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
- Prove angles opposite equal sides are equal.
- Bisector of angle in isosceles triangle bisects base.
- Show sum of any two sides > third side.
- Prove exterior angle theorem.
- 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