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Mathematics · Class 9 · Chapter 12: Sides and Angles of a Triangle

8 min

Key points

Triangle Inequality — The sum of any two sides of a triangle is greater than the third side
Angle Sum Property — The sum of all interior angles of a triangle is 180°
Exterior Angle Property — An exterior angle of a triangle is equal to the sum of two non-adjacent interior angles
In any triangle, the greater side is opposite to the greater angle and vice versa
In any triangle, the smaller side is opposite to the smaller angle and vice versa
The perpendicular from any vertex to the opposite side is the shortest distance from the vertex to that side
If two sides of a triangle are equal, then the angles opposite to them are also equal (Isosceles triangle property)
The sum of any two sides of a triangle is greater than the third side, and the difference of any two sides is less than the third side

Definitions

Triangle Inequality: In any triangle, the sum of any two sides is greater than the third side
Exterior Angle: An angle formed by one side of a triangle and the extension of an adjacent side
Interior Angle: An angle formed by two sides of a triangle inside the triangle
Isosceles Triangle: A triangle in which two sides are equal and the angles opposite to equal sides are also equal
Scalene Triangle: A triangle in which all three sides are of different lengths and all three angles are of different measures

Worked example

In triangle ABC, angle A = 60°, angle B = 45°. Find angle C and state which theorem you used. (4 marks)

By angle sum property of triangle: sum of all interior angles equals 180°. Therefore ∠A + ∠B + ∠C = 180°. Substituting: 60° + 45° + ∠C = 180°, so ∠C = 75°.

Common mistakes

Students often forget to check triangle inequality when given three sides
Confusing interior and exterior angles in theorem statements
Not drawing proper diagrams with correct labeling of angles and sides
Writing angle sum as 360° instead of 180° for triangles
Forgetting to mention 'non-adjacent' when stating exterior angle theorem

Quick recap

Triangle angle sum = 180° always. Exterior angle equals sum of two non-adjacent interior angles. In isosceles triangle, base angles are equal. Triangle inequality: sum of any two sides must be greater than third side.

Generated by AI from Sindh Textbook Board material; review state: ai_unreviewed. MCQs, past-paper references and examiner notes are held back until a teacher has checked them.