Lines, Angles and Shapes
Lines, Angles and Shapes - Theory and Formulas
3.1 Lines and Angles
Types of Angles
Angles can be classified based on their size:
| Type | Size | Example |
|---|---|---|
| Acute angle | Less than 90° | 30°, 45°, 70° |
| Right angle | Exactly 90° | Corner of a book |
| Obtuse angle | More than 90° but less than 180° | 100°, 120°, 150° |
| Reflex angle | More than 180° but less than 360° | 200°, 270°, 300° |
Special Pairs of Angles
- Complementary Angles: Two angles that add up to 90°.
- Supplementary Angles: Two angles that add up to 180°.
Fundamental Angle Rules
- Angles on a Straight Line: The sum of angles on a straight line is 180°.
∑ ∠ on line = 180° - Angles Around a Point: The sum of all angles around a single point is 360°.
∑ ∠ round point = 360° - Vertically Opposite Angles: When two lines intersect, the vertically opposite angles are equal.
Vertically opposite ∠s are equal
Parallel Lines and a Transversal
When a transversal cuts two parallel lines, several pairs of equal or supplementary angles are formed.
- Alternate Angles (Z-shape): a = d, b = c, e = h, f = g. They are equal.
Alt ∠s are equal - Corresponding Angles (F-shape): a = e, b = f, c = g, d = h. They are equal.
Corr ∠s are equal - Co-interior Angles (C or U-shape): c + e = 180°, d + f = 180°. They are supplementary.
Co-int ∠s add up to 180°
Tip: If you find that alternate or corresponding angles are equal, or co-interior angles add up to 180°, then the two lines cut by the transversal are parallel.
3.2 Triangles
Types of Triangles
| Type | Sides | Angles |
|---|---|---|
| Scalene | All different | All different |
| Isosceles | Two equal | Two equal (base angles) |
| Equilateral | All three equal | All 60° |
| Right-angled | - | One 90° angle |
Important Triangle Rules
- Sum of Interior Angles: The sum of the interior angles in any triangle is 180°.
∑ interior ∠s = 180° - Exterior Angle: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Exterior ∠ = Sum of opposite interior ∠s
Important: In an isosceles triangle, if two sides are equal, then the base angles are equal. Conversely, if two angles are equal, the triangle is isosceles.
3.3 Quadrilaterals
What is a Quadrilateral?
A polygon with four sides. The sum of its interior angles is 360°.
∑ interior ∠s = 360°
Types of Quadrilaterals
- Trapezium: One pair of parallel sides.
- Parallelogram: Opposite sides parallel and equal; opposite angles equal.
- Rectangle: All angles 90°; diagonals equal.
- Rhombus: All sides equal; diagonals bisect at 90°.
- Square: All sides equal, all angles 90°.
- Kite: Two pairs of adjacent equal sides; one diagonal bisects the other at 90°.
3.4 Polygons
Polygon Basics
A polygon is a 2D shape with three or more straight sides.
Names based on number of sides:
- Triangle (3)
- Quadrilateral (4)
- Pentagon (5)
- Hexagon (6)
- Heptagon (7)
- Octagon (8)
- Nonagon (9)
- Decagon (10)
Polygon Formulas
- Sum of Interior Angles: For a polygon with n sides:
Sum = (n - 2) × 180° - Interior Angle of a Regular Polygon: (All sides and angles equal)
Each interior ∠ = [(n - 2) × 180°] / n - Sum of Exterior Angles: For any convex polygon:
∑ exterior ∠s = 360° - Exterior Angle of a Regular Polygon:
Each exterior ∠ = 360° / n
Tip: If you know one exterior angle of a regular polygon, you can find the number of sides: n = 360° / (exterior ∠)
3.5 Circles
Key Parts of a Circle
- Centre (O): The fixed point equidistant from all points on the circumference.
- Radius: A line segment from the centre to the circumference.
- Diameter: A line segment passing through the centre, connecting two points on the circumference. Diameter = 2 × Radius
- Circumference: The perimeter of the circle.
- Chord: A line segment joining any two points on the circumference.
- Arc: A part of the circumference.
- Sector: A region bounded by two radii and an arc.
- Segment: A region bounded by a chord and an arc.
- Tangent: A line that touches the circle at exactly one point.
3.6 Construction
Essential Constructions
Using a ruler and a pair of compasses:
- Bisecting an Angle: Dividing an angle into two equal parts.
- Perpendicular Bisector of a Line Segment: Drawing a line perpendicular to a segment at its midpoint.
- Constructing Triangles: Given three sides (SSS), two sides and included angle (SAS), or two angles and a side (ASA).
