When are triangles similar?

Triangles are similar if they have the same shape, but can be different sizes.
(They are still similar even if one is rotated, or one is a mirror image of the other).

AA

AA stands for "angle, angle" and means that the triangles have two of their angles equal.

If two triangles have two of their angles equal, the triangles are similar.

For example, these two triangles are similar:

If two of their angles are equal, then the third angle must also be equal, because angles of a triangle always add to make 180°.
In this case the missing angle is 180° - (72° + 35°) = 83°.
So AA could also be called AAA.

SAS

SAS stands for "side, angle, side" and means that we have two triangles where:

• the ratio between two sides is the same as the ratio between another two sides
• and we we also know the included angles are equal.

If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.

For example:

In this example we can see that:

• one pair of sides is in the ratio of 21 : 14 = 3 : 2
• another pair of sides is in the ratio of 15 : 10 = 3 : 2
• there is a matching angle of 75° in between them

So there is enough information to tell us that the two triangles are similar.

Using Trigonometry

We could also use Trigonometry to calculate the other two sides using the Law of Cosines:

In Triangle ABC:		a2	= b2 + c2 - 2bc cos A = 212 + 152 - 2 × 21 × 15 × Cos75° = 441 + 225 - 630 × 0.2588... = 666 - 163.055... = 502.944...
		Therefore a = √502.94 = 22.426...
In Triangle XYZ:		x2	= y2 + z2 - 2yz cos X = 142 + 102 - 2 × 14 × 10 × Cos75° = 196 + 100 - 280 × 0.2588... = 296 - 72.469... = 223.530...
		Therefore x = √223.530... = 14.950...

Now let us check the ratio of those two sides:

a : x = 22.426... : 14.950... = 3 : 2

the same ratio as before!

Note: you could also use the Law of Sines to show that the other two angles are equal.

SSS

SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.

If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

For example:

In this example, the ratios of sides are:

• a: x = 6 : 7.5 = 12 : 15 = 4 : 5
• b: y = 8 : 10 = 4 : 5
• c: z = 4 : 5

These ratios are all equal, so the two triangles are similar.

Guided Questions:
1) Which Postulate is correct to find the similarities of the triangle below

 

Citations:
 http://www.mathsisfun.com/geometry/triangles-similar-finding.html