Triangle similarity criteria

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:

  • Angle - Angle (AA): two triangles are similar if they have two equal angles.
  • Side - Angle - Side (LAL): Two triangles are similar if the ratio between two pairs of sides and also the angle they form are equal.
  • Side - Side - Side (LLL): Two triangles are similar if the ratio between all their sides (similarity ratio) is equal in both 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:


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). QED.

Let us observe that, if in two triangles there is equivalence between two pairs of corresponding angles, there will necessarily 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 (LAL)

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

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

QED.


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

Side - Side - Side (LLL): 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

QED