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# Similar Triangles

See how to find the side of similar triangles.
12 examples and their solutions.

## Definition

If two triangles are similar,
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

For two triangles,
if 2 angles are congruent
(Angle-Angle),
then those two triangles are similar.

### Example

Given: AB // CD
Prove: △PAB ≌ △PDC

Solution

## SSS Similarity

### Theorem

aa' = bb' =cc'
For two triangles,
if 3 sides of each triangle are proportional,
then those two triangles are similar.

### Example

Show that △ABD ~ △DCB.

Solution

## SAS Similarity

### Theorem

aa' = bb'
For two triangles,
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

Solution

Solution

Solution

## Similar Triangles in a Right Triangle

### How to Find

To find similar triangles from this right triangle,
First draw angles like this.
(m∠[plane] + m∠[dot] = 90)
Then there are 3 similar right triangles.

Solution

Solution

## Midsegment Theorem

### Midsegment of a Triangle

The midsegment of a triangle is a line segment
that connects the midpoints of two sides.

### Theorem

m = 12a
1. The midsegment (m) and the opposite side (a) are parallel.
2. m = [1/2]a

Solution

ab = a'b'

Solution

ab = a'b'

Solution