OxfordAQA International GCSE Mathematics: Angle facts, polygons and quadrilaterals (9260) – Revision Notes

These notes condense references G1 to G5 of section 3.3.1 (Properties and constructions) in the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). All of it is Core content, and Core content can also be assessed on the Extension tier papers. For full explanations see the study guide.

Next, try the practice questions. The course page is OxfordAQA IGCSE Mathematics, the printable list is the checklist, and the free diagnostics show which topics need more work.

G1: Terms and notation

Term or symbol	Meaning
Vertex (vertices)	a corner where edges meet
Edge	a side of a polygon, or where two faces meet
Plane	a flat surface
AB ∥ CD	AB parallel to CD (matching arrowheads)
AB ⊥ CD	AB perpendicular to CD (small square marks 90°)
∠ABC	the angle at B between BA and BC
Side a	the side opposite angle A in triangle ABC
Regular polygon	all sides equal and all angles equal

Tick marks show equal sides. Arrowheads show parallel lines. Diagrams may not be drawn to scale.

G2: Angles at a point and on a line

Fact	Rule
Angles at a point	total 360°
Angles on a straight line	total 180°
Vertically opposite angles	equal
Perpendicular lines	four angles of 90°

Method. Mark every known angle on the diagram. Write the sum as an equation. Solve it. Write the reason beside each line.

Reminder. Angles of 130°, 85° and 70° meet at a point with a fourth angle. The fourth is 360 − 285 = 75°.

G3: Parallel lines

Name	Property	Memory aid
Corresponding	equal	same position at each crossing
Alternate	equal	opposite sides of the transversal, between the lines
Interior (co-interior)	add up to 180°	same side of the transversal, between the lines

Use the full names in your reasons. The specification asks for alternate, corresponding and interior, not colloquial letter shapes.

Method in steps.

1. Mark the parallel lines with arrowheads.
2. Pick an angle you know and find the angle you need in as few steps as possible.
3. Name the property each time: “alternate angles”, “angles on a straight line” and so on.

Reminder. Parallel lines PQ and RS are crossed by a transversal. One interior angle is 107°, so the other is 180 − 107 = 73°.

G3: Triangles

Fact	Rule
Angles in a triangle	total 180°
Exterior angle	equals the sum of the two interior opposite angles
Equilateral	3 equal sides, 3 angles of 60°
Isosceles	2 equal sides; the angles opposite them are equal
Scalene	no equal sides, no equal angles
Right-angled	one angle of 90°
Acute-angled	all angles below 90°
Obtuse-angled	one angle above 90°

Reminder. An isosceles triangle has an apex angle of 100°. Each base angle is (180 − 100) ÷ 2 = 40°.

Reminder. Two angles of a triangle are 52° and 67°. The third is 180 − 119 = 61°, so the triangle is scalene and acute-angled.

G3 and G5: Quadrilaterals

Angles in any quadrilateral add up to 360°.

Shape	Key properties
Square	4 equal sides, 4 right angles; diagonals equal, bisect each other at 90°
Rectangle	opposite sides equal and parallel, 4 right angles; diagonals equal and bisect each other
Parallelogram	opposite sides equal and parallel; opposite angles equal; diagonals bisect each other
Rhombus	4 equal sides; opposite angles equal; diagonals bisect each other at 90° and bisect the angles
Kite	two pairs of adjacent equal sides; one pair of equal opposite angles; diagonals cross at 90°
Trapezium	one pair of parallel sides

Reminder. A kite has angles 84°, 112°, x and 112°. Then x = 360 − 308 = 52°. The two 112° angles are the equal pair, so x is the odd one out.

G4: Polygons

Quantity	Rule
Sum of interior angles	(n − 2) × 180°
Sum of exterior angles	360°
Regular polygon, exterior angle	360° ÷ n
Regular polygon, interior angle	180° − exterior angle
Regular polygon, angle at the centre	360° ÷ n
Interior plus exterior at one vertex	180°

Sides	Name	Interior sum	Regular: exterior	Regular: interior
5	pentagon	540°	72°	108°
6	hexagon	720°	60°	120°
8	octagon	1080°	45°	135°
9	nonagon	1260°	40°	140°
10	decagon	1440°	36°	144°
12	dodecagon	1800°	30°	150°

Method for finding n. If you are given an interior angle of a regular polygon, subtract it from 180° to get the exterior angle, then divide 360° by that.

Reminder. A regular polygon has interior angle 165°. Exterior angle = 15°, so n = 360 ÷ 15 = 24.

Worked reminders

Exterior angle of a triangle. Triangle PQR has angle P = 48° and angle Q = 77°. Side QR is extended beyond R. The exterior angle at R is 48 + 77 = 125°. Check: angle R = 180 − 125 = 55°, and 48 + 77 + 55 = 180.

Missing angle in a pentagon. Four angles of a pentagon are 95°, 110°, 120° and 105°.

Sum for 5 sides = 3 × 180 = 540°
Four angles total 430°
Fifth angle = 540 − 430 = 110°

Trapezium. ABCD has AB ∥ DC and angle DAB = 73°. Angles DAB and ADC are interior angles between the same parallel lines, so angle ADC = 180 − 73 = 107°.

Isosceles triangle in a regular polygon. In a regular 12-sided polygon, the centre O joins two neighbouring vertices to make an isosceles triangle with apex angle 30°. Each base angle is (180 − 30) ÷ 2 = 75°. Two base angles together make the interior angle: 2 × 75 = 150°.

Square and its diagonal. The diagonal of a square bisects the 90° corner, so it makes 45° with each side.

Setting out a reason

A clear line of working has three parts: the angle, its value and the fact used. For example:

Angle BCD = 180 − 112 = 68°    (angles on a straight line)
Angle CDE = 68°                (alternate angles, AB ∥ DE)

Work in the order you find the angles, so each line depends only on lines above it. If a question says “give a reason”, write the fact in words. A bare calculation shows the arithmetic but not why it is allowed.

Must-know distinctions

• Alternate angles are equal. Interior angles add up to 180°.
• Interior angle sum depends on n. Exterior angle sum is always 360°.
• Dividing 360° by the exterior angle gives n. Dividing 360° by the interior angle does not.
• A rhombus has equal sides. A kite has two pairs of adjacent equal sides. A rhombus is a special kite, but a kite is not usually a rhombus.
• A rectangle has equal diagonals. A rhombus has diagonals that cross at 90°. A square has both.
• Vertically opposite angles are equal. Adjacent angles on a line are not, unless both are 90°.

Quick self-test

1. Angles of 130°, 85° and 70° meet at a point with a fourth angle. Find the fourth angle.
2. Two straight lines cross. One angle is 68°. Write down the other three angles.
3. Two alternate angles lie between parallel lines. One is 54°. What is the other?
4. Two interior angles lie between parallel lines. One is 107°. What is the other?
5. Two angles of a triangle are 52° and 67°. Find the third angle.
6. An isosceles triangle has an apex angle of 100°. Find each base angle.
7. Find the sum of the interior angles of a 12-sided polygon.
8. Find the exterior angle of a regular 24-sided polygon.
9. A regular polygon has an interior angle of 165°. How many sides has it?
10. A quadrilateral has diagonals that cross at 90°, one diagonal bisecting the other but not vice versa. Name it.
11. Parallelogram ABCD has angle A = (5k + 10)° and angle B = (3k + 30)°. Find k.

Answers

1. 360 − (130 + 85 + 70) = 75°.
2. 68°, 112°, 112° (vertically opposite, and angles on a straight line).
3. 54°, because alternate angles are equal.
4. 180 − 107 = 73°, because interior angles add up to 180°.
5. 180 − (52 + 67) = 61°.
6. (180 − 100) ÷ 2 = 40°.
7. (12 − 2) × 180 = 1800°.
8. 360 ÷ 24 = 15°.
9. Exterior angle = 180 − 165 = 15°, so n = 360 ÷ 15 = 24.
10. Kite.
11. Adjacent angles add up to 180°: 8k + 40 = 180, so 8k = 140 and k = 17.5. Angle A = 97.5°, angle B = 82.5°.

Where marks are usually lost

• Giving an angle with no reason. Name the fact each time, such as “angles on a straight line”.
• Mixing up alternate and interior angles, so the wrong sum or the wrong equality is used.
• Finding n from the interior angle by dividing 360° by it.
• Forgetting that the interior angle sum uses (n − 2), not n.
• Measuring an angle from the diagram instead of calculating it.
• Describing a shape as “a rhombus” when the working only shows opposite sides parallel.
• Writing the equation correctly but dropping a bracket, such as 3a + a + 35 + 2a − 15 with the wrong sign.
• Leaving the answer as x when the question asks for the size of an angle.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Section 3.3.1, references G1 to G5.