1. Introduction to Triangles

A triangle is a polygon with three sides and three angles...

2. Classification of Triangles

Based on Sides:

• Scalene Triangle: All sides are different.
• Isosceles Triangle: Two sides are equal.
• Equilateral Triangle: All sides and angles are equal.

Based on Angles:

• Acute Triangle: All angles less than 90°.
• Right Triangle: One angle is 90°.
• Obtuse Triangle: One angle more than 90°.

3. Properties of a Triangle

• Sum of interior angles = 180°
• Exterior angle = Sum of two opposite interior angles
• Greater side lies opposite greater angle

4. Congruence of Triangles

Two triangles are congruent if their corresponding sides and angles are equal.

Criteria:

• SSS
• SAS
• ASA
• AAS
• RHS (for right triangles)

5. Theorems on Congruence

• Angles opposite equal sides are equal
• Sides opposite equal angles are equal

6. Properties of Inequality in a Triangle

• Sum of any two sides > third side
• Difference of any two sides < third side
• Greater angle lies opposite greater side

7. Triangle Inequality Theorem

For triangle sides a, b, c:

• a + b > c
• b + c > a
• c + a > b

8. Median and Altitude of a Triangle

Median: Joins vertex to midpoint of opposite side.
Altitude: Perpendicular from vertex to opposite side.

9. Angle Bisector and Perpendicular Bisector

• Angle bisector divides an angle into two equal parts.
• Perpendicular bisector divides a side at 90°.

10. Types of Special Triangles

• Equilateral: All sides and angles equal.
• Isosceles: Two sides and base angles equal.
• Right: One 90° angle; hypotenuse is longest side.

11. Some Important Results

• Perpendicular from vertex of isosceles triangle bisects the base.
• Angle bisector divides opposite side in ratio of adjacent sides.
• Sum of angles = 180°

12. Exterior Angle Theorem

Exterior angle = Sum of two opposite interior angles.

13. Inequalities in a Triangle

If AB > AC, then ∠C > ∠B.

14. Points of Concurrency

Point	Formed By	Name	Property
Centroid	Medians	Centre of gravity	Divides median in 2:1
Orthocentre	Altitudes	-	May lie inside/outside
Circumcentre	Perpendicular bisectors	Centre of circumcircle	Equidistant from vertices
Incentre	Angle bisectors	Centre of incircle	Equidistant from sides

15. Real-life Applications

• Construction
• Navigation
• Art and Design
• Astronomy

16. Practice Problems

1. Prove angles opposite equal sides are equal.
2. Bisector of angle in isosceles triangle bisects base.
3. Show sum of any two sides > third side.
4. Prove exterior angle theorem.
5. Construct triangle and verify inequality theorem.

17. HOTS Questions

• Equal area triangles with common base – prove line joining apexes is parallel.
• AB = AC, ∠B = 50° → Find ∠C and ∠A.
• Three angles of 60° – what kind of triangle?

18. Summary

Triangles are classified by sides/angles, congruent triangles follow certain rules, and special points of concurrency define unique properties. Triangle inequalities help establish if a triangle can form with given dimensions.

19. Keywords

Congruence, Centroid, Altitude, Median, Inequality, Exterior Angle, Angle Sum, Orthocentre, Incentre, Circumcentre