Triangles

Board: CBSE | Class: Class 9

Comprehensive study notes for Triangles by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Congruence of Triangles

Two triangles are congruent if they have the same shape and size. Corresponding parts of congruent triangles are equal (CPCT).

SSS Congruence

If three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent (SSS Rule).

SAS Congruence

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

ASA Congruence

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

RHS Congruence

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and corresponding leg of another right triangle, they are congruent (RHS Rule).

Inequalities in Triangles

In a triangle: (i) The sum of any two sides is greater than the third side. (ii) The angle opposite the longer side is larger. (iii) The side opposite the larger angle is longer.

Important Formulas

SSS	AB = DE, BC = EF, CA = FD ⇒ ΔABC ≅ ΔDEF
SAS	AB = DE, ∠A = ∠D, AC = DF ⇒ ΔABC ≅ ΔDEF
ASA	∠A = ∠D, AB = DE, ∠B = ∠E ⇒ ΔABC ≅ ΔDEF
RHS	∠B = ∠E = 90°, AC = DF (hypotenuse), AB = DE ⇒ ΔABC ≅ ΔDEF
Triangle Inequality	AB + BC > AC, AB + AC > BC, BC + AC > AB

Solved Examples

Example 1: In ΔABC, AB = AC and ∠B = 50°. Find ∠A and ∠C.

Solution: AB = AC ⇒ ∠C = ∠B = 50° (angles opposite equal sides). ∠A = 180 – (50+50) = 80°

Example 2: Check if a triangle can have sides 3 cm, 4 cm, 8 cm.

Solution: 3 + 4 = 7 < 8. Since sum of two sides is NOT greater than the third side, No, such a triangle is not possible.

Example 3: In ΔABC, ∠A = 80° and AB = AC. Find ∠B.

Solution: AB = AC ⇒ ∠B = ∠C. ∠A + ∠B + ∠C = 180 ⇒ 80 + 2∠B = 180 ⇒ ∠B = 50°

Practice Questions

1. In ΔABC, ∠A = 40° and ∠B = 60°. Which side is the longest?
2. Prove that the angles opposite equal sides of a triangle are equal.
3. Show that ΔABC ≅ ΔDEF using SAS rule given AB = DE, AC = DF, ∠A = ∠D.
4. Two sides of a triangle are 5 cm and 8 cm. What can be the range of the third side?
5. In ΔABC, AB > AC and ∠B = 40°. Which angle is larger: ∠C or ∠B?

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