Angles, parallel lines, polygons, circles, constructions

Resources | Subject Notes | Mathematics

IGCSE Mathematics - Geometry

Geometry

This section covers angles, parallel lines, polygons, circles, and constructions. These topics are fundamental to geometry and appear frequently in the IGCSE Mathematics 0580 syllabus.

Angles

Types of Angles

  • Acute Angle: An angle less than 90 degrees.
  • Right Angle: An angle equal to 90 degrees.
  • Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.
  • Straight Angle: An angle equal to 180 degrees.
  • Reflex Angle: An angle greater than 180 degrees but less than 360 degrees.

Angle Sum Rules

  • Angles on a straight line: The sum of angles on a straight line is 180 degrees.
  • Angles in a triangle: The sum of angles in a triangle is 180 degrees.
  • Angles in a quadrilateral: The sum of angles in a quadrilateral is 360 degrees.

Vertically Opposite Angles

Vertically opposite angles are equal.

Corresponding Angles

Corresponding angles are equal when a transversal intersects two parallel lines.

Alternate Angles

Alternate angles are equal when a transversal intersects two parallel lines.

Consecutive Interior Angles

Consecutive interior angles are supplementary (add up to 180 degrees) when a transversal intersects two parallel lines.

Parallel Lines

Properties of Parallel Lines

  • Alternate interior angles are equal.
  • Consecutive interior angles are supplementary.
  • Corresponding angles are equal.

Proof that Parallel Lines are Concurrently Cut by a Transversal

The interior angles on the same side of the transversal are supplementary.

Polygons

Types of Polygons

  • Triangle: 3 sides, 3 angles. Angle sum = 180 degrees.
  • Quadrilateral: 4 sides, 4 angles. Angle sum = 360 degrees.
  • Pentagon: 5 sides, 5 angles. Angle sum = 540 degrees.
  • Hexagon: 6 sides, 6 angles. Angle sum = 720 degrees.

Regular Polygons

A regular polygon has all sides equal in length and all angles equal in measure.

Interior and Exterior Angles of a Regular Polygon

Interior angle: $\frac{(n-2) \times 180}{n}$ where n is the number of sides.

Exterior angle: $\frac{360}{n}$ where n is the number of sides.

Relationship: Interior angle + Exterior angle = 180 degrees.

Circles

Key Terms

  • Radius: The distance from the center of the circle to any point on the circumference.
  • Diameter: The distance across the circle passing through the center (2r).
  • Circumference: The distance around the circle ($2 \pi r$ or $\pi d$).
  • Arc: A portion of the circumference of a circle.
  • Chord: A line segment joining two points on the circumference.
  • Tangent: A line that touches the circle at only one point.
  • Sector: A region bounded by two radii and an arc.

Angles in a Circle

  • Angles subtended by the same arc are equal.
  • The angle subtended by a diameter at the circumference is a right angle.
  • The angle subtended by a semicircle at the circumference is a right angle.
  • The angle subtended by a chord at the circumference is half the angle subtended by the same chord at the center.

Area and Circumference of a Circle

Area: $\pi r^2$

Circumference: $2 \pi r$ or $\pi d$

Constructions

Drawing Angles

  1. Draw a line.
  2. Mark a point on the line.
  3. With the point as the center, draw an arc that intersects the line at two points.
  4. Use a compass to mark the two points on the arc.
  5. Draw a line through the center and the marked points.

Drawing Lines Bisecting Angles

  1. Draw an angle.
  2. With the vertex as the center, draw an arc that intersects both arms of the angle.
  3. Use a compass to mark the points of intersection.
  4. Draw a line through the two points of intersection.

Drawing Lines Parallel to a Given Line

  1. Choose a point on the given line.
  2. With the point as the center, draw an arc that intersects both rays from the given line.
  3. Use a compass to mark the points of intersection.
  4. Draw a line through the two points of intersection.

Drawing Lines Perpendicular to a Given Line

  1. Choose a point on the given line.
  2. Set the compass width to the perpendicular distance from the point to the line.
  3. Draw arcs from the point intersecting both rays from the given line.
  4. Mark the points of intersection.
  5. Draw a line through the two marked points.
Topic Key Concepts
Angles Types of angles, angle sum rules, vertically opposite angles, parallel lines, transversal angles.
Parallel Lines Properties of parallel lines, alternate interior angles, consecutive interior angles.
Polygons Types of polygons, regular polygons, interior and exterior angles.
Circles Key terms, angles in a circle, area, circumference.
Constructions Drawing angles, bisecting angles, parallel lines, perpendicular lines.