Lines, Angles, Triangles, and Their Properties – Complete Notes for Competitive Exams

1. Introduction

Lines, angles, and triangles form the foundation of geometry. These concepts are frequently asked in SSC, Banking, Railways, and other competitive exams.

• Line: Straight path connecting two points, infinitely extending in both directions.
• Angle: Formed when two lines meet at a point (vertex).
• Triangle: A polygon with 3 sides and 3 angles.

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2. Types of Lines

1. Parallel lines: Lines that never meet.
2. Intersecting lines: Lines that meet at a point.
3. Perpendicular lines: Lines meeting at 90° angle.

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3. Types of Angles

1. Acute: < 90°
2. Right: = 90°
3. Obtuse: > 90° and < 180°
4. Straight: = 180°
5. Reflex: > 180° and < 360°
6. Complementary: Sum = 90°
7. Supplementary: Sum = 180°
8. Vertically opposite angles: Equal angles formed by intersecting lines

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4. Triangle Properties

4.1 Types of Triangles

• By sides: Equilateral, Isosceles, Scalene
• By angles: Acute, Right-angled, Obtuse

4.2 Key Properties

1. Sum of angles = 180°
2. Exterior angle = sum of two opposite interior angles
3. Pythagoras theorem: $a^2 + b^2 = c^2$a2+b2=c2 for right-angled triangle
4. Area formulas:
   • $A = \frac{1}{2} \cdot base \cdot height$A=21​⋅base⋅height
   • Heron’s formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$A=s(s−a)(s−b)(s−c)​, $s = \frac{a+b+c}{2}$s=2a+b+c​

4.3 Special Triangles

• Equilateral: All sides = All angles 60°
• Isosceles: Two sides equal → Two angles equal
• Right-angled: Pythagoras theorem applies

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5. Important Tips

• Always draw a figure for clarity
• Use angle sum property for triangles
• Exterior angles can simplify many problems
• Use Pythagoras theorem and trigonometric ratios for right triangles

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Top 25 Practice Questions – Lines, Angles, Triangles

Lines & Angles

Q1. Find the vertically opposite angle if one angle = 70°
Q2. Sum of two supplementary angles = ?
Q3. One angle of a triangle = 50°, another = 60°. Find third angle
Q4. If two lines are parallel and a transversal forms 65°, find alternate angle
Q5. Find the complement of 35°
Q6. Find the supplement of 120°
Q7. In a triangle, one angle = 90°, other = 30°. Find third angle
Q8. Two angles of a triangle are equal. Third angle = 40°. Find equal angles
Q9. Find exterior angle if interior angles = 50° and 60°
Q10. Angles of a triangle in ratio 2:3:4. Find angles

Triangles

Q11. Area of triangle with base 10 cm, height 5 cm
Q12. Area using Heron’s formula: a = 5, b = 6, c = 7
Q13. Find hypotenuse of right triangle with legs 3 cm and 4 cm
Q14. Find one side if perimeter of equilateral triangle = 24 cm
Q15. Find missing side of isosceles triangle with equal sides = 5 cm, base = ?
Q16. Area of equilateral triangle with side 6 cm
Q17. Angles of triangle in ratio 3:4:5. Find angles
Q18. Right-angled triangle, height = 12 cm, base = 5 cm. Find area
Q19. Triangle with sides 3, 4, 5. Find area
Q20. Find third side of triangle using Pythagoras theorem, sides 8 cm and 15 cm

Advanced

Q21. Sum of exterior angles of triangle = ?
Q22. Triangle with angles x, 2x, 3x. Find x
Q23. Find perimeter of equilateral triangle with side 9 cm
Q24. Triangle with angles 45°, 45°, 90°. Find hypotenuse if base = 7 cm
Q25. Triangle with sides 7 cm, 24 cm, 25 cm. Find area

Answer

Answers – Lines, Angles, Triangles

Lines & Angles

Q1. 70°
Q2. 180°
Q3. 70°
Q4. 65°
Q5. 55°
Q6. 60°
Q7. 60°
Q8. 70° each
Q9. 110°
Q10. 40°, 60°, 80°

Triangles

Q11. 25 cm²
Q12. 14.7 cm² (approx)
Q13. 5 cm
Q14. 8 cm
Q15. Depends on area/perimeter; example = 6 cm
Q16. 15.59 cm²
Q17. 60°, 80°, 100°
Q18. 30 cm²
Q19. 6 cm²
Q20. 17 cm

Advanced

Q21. 360°
Q22. x = 30°
Q23. 27 cm
Q24. Hypotenuse = 7√2 = 9.9 cm approx
Q25. 84 cm²