Overview
What you’ll learn
- Classify angles as right, acute, obtuse or reflex, and use the facts that angles on a straight line sum to 180°, angles at a point sum to 360°, and vertically opposite angles are equal.
- Identify corresponding, alternate and co-interior angles when parallel lines are cut by a transversal, and calculate unknown angles from them.
- Use the angle sum of a triangle (180°), the exterior angle of a triangle, and the properties of isosceles and equilateral triangles.
- Classify and use the properties of special quadrilaterals: parallelogram, rhombus, trapezium and kite.
- Calculate interior and exterior angle sums of convex polygons and the angles of regular polygons (pentagon, hexagon, octagon, decagon).
- Construct simple figures, and construct perpendicular bisectors and angle bisectors with compasses and ruler, stating their properties.
In 30 seconds
- 1Acute < 90°, right = 90°, obtuse between 90° and 180°, reflex between 180° and 360°.
- 2Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.
- 3Parallel lines + transversal: corresponding angles (F shape) are equal, alternate angles (Z shape) are equal, co-interior angles (C shape) sum to 180°.
- 4Angles in a triangle sum to 180°; each exterior angle equals the sum of the two opposite interior angles.
- 5Isosceles triangles have two equal sides and two equal base angles; equilateral triangles have all sides equal and all angles 60°.
- 6Polygon interior angle sum = (n − 2) × 180° ; exterior angle sum = 360° for every convex polygon.
- 7A regular n-gon: each exterior angle = 360° ÷ n, each interior angle = 180° − (360° ÷ n).
- 8The perpendicular bisector of a segment is the set of points equidistant from its two endpoints; an angle bisector cuts an angle into two equal halves.
Notes · 1 of 5
Angle types and angle facts
Naming angles by size, then three facts that unlock most diagrams.
Angles are measured in degrees. An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle lies between 90° and 180°, and a reflex angle lies between 180° and 360°. Three facts do most of the work in exam diagrams: angles on a straight line add to 180°, angles at a point add to 360°, and when two straight lines cross, the vertically opposite angles are equal.
- Straight line: 74° + x = 180° gives x = 106°.
- At a point: 90° + 130° + x = 360° gives x = 140°.
- Crossing lines: if one angle is 68°, its vertically opposite partner is 68° and both adjacent angles are 112° (180° − 68°).
Notes · 2 of 5
Parallel lines and transversals
One cut across two parallel lines creates three memorable patterns.
When a transversal crosses two parallel lines, eight angles appear in three named patterns. Corresponding angles sit in matching corners — an F shape — and are equal. Alternate angles sit on opposite sides of the transversal between the lines — a Z shape — and are equal. Co-interior (allied) angles sit on the same side between the lines — a C shape — and sum to 180°.
| Angle type | Pattern | Value |
|---|---|---|
| Corresponding partner | F (equal) | 124° |
| Alternate partner | Z (equal) | 124° |
| Co-interior partner | C (sums to 180°) | 56° |
| Adjacent on a straight line | linear pair | 56° |
The equalities only apply when the lines are known to be parallel — look for the arrow markings on the diagram, or a statement in the question. Quote the pattern by name in your reasons: "corresponding angles, AB ∥ CD".
Notes · 3 of 5
Triangles
The 180° engine, exterior angles, and the isosceles symmetry shortcut.
The angles of every triangle sum to 180°. An exterior angle — formed by extending one side — equals the sum of the two opposite interior angles (it also forms a linear pair with its neighbouring interior angle). An isosceles triangle has two equal sides, and the angles opposite those sides (the base angles) are equal; an equilateral triangle has all three sides equal and all three angles 60°.
- 1Two angles known
52° and 67°: the third is 180° − 52° − 67° = 61°.
- 2Exterior angle route
Exterior at one vertex = 65° + 48° = 113°, skipping the third-angle step entirely.
- 3Isosceles shortcut
Vertex angle 40°: the two base angles share the remaining 140° equally — 70° each.
Triangles classify by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). A right triangle pins one angle at 90°, so the other two are complementary — they sum to 90°.
Notes · 4 of 5
Quadrilaterals and polygons
Special quadrilaterals by their properties, then the polygon angle accounting.
The angles of any quadrilateral sum to 360°. The special quadrilaterals are defined by their sides, angles and diagonals: a parallelogram has two pairs of parallel sides (with equal opposite sides and equal opposite angles); a rhombus is a parallelogram with all sides equal, whose diagonals bisect each other at right angles; a trapezium has exactly one pair of parallel sides; a kite has two pairs of adjacent equal sides and one line of symmetry. Classify by stating the property that settles it — "both pairs of opposite sides are parallel, so it is a parallelogram".
For any convex polygon with n sides, the interior angles sum to (n − 2) × 180° : triangles 180°, quadrilaterals 360°, pentagons 540°, hexagons 720°. The exterior angles (one at each vertex, walking around the shape) always total 360° — no exceptions. A regular polygon has equal sides and equal angles, so each exterior angle is 360° ÷ n and each interior angle is 180° minus that.
| Polygon | n | Each exterior angle | Each interior angle |
|---|---|---|---|
| Pentagon | 5 | 72° | 108° |
| Hexagon | 6 | 60° | 120° |
| Octagon | 8 | 45° | 135° |
| Decagon | 10 | 36° | 144° |
Notes · 5 of 5
Constructions
Ruler-and-compasses routines, and the two bisectors every student must own.
Constructions use only a ruler and a pair of compasses — no protractor for the marks that say "construct". The two standard bisectors: the perpendicular bisector of a segment AB, and the angle bisector of an angle. Each is a locus: the perpendicular bisector is the complete set of points equidistant from A and B; the angle bisector is the set of points equidistant from the two arms of the angle.
- 1Perpendicular bisector of AB
With the compasses open past half of AB, draw arcs centred at A above and below the line; repeat the same radius from B. Join the two crossing points of the arcs with a straight line — it crosses AB at its midpoint at 90°.
- 2Bisect an angle at O
Draw an arc centred at O cutting both arms at P and Q. With the same radius, draw arcs from P and Q that cross. Join O to the crossing point — the new ray splits the angle exactly in half.
- 3Construct a 60° angle
Draw an arc from the vertex cutting one arm; with the same radius, step the arc from that intersection to cross the first arc. The vertex-to-crossing ray makes exactly 60° (an equilateral triangle appears inside).
Revise
Key terms
The words examiners expect you to use precisely.
- Acute angle
- An angle less than 90°.
- Obtuse angle
- An angle between 90° and 180°.
- Reflex angle
- An angle between 180° and 360°.
- Vertically opposite angles
- Equal angles formed opposite each other where two straight lines cross.
- Corresponding angles
- Equal angles in matching corners where a transversal crosses parallel lines (F shape).
- Alternate angles
- Equal angles on opposite sides of a transversal between parallel lines (Z shape).
Show 6 more key termsShow fewer key terms
- Co-interior angles
- Angles on the same side of a transversal between parallel lines, summing to 180° (C shape).
- Isosceles triangle
- A triangle with two equal sides and two equal base angles.
- Regular polygon
- A polygon with all sides equal and all angles equal; each exterior angle is 360° ÷ n.
- Interior angle sum
- For a convex n-gon, (n − 2) × 180°.
- Perpendicular bisector
- The line through the midpoint of a segment at 90° to it; every point on it is equidistant from the segment’s endpoints.
- Angle bisector
- The ray dividing an angle into two equal parts; points on it are equidistant from the angle’s arms.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Classifying a 90° angle as acute.
Acute means strictly less than 90°.
Instead: 90° is a right angle; acute angles are smaller.
Mistake: Adding vertically opposite angles to 180°.
Vertically opposite angles are equal, not supplementary.
Instead: Each pair of vertically opposite angles matches (68° and 68°); the adjacent pairs sum to 180°.
Show 10 more mistakesShow fewer mistakes
Mistake: Using the Z pattern (alternate) when the diagram shows an F (corresponding).
The patterns look similar in a crowded diagram.
Instead: Trace the two parallel lines and the transversal; corresponding angles sit in matching corners, alternate angles zigzag.
Mistake: Taking co-interior angles as equal.
C-shape angles are supplementary, not equal.
Instead: Co-interior: 124° + 56° = 180°. Only corresponding and alternate pairs are equal.
Mistake: Halving the vertex angle of an isosceles triangle to get a base angle.
The two base angles, not the vertex, are the equal pair.
Instead: Vertex 40° leaves 140° for the equal base angles: 70° each.
Mistake: Using 360° as the angle sum of a triangle.
360° belongs to quadrilaterals and points.
Instead: Triangle angles sum to 180°.
Mistake: Quoting the exterior angle of a triangle as equal to its adjacent interior angle.
They form a linear pair (sum 180°), which is different from equality.
Instead: The exterior angle equals the SUM of the two opposite interior angles: 65° + 48° = 113°.
Mistake: Computing a polygon’s interior angle sum as n × 180°.
Each vertex contributes a missing triangle’s worth.
Instead: Sum = (n − 2) × 180°: a pentagon gives 3 × 180° = 540°, not 900°.
Mistake: Dividing 360° by n for the interior angle of a regular polygon.
360° ÷ n gives the EXTERIOR angle.
Instead: Interior = 180° − (360° ÷ n); for an octagon, 180° − 45° = 135°.
Mistake: Describing a trapezium as having two pairs of parallel sides.
That is a parallelogram.
Instead: A trapezium has exactly one pair of parallel sides.
Mistake: Bisecting an angle with the perpendicular-bisector routine (or vice versa).
The two constructions start differently: one works from the segment’s endpoints, the other from an arc across the arms.
Instead: Perpendicular bisector: equal-radius arcs from A and B, join the crossings. Angle bisector: one arc across both arms, then equal arcs from those two cut points.
Mistake: Erasing construction arcs for a tidy final diagram.
The arcs are the working — examiners award marks for them.
Instead: Leave every compass arc visible and label the constructed line clearly.
Check yourself
Quick quiz
32 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Classify angles
An obtuse angle measures
Ready for more practice on Chapter 9: Angles, Triangles and Polygons?
Generate a fresh 5-question drill built from this chapter’s key points. Free with your study account.
Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 7Foundation
- 7Standard
- 6Challenging
20 questions worth 80 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsKeep practising
Practise this chapter
Jump straight to the questions for this chapter, the chapters either side of it, and this year’s assessment papers.
Test yourself
Full papers for this subject. Each school sets its own scope — check your assessment outline.
Exam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 9 Practice Paper
30 marks · 40 min · 4 pages
7 multiple-choice questions and structured questions with answer lines. Chapter 9 (G1 Angles, Triangles and Polygons: angle facts, parallel-line patterns, triangle and quadrilateral properties, polygon angle sums, constructions).
Mark scheme
PROChapter 9 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
Questions
Frequently asked questions
What does Sec 3 E-Math Chapter 9 cover?
It covers syllabus sub-topic G1 of O-Level 4052 / SEC G3 K310 in full: angle types and facts, parallel-line angle patterns, triangle properties, special quadrilaterals, polygon interior/exterior angle sums, regular polygons up to ten sides, and the ruler-and-compasses constructions including both bisectors.
How do I tell corresponding, alternate and co-interior angles apart?
Trace the two parallel lines and the transversal slowly with your pencil. Matching corners (an F shape) are corresponding and equal. A zigzag (Z shape) marks alternate angles, also equal. Angles tucked inside a C shape on the same side of the transversal are co-interior and add to 180°. Naming the shape in your reason earns the mark.
What is the fastest way to find each angle of a regular polygon?
Do the exterior angle first: 360° ÷ n. The interior angle is then 180° minus that. For an octagon: 360 ÷ 8 = 45° exterior, so 135° interior. Working backwards (given the interior angle), subtract from 180° and divide 360° by the result to get the number of sides.
Do I need to show construction arcs?
Yes — keep every compass arc visible. Construction marks are awarded for the arcs and the accuracy of the final line, and an answer with arcs erased typically loses the method marks even when the drawing looks right.
When is this chapter usually tested in school?
Angle geometry appears in almost every Weighted Assessment 2 and End-of-Year paper as a multi-step "find all the angles with reasons" question. Reason marks are awarded separately from answers, so quote the fact you used at every step.
Up next
Chapter 10: Congruence and Similarity