Criteria for Similarity of Triangles (Theorem and Proof) | Examples (2025)

In Mathematics, a triangle is a closed two-dimensional figure or polygon with the least number of sides. A triangle has three sides and three angles. The most important property of a triangle is the sum of the interior angles of a triangle is equal to 180°. In this article, let us discuss the important criteria for the similarity of triangles with their theorem and proof and many solved examples.

Conditions for Similarity of Two Triangles

Two triangles are said to be similar triangles,

  • If their corresponding angles are equal.
  • If their corresponding sides are in the same proportion/ratio.

Consider two triangles ABC and DEF.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (1)

The two triangles are said to be similar triangles, if

  1. ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
  2. AB/DE = BC/EF = CA/FD

In this scenario, we can say that the two triangles ABC and DEF are similar.

Important Criteria for Similarity of Triangles

The four important criteria used in determining the similarity of triangles are

  • AAA criterion (Angle-Angle-Angle criterion)
  • AA criterion (Angle-Angle criterion)
  • SSS criterion (Side-Side-Side criterion)
  • SAS Criterion (Side-Angle-Side criterion)

Now, let us discuss all these criteria for the similarity of triangles in detail.

AAA Similarity Criterion for Two Triangles

The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.

Proof:

Consider two triangles ABC and DEF, such that ∠A = ∠D, ∠B = ∠E and ∠C = ∠F as shown in the figure.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (2)

Now, cut DP= AB and DQ= AC and join PQ. Hence, we can say that a triangle ABC is congruent to the triangle DPQ.

(i.e) ∆ ABC ≅ ∆ DPQ, which gives ∠B = ∠P = ∠E and also, the line PQ is parallel to EF.

Therefore, by using the basic proportionality theorem, we can write

DP/PE = DQ/QF.

(i.e) AB/DE = AC/DF. …(1)

Similarly,

AB/DE = BC/EF …(2)

From (1) and (2), we can write:

AB/DE = BC/EF = AC/DF.

Therefore, the two triangles ABC and DEF are similar.

AA Similarity Criterion for Two Triangles

The Angle-Angle (AA) criterion for similarity of two triangles states that “If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar”.

The AA criterion states that if two angles of a triangle are respectively equal to the two angles of another triangle, we can prove that the third angle will also be equal on both the triangles. This can be done with the help of the angle sum property of a triangle.

SSS Similarity Criterion for Two Triangles

The Side-Side-Side (SSS) criterion for similarity of two triangles states that “If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar”.

Proof:

Consider the same figure as given above. It is observed that DP/PE = DQ/QF and also in the triangle DEF, the line PQ is parallel to the line EF.

So, ∠P = ∠E and ∠Q = ∠F.

Hence, we can write: DP/DE = DQ/DF= PQ/EF.

The above expression is written as

DP/DE = DQ/DF=BC/EF.

It means that PQ = BC.

Hence, the triangle ABC is congruent to the triangle DPQ.

(i.e) ∆ ABC ≅ ∆ DPQ.

Thus, by using the AAA criterion for similarity of the triangle, we can say that

∠A = ∠D, ∠B = ∠E and ∠C = ∠F.

Also, read:
  • Angle Sum Property of Triangles
  • Corresponding Angles
  • Alternate Angles
  • Congruence of Triangles

SAS Similarity Criterion for Two Triangles

The Side-Angle-Side (SAS) criterion for similarity of two triangles states that “If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar”.

Proof:

This theorem can be proved by taking two triangles such as ABC and DEF (Refer to the same figure as given above).

By using the basic proportionality theorem, we get

AB/DE = AC/DF and ∠A = ∠D

In the triangle DEF, the line PQ is parallel to EF.

So, ∆ ABC ≅ ∆ DPQ.

Hence, we can say ∠A = ∠D, ∠B=∠P and ∠C= ∠Q, which means that the triangle ABC is similar to the triangle DEF.

(i.e) ∆ ABC ~ ∆ DEF.

Examples

Now, let us use the criteria for the similarity of triangles to find the unknown angles and sides of a triangle.

Example 1:

Find ∠P in the following triangles.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (3)

Solution:

From the given triangles, ABC and PQR, we can get

AB/RQ = 3.8/7.6 = ½

BC/QP = 6/12 = ½

CA/PR = (3√3)/6√3 = ½

Therefore,

AB/RQ = BC/QP = CA/PR

Hence, by using the SSS similarity criterion for a triangle, we can write

∆ ABC ~ ∆ RQP (i.e) ∆ABC is similar to ∆RQP.

By using corresponding angles of similar triangles,

∠C = ∠P

∠C = 180°- ∠A – ∠B (Using the angle sum property of triangle).

∠C= 180° – 80° – 60°

∠C= 40°.

Since, ∠C= ∠P, the value of ∠p is 40°.

Example 2:

Show that the triangles POQ and SOR are similar triangles, given that PQ is parallel to RS as shown in the figure.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (4)

Solution:

Given that PQ is parallel to RS. (i.e) PQ || RS.

By using alternate angles property, ∠P = ∠S and ∠Q = ∠R.

Also, by using the vertically opposite angles, ∠POQ = ∠SOR.

Hence, we can conclude that a triangle POQ is similar to the triangle SOR.

(i.e) ∆ POQ ~ ∆ SOR (Using AAA similarity criterion for triangles)

Hence, proved.

Video Lesson on BPT and Similar Triangles

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (5)

Practice Problems

1. From the given figure, if ∆ ABE ≅ ∆ ACD, prove that ∆ ADE ~ ∆ ABC.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (6)

2. In the given figure, QR/QS = QT/PR and ∠1= ∠2. Show that ∆ PQS ~ ∆ TQR.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (7)

3. Determine the height of a tower, if a vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long.

Keep visiting BYJU’S – The Learning App and download the app to learn all Maths-related concepts by exploring more videos.

Criteria for Similarity of Triangles (Theorem and Proof) | Examples (2025)

FAQs

Criteria for Similarity of Triangles (Theorem and Proof) | Examples? ›

Two triangles are similar if they meet one of the following criteria. : Two pairs of corresponding angles are equal. : Three pairs of corresponding sides are proportional. : Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

What are the criteria to prove similar triangles? ›

The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.

What postulates and theorems can be used to prove triangle similarity? ›

The AA similarity postulate and theorem makes it even easier to prove that two triangles are similar. In the interest of simplicity, we'll refer to it as the AA similarity postulate. The postulate states that two triangles are similar if they have two corresponding angles that are congruent or equal in measure.

What is the criteria for SAS proof? ›

SAS or Side-Angle-Side Similarity

If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar. Thus, if ∠A = ∠X and AB/XY = AC/XZ then ΔABC ~ΔXYZ.

What are 3 rules that prove two triangles are similar? ›

The triangle similarity criteria are: AA (Angle-Angle) SSS (Side-Side-Side) SAS (Side-Angle-Side)

What are the 5 ways to prove triangles similar? ›

There are five theorems that can be used to show that two triangles are congruent: the Side-Side-Side (SSS) theorem, the Side-Angle-Side (SAS) theorem, the Angle-Angle-Side (AAS) theorem, the Angle-Side-Angle (ASA) theorem, and the Hypotenuse-Leg (HL) theorem.

What is the AAA criteria for similarity of triangles? ›

Euclidean geometry

may be reformulated as the AAA (angle-angle-angle) similarity theorem: two triangles have their corresponding angles equal if and only if their corresponding sides are proportional.

What are the 4 triangle similarity theorems? ›

Similar triangles are easy to identify because you can apply three theorems specific to triangles. These three theorems, known as Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS), are foolproof methods for determining similarity in triangles.

What is the SSS criterion for similarity of triangles proof? ›

The SSS criterion for triangle similarity states that if three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

What is the SAS criterion for similarity of triangles? ›

The SAS criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

How to prove that two triangles are congruent? ›

Congruence of triangles: Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure. These triangles can be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each other.

What is the proof of congruence criteria? ›

SAS Congruence Rule (Side – Angle – Side)

Two triangles are said to be congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle. Proof : In the given figure OA = OB and OD = OC. (i) ∆ AOD ≅ ∆ BOC and (ii) AD || BC.

How can similar triangles be proven? ›

AA (Angle-Angle): If triangles have two of the same angles, then the triangles are similar. SAS (Side-Angle-Side): If triangles have two pairs of proportional sides and equal included angles, then the triangles are similar.

What are AAA criteria? ›

Theorem 6.3 (AAA Criteria) If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangle are similar.Given: Two triangles ∆ABC and ∆DEF such that ∠A = ∠D, ∠B = ∠E & ∠C = ∠F To Prove: ∆ABC ~ ∆DEF Construction: Draw P and Q on DE & DF ...

What is the SSS criteria for similarity of triangles? ›

What is SSS Similarity Criterion for Triangles? The SSS criterion for triangle similarity states that if three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

What are the three shortcuts to prove triangles similar? ›

Similar triangles possess the same characteristics as other similar figures: congruent corresponding angles and proportional corresponding sides. The triangle similarity theorems, which are Angle - Angle (AA), Side - Angle - Side (SAS) and Side - Side - Side (SSS), serve as shortcuts for identifying similar triangles.

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