Geometry — Study Notes for Maharashtra Police Bharti
Overview
Geometry is a fundamental topic in the Mathematics section of the Maharashtra Police Bharti exam. Questions typically test your understanding of basic properties of lines, angles, triangles, and circles. You won't face complex proofs—instead, expect direct application of standard formulas and theorems to find missing angles, side lengths, or identify figure properties.
Most questions can be solved in under a minute if you know the standard results. The key is memorising essential angle relationships and triangle/circle properties, then practising quick calculations.
Focus areas: angle sum properties, types of triangles, Pythagoras theorem, properties of parallel lines with transversal, and basic circle theorems involving chords, tangents, and central angles.
Key Concepts
- Types of Angles: Acute (< 90°), Right (= 90°), Obtuse (> 90° but < 180°), Straight (= 180°), Reflex (> 180° but < 360°)
- Complementary & Supplementary: Two angles are complementary if sum = 90°; supplementary if sum = 180°
- Vertically Opposite Angles: When two lines intersect, opposite angles are equal
- Parallel Lines + Transversal: Creates 8 angles with special relationships — alternate angles are equal, corresponding angles are equal, co-interior angles sum to 180°
- Triangle Angle Sum: Interior angles of any triangle always sum to 180°
- Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles
- Pythagoras Theorem: In a right triangle, (hypotenuse)² = (base)² + (perpendicular)²
- Circle Basics: Angle at centre = 2 × angle at circumference (same arc); angle in semicircle = 90°; tangent is perpendicular to radius at point of contact
Formulas / Key Facts
Lines and Angles
| Property | Formula/Rule |
|---|---|
| Linear pair | Adjacent angles on straight line sum to 180° |
| Vertically opposite | Equal to each other |
| Corresponding angles | Equal when lines are parallel |
| Alternate interior angles | Equal when lines are parallel |
| Co-interior (same-side interior) | Sum = 180° when lines are parallel |
Triangles
| Type | Property |
|---|---|
| Equilateral | All sides equal, all angles = 60° |
| Isosceles | Two sides equal, base angles equal |
| Scalene | All sides and angles different |
| Sum of angles | Always 180° |
| Exterior angle | = Sum of two opposite interior angles |
| Pythagoras | a² + b² = c² (right triangle, c = hypotenuse) |
Common Pythagorean Triplets
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Circles
| Property | Result |
|---|---|
| Diameter | 2 × radius |
| Central angle | = 2 × inscribed angle (same arc) |
| Angle in semicircle | Always 90° |
| Tangent-radius | Perpendicular at contact point |
| Equal chords | Equidistant from centre |
| Tangents from external point | Equal in length |
Worked Examples
Example 1: Finding an Unknown Angle
Problem: In a triangle, two angles are 65° and 48°. Find the third angle.
Solution:
- Sum of angles in triangle = 180°
- Third angle = 180° − 65° − 48°
- Third angle = 67°
Example 2: Parallel Lines with Transversal
Problem: Two parallel lines are cut by a transversal. One angle is 115°. Find its co-interior angle.
Solution:
- Co-interior angles are supplementary (sum = 180°)
- Required angle = 180° − 115°
- Required angle = 65°
Example 3: Pythagoras Theorem
Problem: A right triangle has base 6 cm and perpendicular 8 cm. Find the hypotenuse.
Solution:
- Using a² + b² = c²
- 6² + 8² = c²
- 36 + 64 = c²
- c² = 100
- c = 10 cm
Note: This is the 3-4-5 triplet multiplied by 2 (6-8-10)
Example 4: Circle — Angle at Centre
Problem: An inscribed angle on a circle is 35°. Find the central angle subtending the same arc.
Solution:
- Central angle = 2 × inscribed angle
- Central angle = 2 × 35°
- Central angle = 70°
Example 5: Exterior Angle
Problem: In triangle ABC, exterior angle at C is 120°. If angle A = 45°, find angle B.
Solution:
- Exterior angle = sum of two opposite interior angles
- 120° = angle A + angle B
- 120° = 45° + angle B
- Angle B = 75°
Common Mistakes
- Confusing complementary and supplementary
- Wrong: Assuming complementary means 180°
- Correct: Complementary = 90°, Supplementary = 180°
- Mixing up alternate and co-interior angles
- Wrong: Thinking co-interior angles are equal
- Correct: Alternate angles are equal; co-interior angles are supplementary (sum = 180°)
- Applying Pythagoras to non-right triangles
- Wrong: Using a² + b² = c² for any triangle
- Correct: Pythagoras works ONLY for right-angled triangles
- Forgetting to double or halve in circle problems
- Wrong: Central angle = inscribed angle
- Correct: Central angle = 2 × inscribed angle for the same arc
- Exterior angle confusion
- Wrong: Exterior angle = one interior angle
- Correct: Exterior angle = sum of the TWO non-adjacent interior angles
Quick Reference
- Triangle angles: Always sum to 180°
- Exterior angle of triangle: = Sum of two opposite interior angles
- Pythagoras triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25
- Parallel lines: Alternate angles equal, co-interior angles = 180°
- Angle in semicircle: Always 90°
- Tangent rule: Perpendicular to radius; two tangents from external point are equal