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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

Similar Triangles

Example

Solution

Example

Solution

Example

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.

Example

Solution

Example

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

Example

Solution

Basic Proportionality Theorem

Theorem


ab = a'b'

Example

Solution

Theorem: Upgrade Version 1


ab = a'b'

Example

Solution

Theorem: Upgrade Version 2


a + ba = a' + b'a'
a + bb = a' + b'b'

Example

Solution