Chapter Overview & Weightage
Triangles is Chapter 7 of CBSE Class 9 Maths. It covers the congruence of triangles, properties of isosceles triangles, and inequalities in triangles. This chapter provides the geometric reasoning foundations for circles, quadrilaterals, and coordinate geometry in higher classes.
| Exam Year | Marks Allocated | Question Types |
|---|---|---|
| 2024 | 12–15 marks | 2 short + 2 long + 1 HOTS |
| 2023 | 10–12 marks | 1 MCQ + 2 short + 1 long |
| 2022 | 12 marks | 2 short + 2 long |
| 2021 | 10 marks | 2 short + 1 long + 1 proof |
| 2020 | 12 marks | 1 short + 2 long + 1 proof |
Triangles is one of the highest-weightage chapters in Class 9, contributing 10–15 marks consistently. Proof questions are the most common format — you need to know all five congruence rules (SAS, ASA, AAS, SSS, RHS) and the properties of isosceles triangles thoroughly. Most marks come from correctly stating which rule you’re applying and why.
Key Concepts You Must Know
Congruent triangles: Two triangles are congruent if their corresponding sides and angles are equal. Congruent triangles have the same shape AND size.
Congruence rules (criteria):
- SAS (Side-Angle-Side): Two sides and the included angle of one triangle equal corresponding sides and included angle of another.
- ASA (Angle-Side-Angle): Two angles and the included side of one triangle equal corresponding angles and included side of another.
- AAS (Angle-Angle-Side): Two angles and a non-included side.
- SSS (Side-Side-Side): All three sides of one triangle equal the corresponding three sides of another.
- RHS (Right angle-Hypotenuse-Side): Only for right-angled triangles — right angle, hypotenuse, and one other side are equal.
Important theorems:
- Theorem 7.2 (Isosceles triangle): If two sides of a triangle are equal, the angles opposite to those sides are equal. (Converse also holds)
- Theorem 7.3: The bisector from the vertex angle of an isosceles triangle bisects the base and is perpendicular to the base.
- Theorem 7.6: If two sides of a triangle are unequal, the greater angle is opposite to the longer side (and converse).
- Theorem 7.7: Sum of any two sides of a triangle is greater than the third side (Triangle inequality).
- Theorem 7.8: Of all line segments from an external point to a line, the perpendicular is the shortest.
CPCT (Corresponding Parts of Congruent Triangles): Once you prove two triangles are congruent, you can state any other pair of corresponding sides or angles are equal using “by CPCT.”
Important Formulas
SAS: 2 sides + included angle between them
ASA: 2 angles + included side between them
AAS: 2 angles + any corresponding non-included side
SSS: all 3 sides
RHS: right angle + hypotenuse + one leg (right triangles only)
NOT valid: AAA (proves similarity, not congruence) and SSA/ASS (ambiguous)
If AB = AC in △ABC, then ∠B = ∠C
Conversely: If ∠B = ∠C, then AB = AC
The altitude from vertex A to base BC: bisects BC AND is perpendicular to BC (for isosceles only)
Equilateral triangle: all three sides equal → all three angles = 60°
Solved Previous Year Questions
PYQ 1: (CBSE 2023, 4 marks)
Q: In △ABC, the bisector AD of ∠A is perpendicular to BC. Show that △ABC is isosceles.
Solution:
Given: AD bisects ∠A (so ∠BAD = ∠CAD) and AD ⊥ BC (so ∠ADB = ∠ADC = 90°)
In △ABD and △ACD:
- ∠ADB = ∠ADC = 90° (given, AD ⊥ BC)
- AD = AD (common side)
- ∠BAD = ∠CAD (AD bisects ∠A)
By ASA congruence: △ABD ≅ △ACD
Therefore AB = AC (by CPCT)
Since AB = AC, △ABC is isosceles. (Proved)
PYQ 2: (CBSE 2024, 3 marks)
Q: Prove that the angles opposite to equal sides of an isosceles triangle are equal.
Solution:
Given: △ABC where AB = AC. Prove: ∠ABC = ∠ACB.
Construction: Draw AD, the bisector of ∠A, meeting BC at D.
In △ABD and △ACD:
- AB = AC (given)
- ∠BAD = ∠CAD (AD bisects ∠A)
- AD = AD (common)
By SAS: △ABD ≅ △ACD
Therefore ∠ABD = ∠ACD (by CPCT), i.e., ∠ABC = ∠ACB. (Proved)
PYQ 3: (CBSE 2022, 5 marks)
Q: ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that △ABD ≅ △BAC. Hence show that BD = AC.
Solution:
In △ABD and △BAC:
- AD = BC (given)
- ∠DAB = ∠CBA (given)
- AB = BA (common side)
By SAS: △ABD ≅ △BAC
Therefore BD = AC (by CPCT). (Proved)
Difficulty Distribution
| Difficulty | % | Question Type |
|---|---|---|
| Easy (30%) | State congruence rule, MCQ on properties | 1–2 mark |
| Medium (40%) | Prove triangles congruent using CPCT for one result | 3–4 mark |
| Hard (30%) | Multi-step proofs, inequality theorems, HOTS | 5 mark + HOTS |
Expert Strategy
For proof questions, always write the answer in the standard format:
Given: State all given conditions
To Prove: State what needs to be proved
Construction: (if any) State any additional lines drawn
Proof: Step-by-step with reasons in parentheses
Conclusion: End with “Hence proved” or “(Proved)”
This format earns full marks for structure, even if your reasoning has minor gaps.
Before writing a proof, scan which congruence rule applies. A quick mental checklist: Do I know the angles? → ASA or AAS. Do I know sides? → SSS. Do I know one angle between two sides? → SAS. Is it right-angled with hypotenuse given? → RHS. Identifying the right rule first prevents you from going in circles.
In inequality problems (longer side, bigger angle), remember: in any triangle, the side opposite the largest angle is the longest side. If ∠A > ∠B, then BC > AC (the side opposite A is larger than the side opposite B).
Common Traps
Trap 1: Trying to use SSA (or ASS) as a congruence criterion. Two sides and a non-included angle do NOT guarantee congruence — this is the “ambiguous case.” Only SAS (included angle) is valid. If you see two sides and an angle, check whether the angle is between the two sides (SAS — valid) or not (SSA — not a valid criterion). Students frequently make this mistake in proof problems.
Trap 2: Using AAA (three equal angles) to prove congruence. AAA proves triangles are similar (same shape), not congruent (same shape AND size). You cannot conclude equal sides from AAA alone. This is a fundamental distinction between similarity and congruence.
Trap 3: Forgetting to state the congruence rule. In proofs, after listing the three matching criteria, you must explicitly write “By SAS” (or the applicable rule) before stating “△ABC ≅ △DEF.” Omitting the rule name costs 1 mark in board exams. The pattern is: three matching conditions → state rule → conclude congruence ��� use CPCT.