CBSE Class 9 Maths Chapter 7 Triangles Notes | Revision Sheet

Key Points to Remember

• Congruence means 'equal in all respects' – same shape and same size.
• Two geometric figures are congruent if they superimpose exactly on each other.
• If two triangles are congruent, their corresponding parts are equal (CPCTC – Corresponding Parts of Congruent Triangles are Congruent).
• There are four main congruence criteria for triangles: SSS, SAS, ASA, and RHS.
• SSS Congruence Rule: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
• SAS Congruence Rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
• ASA Congruence Rule: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
• RHS Congruence Rule: If the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the two triangles are congruent.
• Isosceles Triangle Property: Angles opposite to equal sides of an isosceles triangle are equal. (Theorem 7.2)
• Converse of Isosceles Triangle Property: Sides opposite to equal angles of a triangle are equal. (Theorem 7.3)
• Triangle Inequality Theorem: The sum of any two sides of a triangle is greater than the third side.

Key Definitions

Congruent Figures

Figures that have exactly the same shape and size. They can be superposed on each other to cover each other exactly.

Congruent Triangles

Triangles whose corresponding sides and corresponding angles are equal. If two triangles are congruent, they are denoted by the symbol '≅'.

Corresponding Parts of Congruent Triangles (CPCTC)

A fundamental property stating that if two triangles are congruent, then all their corresponding sides and angles are equal.

Included Angle

The angle formed by two sides of a triangle. For example, in ΔABC, ∠B is the included angle between sides AB and BC.

Included Side

The side of a triangle that is common to two angles. For example, in ΔABC, side BC is the included side between ∠B and ∠C.

Isosceles Triangle

A triangle with at least two sides of equal length. The angles opposite the equal sides are also equal.

Equilateral Triangle

A triangle in which all three sides are of equal length. All three angles in an equilateral triangle are also equal, each measuring 60°.

Median of a Triangle

A line segment joining a vertex to the midpoint of the opposite side.

Altitude of a Triangle

A perpendicular line segment from a vertex to the opposite side (or its extension).

Understanding Congruence of Triangles

In geometry, congruence is a crucial concept, especially when dealing with triangles. Two figures are said to be congruent if they are exact copies of each other, meaning they have the same shape and the same size. Imagine placing one figure precisely on top of another; if they match perfectly, they are congruent. For triangles, we don't always need to check all six corresponding parts (three sides and three angles) to prove congruence. Mathematicians have developed specific criteria, often called congruence rules or postulates, which allow us to prove two triangles are congruent by checking only three specific pairs of corresponding parts.

The most common congruence criteria are:

1. SSS (Side-Side-Side) Congruence Rule: If all three sides of one triangle are respectively equal to the three corresponding sides of another triangle, then the two triangles are congruent. For example, if in ΔABC and ΔPQR, AB = PQ, BC = QR, and CA = RP, then ΔABC ≅ ΔPQR.

1. SAS (Side-Angle-Side) Congruence Rule: This rule applies if **two sides and the *included angle*** (the angle formed by those two sides) of one triangle are respectively equal to two sides and the included angle of another triangle. The term 'included angle' is vital here; if the angle is not between the two sides, the SAS rule cannot be applied directly. For instance, if AB = PQ, BC = QR, and ∠B = ∠Q, then ΔABC ≅ ΔPQR.

1. ASA (Angle-Side-Angle) Congruence Rule: This criterion states that if **two angles and the *included side*** (the side common to both angles) of one triangle are respectively equal to two angles and the included side of another triangle, then the triangles are congruent. Similar to SAS, the 'included side' is key. If ∠B = ∠Q, ∠C = ∠R, and BC = QR, then ΔABC ≅ ΔPQR.

1. RHS (Right Angle-Hypotenuse-Side) Congruence Rule: This rule is specifically for right-angled triangles. It states that if the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the two triangles are congruent. Here, the right angle is inherently part of the condition. For example, if in right-angled ΔABC and ΔPQR (right-angled at B and Q respectively), AC = PR (hypotenuses) and AB = PQ (one side), then ΔABC ≅ ΔPQR.

It's important to note that AAA (Angle-Angle-Angle) is not a congruence criterion for triangles (it implies similarity, not congruence), and SSA (Side-Side-Angle) is also generally not a congruence criterion because it can lead to two possible triangles, unless it's a specific case like RHS.

Steps to Prove Triangle Congruence

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Exam Tips for Triangles

• Draw Neat Diagrams: Always draw clear, labeled diagrams for geometry problems. This helps visualise the problem and identify corresponding parts.
• Identify Given Information: Carefully read the problem statement and mark all given equal sides or angles on your diagram.
• Look for Hidden Information: Often, problems include common sides/angles, vertically opposite angles, or angles/lines formed by parallel lines that can be used as equal parts.
• Correct Order of Vertices: When writing congruence statements (e.g., ΔABC ≅ ΔPQR), ensure the order of vertices reflects the corresponding parts accurately. A common mistake is to write ΔABC ≅ ΔQPR when it should be ΔABC ≅ ΔPQR.
• Don't Assume: Never assume parts are equal unless they are explicitly given or can be proven using a theorem.
• Practice Proofs: The best way to master this chapter is to practice various types of proofs repeatedly. Understand the logic behind each step.
• Memorise Theorems: Know the properties of isosceles triangles and angle/side relationships by heart.

Solved Examples

• Example 1: Using SAS Congruence Problem: In quadrilateral ABCD, AC bisects ∠A and AC bisects ∠C. Prove that ΔADC ≅ ΔABC. Solution: In ΔADC and ΔABC: 1. ∠DAC = ∠BAC (Given, AC bisects ∠A) 2. AC = AC (Common side) 3. ∠DCA = ∠BCA (Given, AC bisects ∠C) Therefore, ΔADC ≅ ΔABC (By ASA Congruence Rule).
• Example 2: Isosceles Triangle Property Problem: In ΔABC, AB = AC. BD and CE are medians. Prove that BD = CE. Solution: Since AB = AC, ΔABC is an isosceles triangle. Thus, ∠ABC = ∠ACB (Angles opposite equal sides). Since BD and CE are medians, D is the midpoint of AC and E is the midpoint of AB. So, AE = EB = AB/2 and AD = DC = AC/2. Since AB=AC, AE=AD. Now, consider ΔEBC and ΔDCB: 1. EB = DC (Since AE=AD and AB=AC, then AB-AE = AC-AD, which means EB=DC is incorrect here. Let's restart this example with a clearer approach. Focus on sides and angles directly for congruence.) Revised Solution for Example 2: Problem: In ΔABC, AB = AC. E and D are midpoints of AB and AC respectively. Prove BD = CE. Solution: In ΔABD and ΔACE: 1. AB = AC (Given) 2. ∠A = ∠A (Common angle) 3. AD = AE (Since D and E are midpoints of equal sides AC and AB respectively, then AC/2 = AB/2) Therefore, ΔABD ≅ ΔACE (By SAS Congruence Rule). By CPCTC, BD = CE. (Hence Proved)
• Example 3: Using RHS Congruence Problem: In two right-angled triangles, one side and the hypotenuse of one triangle are equal to the corresponding side and hypotenuse of the other triangle. Prove the triangles are congruent. Solution: Let the two right-angled triangles be ΔABC and ΔPQR, right-angled at B and Q respectively. Given: 1. AC = PR (Hypotenuse) 2. AB = PQ (One side) 3. ∠ABC = ∠PQR = 90° (Right angle) Therefore, ΔABC ≅ ΔPQR (By RHS Congruence Rule).

Quick Revision Checks

• Q: What does the acronym CPCTC stand for? A: Corresponding Parts of Congruent Triangles are Congruent.
• Q: Can two triangles be congruent if all their angles are equal? A: No, AAA is a criterion for similarity, not congruence. Sides must also be proportional or equal.
• Q: In ΔPQR, if PQ = PR, what can you say about ∠Q and ∠R? A: ∠Q = ∠R, because angles opposite to equal sides of an isosceles triangle are equal.
• Q: State the difference between SAS and SSA congruence rules. A: SAS requires the angle to be included between the two sides, while SSA does not, and SSA is generally not a valid congruence rule as it can lead to ambiguous cases.

Frequently Asked Questions