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
x2 = 9⋅16 - [1]
= 32⋅42
= (3⋅4)2 - [2]
= 122
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