Class 9 Maths Chapter 7: Triangles

Triangles is Chapter 7 of NCERT Class 9 Maths. It is all about congruence: when two triangles have exactly the same shape and size, the rules used to prove it, and the special properties of isosceles triangles that follow from these rules. These Class 9 Maths Chapter 7 Triangles notes cover every rule with simple proofs and solved examples.

What you will learn

• The meaning of congruent figures and how to write a correct congruence statement
• The SAS, ASA, AAS, SSS and RHS congruence rules and when each one can be used
• Using CPCT to prove that sides or angles are equal
• The properties of an isosceles triangle and their converses
• How to write neat, step-by-step geometric proofs that score full marks

Key Concepts and Formulas

Congruence and CPCT

Two figures are congruent if one can be placed exactly on top of the other so that they cover each other completely. For two triangles, this means all three pairs of sides and all three pairs of angles are equal.

△ABC ≅ △PQR means A ↔ P, B ↔ Q, C ↔ R
So AB = PQ, BC = QR, CA = RP
and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R

• The order of the letters in a congruence statement tells you which parts match.
• CPCT stands for Corresponding Parts of Congruent Triangles. Once two triangles are proved congruent, every pair of matching parts is equal.
• Congruent triangles always have equal areas, but two triangles with equal areas need not be congruent.

SAS Congruence Rule

This is the starting point of the chapter and is accepted as an axiom (it is not proved).

AB = PQ, ∠B = ∠Q, BC = QR
⇒ △ABC ≅ △PQR (SAS)

• SAS (Side-Angle-Side): if two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent.
• The angle must lie between the two sides. Two sides and a non-included angle (SSA) do not guarantee congruence.

ASA and AAS Congruence Rules

ASA is proved as a theorem using SAS, and AAS follows from ASA because the angle sum of a triangle is fixed.

ASA: ∠B = ∠Q, BC = QR, ∠C = ∠R ⇒ congruent
AAS: ∠A = ∠P, ∠B = ∠Q, BC = QR ⇒ congruent
∠A + ∠B + ∠C = 180°

• ASA (Angle-Side-Angle): two angles and the included side are equal.
• AAS (Angle-Angle-Side): two angles and any non-included side are equal. If two angles match, the third angles also match.
• AAA is not a congruence rule. Triangles with equal angles can be of different sizes.

Isosceles Triangle Properties

An isosceles triangle has two equal sides. The angles at the ends of the third side (the base) are called base angles.

If AB = AC, then ∠B = ∠C
If ∠B = ∠C, then AB = AC
Equilateral: each angle = 180° ÷ 3 = 60°

• Theorem: angles opposite to equal sides of a triangle are equal.
• Converse: sides opposite to equal angles of a triangle are equal.
• The bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base.

SSS and RHS Congruence Rules

SSS needs all three sides, while RHS is a special rule only for right triangles.

SSS: AB = PQ, BC = QR, CA = RP ⇒ congruent
RHS: ∠B = ∠Q = 90°, AC = PR, AB = PQ
     ⇒ △ABC ≅ △PQR

• SSS (Side-Side-Side): if all three sides of one triangle are equal to the three sides of another, the triangles are congruent.
• RHS (Right angle-Hypotenuse-Side): in two right triangles, if the hypotenuse and one side are respectively equal, the triangles are congruent.
• RHS can only be used when both triangles have a right angle.

Solved Examples

Example 1: In △ABC, AB = AC and ∠B = 68°. Find ∠A.

AB = AC ⇒ ∠C = ∠B = 68°
∠A = 180° − 68° − 68°
∴ ∠A = 44°

Example 2: In isosceles △PQR with PQ = PR, the vertex angle ∠P = 50°. Find the base angles.

PQ = PR ⇒ ∠Q = ∠R
∠Q + ∠R = 180° − 50° = 130°
∠Q = 130° ÷ 2
∴ ∠Q = ∠R = 65°

Example 3: In △XYZ, ∠Y = ∠Z = 55° and XY = 7.4 cm. Find XZ.

XZ is opposite ∠Y, XY is opposite ∠Z
∠Y = ∠Z ⇒ XZ = XY
∴ XZ = 7.4 cm

Example 4: △ABC ≅ △DEF with AB = 6 cm, ∠B = 40° and ∠C = 75°. Find ∠D and DE.

∠A = 180° − 40° − 75° = 65°
∠D = ∠A (CPCT), DE = AB (CPCT)
∴ ∠D = 65°, DE = 6 cm

Example 5: Line segments AB and CD bisect each other at O. Show that AC = BD.

In △AOC and △BOD:
OA = OB (O is mid-point of AB)
OC = OD (O is mid-point of CD)
∠AOC = ∠BOD (vertically opposite)
△AOC ≅ △BOD (SAS)
∴ AC = BD (CPCT)

Example 6: △PQR and △STU are right-angled at Q and T, with PR = SU = 13 cm and PQ = ST = 5 cm. Name the congruence rule and find QR.

∠Q = ∠T = 90°, PR = SU, PQ = ST
△PQR ≅ △STU (RHS) ⇒ QR = TU
QR = √(13² − 5²) = √(169 − 25)
QR = √144
∴ QR = TU = 12 cm

Example 7: In △ABC, AB = AC and the bisector of ∠A meets BC at D. If BC = 10 cm, find BD and ∠ADB.

AB = AC, ∠BAD = ∠CAD, AD = AD
△ABD ≅ △ACD (SAS)
BD = DC = 10 ÷ 2 = 5 cm (CPCT)
∠ADB = ∠ADC and ∠ADB + ∠ADC = 180°
∴ BD = 5 cm, ∠ADB = 90°

Common Mistakes to Avoid

• Writing the vertices in the wrong order, for example △ABC ≅ △QPR when A actually matches P. Always match the letters before writing the statement.
• Using SSA or AAA as if they were congruence rules. They are not, and examiners deduct marks for this.
• Using CPCT before proving the triangles congruent, or using ASA when the side is not between the two angles (that case is AAS).
• Skipping reasons. In CBSE exams, every step of a proof needs a reason in brackets such as (given), (common) or (vertically opposite angles).

Quick Revision

Rules: SAS, ASA, AAS, SSS, RHS
Not rules: SSA, AAA
AB = AC ⇔ ∠C = ∠B
Equilateral triangle: each angle = 60°
Angle sum of a triangle = 180°

• CPCT is used only after congruence is proved
• RHS works only for right triangles

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