Geometry Basics

Geometry deals with properties of shapes, sizes, positions, and dimensions of figures.

Important Topics for SSC:

1. Lines and Angles
2. Triangles (Most Important!)
3. Circles
4. Quadrilaterals
5. Polygons

Lines and Angles

Basic Definitions

• Complementary Angles: Sum = 90°
• Supplementary Angles: Sum = 180°
• Linear Pair: Sum = 180° (adjacent angles)
• Vertically Opposite Angles: Equal

Parallel Lines with Transversal

When a transversal cuts parallel lines:
- Corresponding angles are equal
- Alternate interior angles are equal
- Co-interior angles sum to 180°

Triangles (HIGH Priority!)

Types of Triangles

By Sides:

• Equilateral: All sides equal
• Isosceles: 2 sides equal
• Scalene: All sides different

By Angles:

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

Important Triangle Theorems

1. Angle Sum Property

Sum of all angles in triangle = 180°

2. Exterior Angle Theorem

Exterior angle = Sum of two opposite interior angles

3. Pythagoras Theorem (Right Triangle)

(Hypotenuse)² = (Base)² + (Height)²
h² = b² + p²

4. Important Pythagorean Triplets

3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
9, 40, 41

5. Triangle Inequality

Sum of any 2 sides > Third side
Difference of any 2 sides < Third side

Triangle Centers (SSC Favorite!)

1. Centroid

• Intersection of medians
• Divides median in ratio 2:1

2. Circumcenter

• Intersection of perpendicular bisectors
• Equidistant from all vertices

3. Incenter

• Intersection of angle bisectors
• Equidistant from all sides

4. Orthocenter

• Intersection of altitudes

Similar Triangles

When triangles are similar:

• Corresponding angles are equal
• Corresponding sides are proportional

AAA, SAS, SSS similarity criteria

Circles

Basic Formulas

Circumference = 2πr = πd
Area = πr²

Arc length = (θ/360°) × 2πr
Sector area = (θ/360°) × πr²

Important Circle Theorems

1. Chord Theorems

• Equal chords subtend equal angles at center
• Perpendicular from center bisects chord

2. Tangent Theorems

• Tangent ⊥ radius at point of contact
• Two tangents from external point are equal

3. Angle in Semicircle

• Angle in semicircle = 90°

4. Cyclic Quadrilateral

• Opposite angles sum to 180°

Quadrilaterals

Types & Properties

Rectangle:

• Opposite sides equal
• All angles = 90°
• Diagonals equal and bisect each other

Square:

• All sides equal
• All angles = 90°
• Diagonals equal, perpendicular, bisect each other

Parallelogram:

• Opposite sides equal and parallel
• Opposite angles equal
• Diagonals bisect each other

Rhombus:

• All sides equal
• Opposite angles equal
• Diagonals perpendicular and bisect each other

Trapezium:

• One pair of parallel sides

Important Shortcuts

Shortcut 1: Triangle Area (Heron’s Formula)

s = (a+b+c)/2
Area = √[s(s-a)(s-b)(s-c)]

Shortcut 2: 30-60-90 Triangle

Sides ratio = 1 : √3 : 2
If smallest side = x
Then sides = x, x√3, 2x

Shortcut 3: 45-45-90 Triangle

Sides ratio = 1 : 1 : √2
If equal sides = x
Then hypotenuse = x√2

Shortcut 4: Number of Diagonals

n-sided polygon has n(n-3)/2 diagonals
Triangle (n=3): 0 diagonals
Square (n=4): 2 diagonals
Pentagon (n=5): 5 diagonals

Solved Examples

Q1: Two angles of triangle are 50° and 60°. Find third angle.

Solution:

Sum of angles = 180°
Third angle = 180° - 50° - 60° = 70°

Q2: In right triangle, if base = 6 cm, height = 8 cm, find hypotenuse.

Solution:

h² = 6² + 8²
h² = 36 + 64 = 100
h = 10 cm

Q3: Find angle subtended by 60° sector at center.

Solution:

Given directly: 60°
(Trick question - sector angle IS the central angle!)

Related Topics

💡 Pro Tip: Geometry needs diagram practice! Draw figures for every problem. Memorize Pythagorean triplets for instant solutions.