Triangle similarity criteria

🏆Practice similarity theorems

Conditions for the similarity between two triangles

To demonstrate the similarity between triangles it is not necessary to show again and again the relationship between the three pairs of sides and the equivalence between all the corresponding angles. This would require too much unnecessary work.

There are three criteria by which we can see the similarity between triangles:

Conditions for the similarity between two triangles
  • Angle - Angle (AA): two triangles are similar if they have two equal angles.
  • Side - Angle - Side (SAS): Two triangles are similar if the ratio between two pairs of sides and also the angle they form are equal.
  • Side - Side - Side (SSS): Two triangles are similar if the ratio between all their sides (similarity ratio) is equal in both triangles.
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Test yourself on similarity theorems!

einstein

Angle B is equal to 60°

Angle C is equal to 55°

Angle E is equal to 60°

Angle F is equal to 50°

Are these triangles similar?

AAABBBCCCDDDEEEFFF

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There are three criteria by which we can see the similarity between triangles:


Similarity criterion 1 - Angle - Angle (AA)

Definition: Two triangles are similar if they have two respectively equal angles.


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Example 1

Given the two triangles:

ΔABCΔ ABC
ΔDEFΔ DEF

Image given the two triangles

Likewise we know that

A=D∢A=∢D
B=D∢B=∢D

Prove that the two triangles are similar.

The proof is immediate. We will rely on the criterion we have just learned. Let's note that we have been given two respectively equal angles in the two triangles. We will write it down as follows:

A=D∢A=∢D (angle)
B=E∢B=∢E (angle).

It follows that:
ΔABCΔ ABC ~ ΔDEFΔ DEF (According to the similarity criterion Angle - Angle).

We observe that, if in two triangles there is equivalence between two pairs of corresponding angles, there must be equivalence between the third pair, since the sum of the internal angles of a triangle is always180° 180° . This is the explanation of the first criterion of similarity: Angle - Angle.


Similarity criterion 2 - Side, Angle, Side (SAS)

Definition: Side - Angle - Side (SAS): Two triangles are similar if the ratio between two pairs of sides and also the angle they form are equal.

Do you know what the answer is?

Example 2

Given the two triangles
ΔABCΔ ABC
ΔDEFΔ DEF

Example 2 Given the two triangles

We also know that:
BC=4BC = 4
CA=6CA = 6
EF=2EF = 2
FD=3FD = 3
F=C∢F=∢C

Prove that the triangles are similar and calculate the similarity ratio.

Demonstration:
We will rely on the second criterion of similarity that we just learned: Side - Angle - Side.

Let's pay attention to the relationship between the length of the sides:
FDCA=36=12\frac{FD}{CA}=\frac{3}{6}=\frac{1}{2}

EFBC=24=12\frac{EF}{BC}=\frac{2}{4}=\frac{1}{2}

That is, we see that the ratio between two pairs of sides is equal.
Also the datum
F=C∢F=∢C

That is, the angles formed between the two pairs of sides are equal.
From the above we can deduce that

ΔABCΔ ABC ~ ΔDEFΔ DEF (According to the similarity criterion Side - Angle - Side).

The similarity ratio is 1:2 1:2


Similarity criterion 3 - Side, Side, Side (SSS)

Side - Side - Side (SSS): Two triangles are similar if the proportion between all their sides (similarity ratio) is equal in both triangles.
Given the two triangles:
ΔABCΔ ABC
ΔDEFΔ DEF

Likewise we know that:
DE=2DE = 2
EF=3EF = 3
FD=5FD = 5
AB=3AB = 3
BC=4.5BC = 4.5
AC=7.5AC = 7.5

All data are shown in the illustration:

two triangles with all the data shown in the illustration

Prove that the two triangles are similar.

Proof: We will use the similarity criterion we just learned, Side, Side, Side.

We will see that the following is true

FDCA=57.5=23\frac{FD}{CA}=\frac{5}{7.5}=\frac{2}{3}

DEAB=23\frac{DE}{AB}=\frac{2}{3}

EFBC=34.5=23 \frac{EF}{BC}=\frac{3}{4.5}=\frac{2}{3}

That is to say that the proportion between the three sides of a triangle and the three sides of the other is equal.

Therefore, we can deduce that ΔABCΔ ABC ~ ΔDEF Δ DEF (According to the similarity criterion Side - Side - Side).

The similarity ratio is the relation we saw, that is, 2:3 2:3


Examples and exercsies with solutions

Exercise #1

Look at the following two triangles:

AAABBBCCCDDDEEEFFFAngles B and D are equal.
Angles A and F are equal.

Which side corresponds to AB?

Video Solution

Step-by-Step Solution

As we have two equal angles, we will use the angle-angle theorem to simulate triangles.

We will compare the vertices:A=F,B=D A=F,B=D

According to the data it seems that:

Side AC corresponds to side EF.

Side BC corresponds to side DE.

Therefore, side AB corresponds to side FD.

Answer

FD FD

Exercise #2

Look at the two triangles below:

AAABBBCCCDDDEEEFFF

Angle B is equal to angle E.
Angle A is equal to angle D.

Which angle corresponds to angle C?

Video Solution

Step-by-Step Solution

As we have two pairs of corresponding angles, we will use the angle-angle theorem for triangle similarity.

Now that we know all angles are equal to each other, we note that the remaining angle that is equal and corresponds to angle C is angle F.

Answer

F F

Exercise #3

Angle B is equal to 40°

Angle C is equal to 60°

Angle E is equal to 40°

Angle F is equal to 60°

Are the triangles similar?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Given that the data shows that there are two pairs with equal angles:

B=E=40 B=E=40

C=F=60 C=F=60

The triangles are similar according to the angle-angle theorem, therefore triangle ABC is similar to triangle DEF.

Answer

Yes

Exercise #4

Angle B is equal to 70 degrees

Angle C is equal to 35 degrees

Angle E is equal to 70 degrees

Angle F is equal to 35 degrees

Are the triangles similar?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

The triangles are similar according to the angle-angle theorem.

Having two pairs of equal angles is sufficient to conclude that the triangles are similar.

Answer

Yes

Exercise #5

Look at the two triangles below:

AAABBBCCCDDDEEEFFF

Angle B is equal to angle F.

Angle C is equal to angle D.

Which angle corresponds to angle A?

Video Solution

Step-by-Step Solution

We use the angle-angle theorem to simulate triangles.

Let's observe the data we already have:

Angles B and F are equal.

Angle C is equal to angle D.

Therefore, the remaining angles must also be equal: angles A and E.

Answer

E E

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