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Core + Extended Geometry

Angle Properties of Triangles and Polygons for IGCSE Maths

Interior and exterior angles of triangles and polygons. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding angle propertie

What You Need to Know

Interior and exterior angles of triangles and polygons. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding angle properties of triangles and polygons is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Angle Properties of Triangles and Polygons

Basic angle facts are the foundation of all geometry in IGCSE 0580: angles on a straight line sum to 180°, angles at a point sum to 360°, angles in a triangle sum to 180°, and vertically opposite angles are equal. Exterior angle of a triangle equals the sum of the two non-adjacent interior angles. For regular polygons: interior angle = (n−2)×180°/n; exterior angle = 360°/n. These appear on every Core and Extended paper.

Step-by-Step Method

  1. 1

    Identify the angle relationship

    State which rule applies: straight line (180°), point (360°), triangle (180°), vertically opposite (equal), exterior angle theorem.

  2. 2

    Write the equation

    Express the angle relationship as an equation. E.g. 'angles on a straight line: x + 55° + 75° = 180°'.

  3. 3

    Solve for the unknown angle

    Simplify the equation and find x.

  4. 4

    State the reason

    Always write the geometric reason: 'angles in a triangle', 'vertically opposite', etc. Reasons earn marks.

  5. 5

    For polygons

    For an n-sided polygon: sum of interior angles = (n−2)×180°. Each exterior angle sums to 360°.

Worked Example

Question

In triangle PQR, angle P = 58°, angle Q = 73°. Find angle R. Also, find the interior angle of a regular hexagon.

Solution

Part 1: Angles in a triangle sum to 180°. Angle R = 180° − 58° − 73° = 49° Part 2: Regular hexagon (n=6). Sum of interior angles = (6−2) × 180° = 720° Each interior angle = 720° ÷ 6 = 120° Or: exterior angle = 360°/6 = 60°, so interior = 180° − 60° = 120°. Answers: Angle R = 49°; Interior angle of regular hexagon = 120°

Exam Tips for Angle Properties of Triangles and Polygons

  • Always state the geometric reason alongside each angle calculation — this is required for full marks.
  • Angles on a straight line = 180°, at a point = 360°, in a triangle = 180° — memorise these three.
  • Exterior angle = sum of two opposite interior angles — this shortcut avoids finding all three angles separately.
  • For regular polygons: exterior angle = 360°/n is the quickest formula.

Practice Questions

Q1: Two angles on a straight line are (3x + 10)° and (x + 30)°. Find x.

Show hint

(3x+10)+(x+30)=180 → 4x+40=180 → x=35.

Q2: Find the number of sides of a regular polygon if each exterior angle is 24°.

Show hint

n = 360/24 = 15 sides.

Q3: In a triangle, one angle is double the smallest, and the third is 20° more than the smallest. Find all three angles.

Show hint

Let smallest = x: x + 2x + (x+20) = 180 → 4x = 160 → x = 40°. Angles: 40°, 80°, 60°.

Frequently Asked Questions

What is angle properties of triangles and polygons in IGCSE Maths?

Interior and exterior angles of triangles and polygons.

Is angle properties of triangles and polygons in the Core or Extended syllabus?

Angle Properties of Triangles and Polygons is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise angle properties of triangles and polygons effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on angle properties of triangles and polygons rather than trying to cover everything at once.

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