Triangles Notes: Class 9 Maths Chapter 7

Class 9 Maths Chapter 7 Triangles covers congruence of triangles, the SAS, ASA, AAS, SSS and RHS congruence rules, properties of isosceles triangles, and inequalities in a triangle.

Overview of Topics & Subtopics (Click to Jump)

1. Congruence of Triangles
2. Properties of a Triangle
3. Inequalities in a Triangle

1. Congruence of Triangles

Short Note:

• Two figures are congruent if they are of the same shape and same size.

• SAS Rule: two sides and the included angle of one triangle equal to the corresponding two sides and included angle of another.

• ASA Rule: two angles and the included side equal. AAS Rule: two angles and a non-included side equal.

• SSS Rule: all three sides equal. RHS Rule: hypotenuse and one side of a right triangle equal to the hypotenuse and corresponding side of another right triangle.

2. Properties of a Triangle

Short Note:

• Angles opposite to equal sides of an isosceles triangle are equal, and conversely, sides opposite to equal angles are equal.

• In any triangle, the side opposite the larger angle is longer, and the angle opposite the longer side is larger.

3. Inequalities in a Triangle

Short Note:

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

NCERT Textbook Exercise Questions

In quadrilateral ACBD, AC = AD and AB bisects ∠A. Show that ΔABC ≅ ΔABD.

Answer: In ΔABC and ΔABD: AC = AD (given), ∠CAB = ∠DAB (AB bisects ∠A), AB = AB (common). So by SAS rule, ΔABC ≅ ΔABD.

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that AD bisects BC.

Answer: In ΔADB and ΔADC: ∠ADB = ∠ADC = 90°, AB = AC (given), AD = AD (common). By RHS rule, ΔADB ≅ ΔADC, so by CPCT, BD = DC, i.e., AD bisects BC.

ABC is a triangle in which ∠B = ∠C. Show that AB = AC (i.e., prove the converse of the isosceles triangle property).

Answer: Draw AD ⊥ BC. In ΔABD and ΔACD: ∠B = ∠C (given), ∠ADB = ∠ADC = 90°, AD = AD (common). By AAS rule, ΔABD ≅ ΔACD, so by CPCT, AB = AC.

Show that in a right-angled triangle, the hypotenuse is the longest side.

Answer: In right ΔABC with ∠B = 90°, the sum of angles A and C is 90°, so each of ∠A and ∠C is less than 90°, making ∠B the largest angle. Since the side opposite the largest angle is the longest, the hypotenuse (opposite ∠B) is the longest side.