Class 10 - Similarity of Triangles: Rules, Proofs & Examples

The similarity of triangles is an important concept in geometry that helps compare triangles based on their angles and ratios of their sides. Two triangles are said to be similar if they have the same shape, even if their sizes are different. The similarity of triangles is used in solving problems related to proportions and geometric proofs. In this guide, you'll learn the criteria for triangle similarity, along with clear explanations and examples to build a strong understanding of the topic.

Table of Contents

• What are Similar Triangles?
• Condition for Similarity of Triangles
• AA similarity criteria
• SAS similarity criteria
• SSS similarity criteria
• Properties of Similar Triangles
• Solved Examples on Similarity of Triangles

What are Similar Triangles?

Two triangles are said to be similar when they have the same shape, even if their sizes are different. This happens when their corresponding angles are equal and their corresponding sides are in the same ratio; that is, they are proportional.

In simple terms, similar triangles may not have the same side lengths, but the relationship between their sides remains consistent, and their angles match exactly.

To represent similarity, we use the symbol “∼”. For example, if triangle ABC is similar to triangle PQR, we write: △ABC∼△PQR

This notation shows that both triangles have the same shape and proportional sides.

Read more: Important Question on Triangles - Class 10

Condition for Similarity of Triangles

Two triangles are similar if:

(i) Their corresponding angles are equal, and

(ii) their corresponding sides are in the same ratio (or proportion).

AA similarity criteria

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

If ∠A=∠D, ∠B=∠E Then: △ABC∼△DEF. The third angle automatically becomes equal because of the angle sum property of a triangle.

SAS similarity criteria

If:

• Two sides are proportional, and
• The included angle is equal
• Then triangles are similar.
• △ABC∼△XYZ

SSS similarity criteria

If all three sides of one triangle are proportional to another:

ABXY=BCYZ=ACXZ

Then triangles are similar. △ABC∼△XYZ

Properties of Similar Triangles

Here are a few important properties of similar triangles:

• Corresponding angles are equal
• Corresponding sides are proportional
• Ratio of areas = square of ratio of sides
• Area of ΔDEFArea of ΔABC=[DEAB]2

Solved Examples on Similarity of Triangles

Example 1: In the ΔABC length of the sides are given as AP = 4 cm , PB = 8 cm and BC = 18 cm. Also PQ||BC. Find PQ.

Solution: Given AP = 4 cm , PB = 8 cm and BC = 18 cm and PQ||BC

In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles, AP transversal cuts PQ||BC)
= ΔABC ~ ΔAPQ (AA criterion for similar triangles)
= AP/AB = PQ/BC
= 4/12 = PQ/18
= PQ = 6 cm

Example 2: In triangles PQR and XYZ:

∠P=70°, ∠Q=50°, ∠X=70°, and ∠Z=60°. Check whether triangles are similar.

Solution: In triangle PQR, by angle sum property;

∠P + ∠Q + ∠R = 180°,
70° + 50° + ∠R = 180°
120° + ∠R = 180°
⇒ ∠R = 60°
In triangle XYZ, by angle sum property; ∠X + ∠Y + ∠Z = 180°
70° + ∠Y + 60° = 180°
130° + ∠Y = 180°
⇒ ∠Y = 50°
∠Q = ∠Y = 50° and ∠Z = ∠R = 60°
Therefore, by the Angle-Angle (AA) rule, ΔPQR~ΔXYZ.

Example 3: Show that the given triangles are equal to each other.

Solution: Sides of triangle ABC = 4 cm, 6 cm, 8 cm.
Sides of triangle DEF = 2 cm, 3 cm, 4 cm.
Compare ratios of corresponding sides: ABDE=42   BCEF=63   ACDF=84.
All three ratios are equal. ABDE=BCEF=ACDF.
Therefore, the triangles are similar by SSS similarity criterion.