Quick Revision Notes: Class 9 Mathematics

Chapter: Triangles (त्रिभुज)

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### Concept 1: Introduction to Triangles

A triangle is a closed figure formed by three intersecting sides. It has three sides, three angles, and three vertices.

• Congruent Figures (सर्वांगसम आकृतियाँ): Figures which have the same shape and the same size.
• Congruence of Triangles: Two triangles are congruent if they are exact copies of each other and when superposed, they cover each other completely.
• Symbol for congruence: ≅

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### Key Formulas & Criteria for Triangle Congruence

To prove two triangles congruent, we use specific criteria. (Note: ASS or SSA is not a valid criterion).

1. SAS (Side-Angle-Side) Congruence Rule

• Statement: Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
• Notation: AB = PQ, ∠B = ∠Q, BC = QR $\implies$ ΔABC ≅ ΔPQR

2. ASA (Angle-Side-Angle) Congruence Rule

• Statement: Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.
• Note: AAS (Angle-Angle-Side) is also valid as a corollary to ASA.

3. SSS (Side-Side-Side) Congruence Rule

• Statement: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

4. RHS (Right angle-Hypotenuse-Side) Congruence Rule

• Statement: If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

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### Concept 2: Properties of an Isosceles Triangle

An isosceles triangle (समद्विबाहु त्रिभुज) has two equal sides.

1. Theorem: Angles opposite to equal sides of an isosceles triangle are equal.
   • If AB = AC, then ∠B = ∠C.
2. Theorem: The sides opposite to equal angles of a triangle are equal.
   • If ∠B = ∠C, then AB = AC.
3. An equilateral triangle (समबाहु त्रिभुज) has all three sides equal and each angle measures 60°.

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### Concept 3: Inequalities in a Triangle

1. Angle-Side Inequality:
   • If two sides of a triangle are unequal, the angle opposite to the longer side is larger (greater).
   • If AC > AB, then ∠B > ∠C.
2. Side-Angle Inequality:
   • In any triangle, the side opposite to the larger (greater) angle is longer.
   • If ∠B > ∠C, then AC > AB.
3. Triangle Inequality Theorem:
   • The sum of any two sides of a triangle is greater than the third side.
   • AB + BC > AC
   • BC + CA > AB
   • CA + AB > BC
   • (Note: The difference of any two sides is less than the third side).

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### Important Exam Tips for MP Board Students

1. Corresponding Parts: Always write congruent triangles in correct corresponding order (e.g., if ΔABC ≅ ΔPQR, then AB = PQ, BC = QR, AC = PR). Mention the rule used (like SAS, SSS) in brackets.
2. Construction/Diagram: Always draw a neat, labelled pencil diagram for geometry questions.
3. Reasons: While writing proofs in theorems, always write the reason in brackets (e.g., Given, Common side, Alternate interior angles, etc.).