THE TRIANGLES & ITS PROPERTIES

UNFOLDING THE SECRETS OF TRIANGLES 🔺

Introduction

Triangles are one of the most important shapes in geometry. From bridges to buildings, triangles provide strength, balance, and structure.

In this chapter, we explore their types, properties, and hidden rules that make geometry powerful and logical.

Key Topics Covered

• Definition of Triangle
• Types of Triangles
• Properties of Triangles
• Special Lines (Median, Altitude, etc.)
• Pythagoras Theorem
• Triangle Inequality

1. Definition of a Triangle

A triangle is a closed figure with three sides, three vertices, and three angles.

👉 Important Rule:
Sum of all interior angles = 180°

2. Types of Triangles

Based on Sides

• Scalene: All sides different
• Isosceles: Two sides equal
• Equilateral: All sides equal (each angle = 60°)

Based on Angles

• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°

3. Properties of Triangles

• Angle Sum Property: Sum = 180°
• Exterior Angle Property: Exterior = sum of opposite interior angles
• Triangle Inequality: Sum of any two sides > third side

4. Special Properties

• Median → Centroid
• Altitude → Orthocenter
• Perpendicular Bisector → Circumcenter
• Angle Bisector → Incenter

5. Pythagoras Theorem

👉 Applies only to right triangles

6. Inequalities in Triangle

• Sum of two sides > third side
• Difference of two sides < third side

📝 Practice MCQs (50 Questions)

Basic Concepts (1–10)

1. A triangle has:
   A) 2 sides
   B) 3 sides
   C) 4 sides
   D) Infinite sides
2. Sum of angles in triangle:
   A) 90°
   B) 180°
   C) 360°
   D) 270°
3. Triangle has how many vertices?
   A) 2
   B) 3
   C) 4
   D) 5
4. Triangle is a:
   A) Open figure
   B) Closed figure
   C) Curve
   D) Line
5. A triangle has how many angles?
   A) 2
   B) 3
   C) 4
   D) 5
6. Shape with 3 sides is:
   A) Square
   B) Triangle
   C) Circle
   D) Rectangle
7. Interior angles are:
   A) Outside
   B) Inside
   C) Midpoint
   D) None
8. Exterior angle is:
   A) Inside
   B) Outside
   C) Vertex
   D) Side
9. Triangle sides meet at:
   A) Edges
   B) Vertices
   C) Curves
   D) Lines
10. Triangle is:
    A) 1D
    B) 2D
    C) 3D
    D) None

Types of Triangles (11–20)

11. All sides unequal →
    A) Isosceles
    B) Scalene
    C) Equilateral
    D) Right
12. Two sides equal →
    A) Scalene
    B) Isosceles
    C) Equilateral
    D) Acute
13. All sides equal →
    A) Equilateral
    B) Scalene
    C) Isosceles
    D) Right
14. All angles 60° →
    A) Right
    B) Acute
    C) Equilateral
    D) Obtuse
15. One angle = 90° →
    A) Acute
    B) Right
    C) Obtuse
    D) Reflex
16. All angles < 90° →
    A) Acute
    B) Right
    C) Obtuse
    D) Reflex
17. One angle > 90° →
    A) Acute
    B) Right
    C) Obtuse
    D) Straight
18. Triangle (3,4,5) is:
    A) Right
    B) Acute
    C) Obtuse
    D) Equilateral
19. Equal angles →
    A) Scalene
    B) Isosceles
    C) Equilateral
    D) Both B & C
20. Equilateral triangle angles:
    A) 90°
    B) 60°
    C) 45°
    D) 120°

Properties (21–35)

21. Angle sum =
    A) 90°
    B) 180°
    C) 360°
    D) 270°
22. Exterior angle equals:
    A) Adjacent
    B) Opposite angles sum
    C) 180°
    D) 90°
23. If two angles are 50° & 60°, third is:
    A) 70°
    B) 80°
    C) 90°
    D) 60°
24. Triangle inequality:
    A) a+b<c
    B) a+b>c
    C) a+b=c
    D) None
25. 3,4,8 forms triangle?
    A) Yes
    B) No
    C) Maybe
    D) Equal
26. 5,7,8 forms triangle?
    A) Yes
    B) No
    C) Maybe
    D) Equal
27. Exterior angle 120°, one interior 50°, other is:
    A) 70°
    B) 60°
    C) 50°
    D) 80°
28. Sum of any two sides is:
    A) Less
    B) Equal
    C) Greater
    D) None
29. Difference rule:
    A) > third
    B) < third
    C) = third
    D) None
30. Angle opposite bigger side is:
    A) Smaller
    B) Bigger
    C) Equal
    D) None
31. Largest side opposite:
    A) Smallest angle
    B) Largest angle
    C) Equal
    D) None
32. Triangle sides always:
    A) Equal
    B) Unequal
    C) Follow inequality
    D) Same
33. Exterior angle always:
    A) Smaller
    B) Greater
    C) Equal
    D) None
34. Sum of 3 angles:
    A) 180°
    B) 90°
    C) 360°
    D) 270°
35. Triangle always has:
    A) 2 sides
    B) 3 sides
    C) 4 sides
    D) 5 sides

Special Lines & Pythagoras (36–50)

36. Median meets at:
    A) Incenter
    B) Centroid
    C) Orthocenter
    D) Circumcenter
37. Altitude meets at:
    A) Centroid
    B) Orthocenter
    C) Incenter
    D) None
38. Perpendicular bisector →
    A) Centroid
    B) Circumcenter
    C) Incenter
    D) Orthocenter
39. Angle bisector →
    A) Incenter
    B) Centroid
    C) Orthocenter
    D) Circumcenter
40. Hypotenuse is:
    A) Smallest side
    B) Largest side
    C) Equal
    D) None
41. 6,8 → hypotenuse =
    A) 10
    B) 12
    C) 14
    D) 8
42. 5,12 → hypotenuse =
    A) 13
    B) 15
    C) 17
    D) 10
43. Right triangle follows:
    A) a+b=c
    B) a²+b²=c²
    C) a-b=c
    D) None
44. Pythagoras works in:
    A) All triangles
    B) Right triangle
    C) Acute
    D) Obtuse
45. Centroid divides median in:
    A) 1:1
    B) 2:1
    C) 3:1
    D) 1:2
46. Altitude is:
    A) Slanted
    B) Perpendicular
    C) Curve
    D) Parallel
47. Angle bisector divides angle into:
    A) Equal parts
    B) Unequal
    C) Random
    D) None
48. Perpendicular bisector divides side into:
    A) Equal parts
    B) Unequal
    C) 3 parts
    D) None
49. Hypotenuse opposite:
    A) 45°
    B) 60°
    C) 90°
    D) 30°
50. Triangle inequality fails when:
    A) Sum > side
    B) Sum = side
    C) Sum < side
    D) Both B & C

✅ Answer Key

1-B, 2-B, 3-B, 4-B, 5-B
6-B, 7-B, 8-B, 9-B, 10-B
11-B, 12-B, 13-A, 14-C, 15-B
16-A, 17-C, 18-A, 19-D, 20-B
21-B, 22-B, 23-A, 24-B, 25-B
26-A, 27-A, 28-C, 29-B, 30-B
31-B, 32-C, 33-B, 34-A, 35-B
36-B, 37-B, 38-B, 39-A, 40-B
41-A, 42-A, 43-B, 44-B, 45-B
46-B, 47-A, 48-A, 49-C, 50-D

🎯 Smart Tips (Quick Revision)

• 🔺 “All Angles = 180°”
• 🔺 “Big Side → Big Angle”
• 🔺 “Pythagoras = Right Triangle Only”
• 🔺 “Sum of two sides > third side”