What triangle similarity theorem states that if two angles of a triangle are congruent to the two angles of the second triangle?

Two triangles are similar if they have:

• all their angles equal
• corresponding sides are in the same ratio

But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough.

There are three ways to find if two triangles are similar: AA, SAS and SSS:

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.

So AA could also be called AAA (because when two angles are equal, all three angles must be equal).

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.

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
• a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
• a2 = 441 + 225 - 630 × 0.2588...
• a2 = 666 - 163.055...
• a2 = 502.944...
• So a = √502.94 = 22.426...

In Triangle XYZ:

• x2 = y2 + z2 - 2yz cos X
• x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
• x2 = 196 + 100 - 280 × 0.2588...
• x2 = 296 - 72.469...
• x2 = 223.530...
• So 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: we can 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.

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.

Using Trigonometry

Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:

In Triangle ABC:

• cos A = (b2 + c2 - a2)/2bc
• cos A = (82 + 42 - 62)/(2× 8 × 4)
• cos A = (64 + 16 - 36)/64
• cos A = 44/64
• cos A = 0.6875
• So Angle A = 46.6°

In Triangle XYZ:

• cos X = (y2 + z2 - x2)/2yz
• cos X = (102 + 52 - 7.52)/(2× 10 × 5)
• cos X = (100 + 25 - 56.25)/100
• cos X = 68.75/100
• cos X = 0.6875
• So Angle X = 46.6°

So angles A and X are equal!

Similarly we can show that angles B and Y are equal, and angles C and Z are equal.

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There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

is congruent to:

(See Solving SAS Triangles to find out more)

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to:

(See Solving ASA Triangles to find out more)

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

(See Solving AAS Triangles to find out more)

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL   (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:

is congruent to:

(See Pythagoras' Theorem to find out more)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.

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