∠DPC | Vertical Angles |

4. △PAB ≌ △PDC | AA Similarity |

Close

## SSS Similarity

### Theorem

aa' = bb' =cc'

if 3 sides of each triangle are proportional,

then those two triangles are similar.

### Example

Show that △ABD ~ △DCB.

Solution 63 = 42 = 84( o ) - [2]

By SSS similarity,

△ABD ~ △DCB. - [3]

[1]

Draw △ABD and △DCB.

Make their shapes the same.

Make their shapes the same.

[2]

See if 6/3 = 4/2 = 8/4 is true.

[3]

6/3 = 4/2 = 8/4 is true.

So, by SSS similarity,

△ABD is similar to △DCB.

So, by SSS similarity,

△ABD is similar to △DCB.

Close

## SAS Similarity

### Theorem

aa' = bb'

if 2 sides of each triangle are proportional

and if 1 angle of each triangle is congruent,

then those two triangles are similar.

### Example

Show that △APQ ~ △ACB.

Solution 36 = 48( o )

∠A ≅ ∠A

By SAS similarity,

△APQ ~ △ACB. - [1]

[1]

3/6 = 4/8 is true.

∠A ≅ ∠A

So, by SAS similarity,

△APQ is similar to △ACB.

∠A ≅ ∠A

So, by SAS similarity,

△APQ is similar to △ACB.

Close

## Similar Triangles

### Example

46 = 5x - [1]

23 = 5x

2x = 15

x = 152

[1]

Close

### Example

3x + 216 = 714

3x + 216 = 12

2(3x + 2) = 16

3x + 2 = 8

3x = 6

x = 2

[1]

Vertical angles are congruent.

[2]

Alternate interior angles in parallel lines

are congruent.

are congruent.

Close

### Example

37 = 55 + x

3(5 + x) = 35

15 + 3x = 35

3x = 20

x = 203

Close

## Similar Triangles in a Right Triangle

### How to Find

(m∠[plane] + m∠[dot] = 90)

### Example

9x = x16

x

^{2}= 9⋅16 - [1]

= 3

^{2}⋅4

^{2}

= (3⋅4)

^{2}- [2]

= 12

^{2}

x = 12 - [3]

[1]

Close

### Example

## Midsegment Theorem

### Midsegment of a Triangle

### Theorem

m = 12a

2. m = [1/2]a

### Example

x = 12⋅12

= 6

Close

## Basic Proportionality Theorem

### Theorem

ab = a'b'

### Example

### Theorem: Upgrade Version 1

ab = a'b'

### Example

816 = x18

12 = x18

2x = 18

x = 9

Close

### Theorem: Upgrade Version 2

a + ba = a' + b'a'

a + bb = a' + b'b'

### Example

11x = 104

11x = 52

5x = 22

x = 225

Close