Geometry (Lines, Angles, Triangles, Circles, etc.)

🔹Basic Definitions

• Point: A location with no size or shape.

• Line: A straight path extending in both directions infinitely.

• Line Segment: A part of a line bounded by two endpoints.

• Ray: A part of a line that starts at a point and extends infinitely in one direction.

• Angle: Formed when two rays meet at a point.

🔹Types of Angles

• Acute Angle: Less than 90°

• Right Angle: Exactly 90°

• Obtuse Angle: Between 90° and 180°

• Straight Angle: Exactly 180°

• Reflex Angle: Between 180° and 360°

• Complementary Angles: Sum = 90°

• Supplementary Angles: Sum = 180°

• Vertically Opposite Angles: Equal when two lines intersect.

Example: If ∠A and ∠B are vertically opposite, then ∠A = ∠B

🔹Lines and Angles Properties

• Sum of angles on a straight line = 180°

• Sum of angles at a point = 360°

• If two parallel lines are cut by a transversal:
  ▪ Corresponding Angles = Equal
  ▪ Alternate Interior Angles = Equal
  ▪ Co-interior Angles = Supplementary

🔹Triangles

A triangle is a polygon with 3 sides and 3 angles.

Types based on Sides:

• Equilateral Triangle: All sides and angles equal.

• Isosceles Triangle: Two sides equal.

• Scalene Triangle: All sides different.

Types based on Angles:

• Acute Triangle: All angles < 90°

• Right Triangle: One angle = 90°

• Obtuse Triangle: One angle > 90°

Important Properties:

• Sum of angles = 180°

• Exterior Angle = Sum of opposite interior angles

• In a right triangle, Pythagoras Theorem:
  Hypotenuse² = Base² + Height²

Area = (1/2) × base × height
Perimeter = Sum of all sides
Heron’s Formula:
If a, b, c are sides and s = (a + b + c)/2
Then Area = √[s(s−a)(s−b)(s−c)]

🔹Quadrilaterals

A 4-sided polygon. Types:

• Square: All sides equal, all angles 90°

• Rectangle: Opposite sides equal, all angles 90°

• Rhombus: All sides equal, opposite angles equal

• Parallelogram: Opposite sides equal and parallel

• Trapezium: Only one pair of sides is parallel

Formulas:

• Area of Square = side²

• Area of Rectangle = length × breadth

• Area of Rhombus = ½ × diagonal₁ × diagonal₂

• Area of Parallelogram = base × height

• Area of Trapezium = ½ × (a + b) × height

🔹Circles

A set of points equidistant from the center.

Key terms:

• Radius (r): Distance from center to any point on circle.

• Diameter = 2 × radius

• Circumference = 2πr or πd

• Area = πr²

• Chord: A line joining two points on the circle.

• Arc: A part of the circumference.

• Sector: Region enclosed by two radii and an arc.

• Segment: Region between a chord and arc.

Formulas:

• Length of Arc = (θ/360) × 2πr

• Area of Sector = (θ/360) × πr²

🔹Polygons

A polygon is a closed figure with three or more straight sides.

Sum of Interior Angles = (n – 2) × 180°
Each Interior Angle of a Regular Polygon = [(n – 2) × 180°]/n
Sum of Exterior Angles = Always 360°

Examples:

• Triangle: 3 sides

• Quadrilateral: 4 sides

• Pentagon: 5 sides

• Hexagon: 6 sides, etc.

🔹Coordinate Geometry (Basics)

• Distance between two points A(x1, y1) and B(x2, y2) = √[(x2 – x1)² + (y2 – y1)²]

• Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

• Slope of line joining two points = (y2 – y1)/(x2 – x1)

• Area of triangle with vertices (x1, y1), (x2, y2), (x3, y3):
  = ½ |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

🔹Important Theorems and Facts

• Pythagoras Theorem: For a right triangle, hypotenuse² = base² + height²

• Basic Proportionality Theorem (Thales): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

• Angle in a semicircle is always 90°

• Sum of angles in any polygon = (n – 2) × 180°

• Exterior angle of any polygon = 360° / n

🔹Practice Examples

Q1. What is the area of a circle with radius 7 cm?
A. 49π cm²
B. 154 cm²
C. 44 cm²
D. 77π cm²
✔ Correct: A
Area = πr² = π × 7² = 49π

Q2. If one angle of a triangle is 90°, and other two angles are equal, find each.
✔ Solution: Remaining sum = 180° – 90° = 90°
⇒ Each of other two angles = 90° ÷ 2 = 45°

Q3. Find the length of the diagonal of a rectangle of length 6 and breadth 8.
✔ Solution: Diagonal² = 6² + 8² = 36 + 64 = 100 ⇒ Diagonal = √100 = 10

Q4. A parallelogram has base 12 cm and height 5 cm. Find area.
✔ Area = base × height = 12 × 5 = 60 cm²

🔹Common Mistakes to Avoid

✘ Forgetting to use Pythagoras only in right-angled triangles
✘ Mixing up radius and diameter
✘ Using wrong formulas for area
✘ Not labeling diagrams in coordinate geometry
✔ Always draw diagrams wherever possible
✔ Memorize key angle and area formulas