OCR GCSE Mathematics: Angle facts, polygons, triangle and quadrilateral properties, and symmetry (J560) – Revision Notes

These revision notes condense OCR GCSE (9-1) Mathematics, specification Version 2.1 (August 2026), sections 8.03 and 8.04. Full explanations and derivations are in the study guide. When you are ready, move on to the practice questions. Content that the specification gives to Higher tier learners only is labelled Higher tier only. See the course hub, the checklist and the free diagnostics.

Core angle facts (8.03a to 8.03c)

Fact	Value	Reason to write
Angles at a point	360°	Angles at a point sum to 360°
Angles on a straight line	180°	Angles on a line sum to 180°
Vertically opposite	Equal	Vertically opposite angles
Alternate (Z)	Equal, parallel lines	Alternate angles
Corresponding (F)	Equal, parallel lines	Corresponding angles
Co-interior (C)	Sum to 180°, parallel lines	Co-interior angles (follows from the facts above)

Method for angle chasing

1. Mark every angle you know on the diagram.
2. Find an angle that follows from one fact. Write the fact next to it.
3. Repeat, using each new angle, until you reach the target.
4. If the target is in algebra, form an equation from the fact and solve it.

Worked reminder. Parallel lines are crossed by a transversal. An interior angle is (4k + 6)° and its alternate partner is (6k − 34)°.

4k + 6 = 6k − 34   (alternate angles)
40 = 2k, k = 20
Angle = 86°

Polygon angles (8.03d)

Result	Formula
Angles of a triangle	180°
Exterior angles of any polygon	360°
Interior angle sum, n sides	(n − 2) × 180°
Regular polygon, exterior angle	360° ÷ n
Regular polygon, interior angle	180° − exterior angle
Interior plus exterior at a vertex	180°

Derivations in one line each

• Triangle: a line through one vertex parallel to the opposite side makes two alternate pairs, and the three angles then sit on a straight line.
• Exterior angles: walking round the polygon, you turn through each exterior angle and finish after one full turn.
• Interior sum: n interior angles plus n exterior angles make 180n, and the exterior angles take 360, so 180n − 360.

Worked reminder. A regular decagon has 10 sides. Exterior angle = 36°, interior angle = 144°, interior sum = 8 × 180 = 1440°.

Worked reminder (finding n). The interior angle sum of a polygon is 2340°.

(n − 2) × 180 = 2340
n − 2 = 13
n = 15

Regular polygon, given the exterior angle: n = 360 ÷ exterior angle. A regular polygon cannot have 7° as its exterior angle if 360 ÷ 7 is not a whole number.

Triangles (8.04a)

Triangle	Properties	Reason
Isosceles	Two equal sides; base angles equal	Symmetry about the line from apex to base midpoint
Equilateral	Three equal sides; all angles 60°	Equal angles that sum to 180°
Right-angled	One angle 90°; other two sum to 90°	Angle sum of a triangle

Exterior angle of a triangle = sum of the two opposite interior angles.

Worked reminder. Isosceles triangle with apex angle 38°. Base angles = (180 − 38) ÷ 2 = 71° each.

Quadrilaterals (8.04b)

Angle sum of any quadrilateral: 360°.

Shape	Key facts to quote
Square	4 equal sides, 4 right angles, diagonals equal and perpendicular, bisecting each other
Rectangle	Opposite sides equal and parallel, 4 right angles, diagonals equal and bisect each other
Parallelogram	Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other
Rhombus	4 equal sides, opposite angles equal, diagonals perpendicular, bisect each other and the angles
Kite	Two pairs of adjacent equal sides, one pair of equal angles, diagonals perpendicular
Trapezium	One pair of parallel sides; angles between the parallel sides (co-interior) sum to 180°

Worked reminder. A kite has angles 100° and 40° at the two ends of its axis of symmetry. The other two angles are equal: (360 − 140) ÷ 2 = 110° each.

Symmetry (8.04c)

Order of rotation symmetry = number of positions in a full turn. Order 1 means none. Angle of turn = 360° ÷ order.

Shape	Lines	Order
Equilateral triangle	3	3
Isosceles triangle	1	1
Rectangle	2	2
Rhombus	2	2
Parallelogram	0	2
Kite	1	1
Isosceles trapezium	1	1
Regular n-gon	n	n

Proof checklist

A proof is a list of statements, each with a reason. Both tiers justify results in simple proofs.

1. State what you are proving.
2. Name the angles with three letters, or with a letter you define.
3. Give a reason in the same line as every statement.
4. Finish by stating the result you were asked to show.

Higher tier only: more formal proofs of geometrical results. The specification names circle theorems as an example. Typical tools are isosceles triangles (base angles), parallel lines (alternate angles) and the triangle angle sum. For example, to show that the angle at the centre of a circle is twice the angle at the circumference, join the circumference point to the centre, use the equal radii to get base angles, and use the exterior angle of a triangle.

Must-know distinctions

• Alternate or corresponding? Alternate angles sit on opposite sides of the transversal, between the parallel lines. Corresponding angles sit in matching positions at the two crossings. Both are equal, but the reason you write must match the shape.
• Interior or exterior? An exterior angle is formed by extending one side. Interior plus exterior is 180° at the same vertex. Regular polygon questions that give an interior angle are usually solved through the exterior angle.
• Sum or one angle? (n − 2) × 180 is the total of the interior angles. Divide by n only for a regular polygon.
• Rhombus or square? Both have four equal sides and perpendicular diagonals. Only the square has four right angles and equal diagonals.
• Parallelogram or rectangle? Both have opposite sides equal and parallel. Only the rectangle has right angles and equal diagonals.
• Line symmetry or rotation symmetry? A parallelogram has no lines of symmetry but has rotation symmetry of order 2.
• Trapezium or kite? A trapezium is defined by a pair of parallel sides. A kite is defined by two pairs of adjacent equal sides.

More worked reminders

Angles on a line and at a point together. Three angles at a point are 2d, 3d and a right angle, and the remaining angle is 80°.

2d + 3d + 90 + 80 = 360
5d = 190
d = 38

Exterior angle of a triangle. Triangle angles are 47° and 68°. The exterior angle at the third vertex is 47 + 68 = 115°, since the interior angle there is 65° and 65 + 115 = 180.

Hidden isosceles triangle. A regular octagon ABCDEFGH has interior angle 135°. In triangle ABC, AB = BC, so angle BAC = (180 − 135) ÷ 2 = 22.5°.

Parallelogram with algebra. Adjacent angles are (5w + 10)° and (3w + 30)°.

8w + 40 = 180   (co-interior angles)
w = 17.5
Angles: 97.5° and 82.5°

Quick self-test

1. Angles at a point are 90°, 135°, and x. Find x.
2. Two angles on a line are (2m + 10)° and (3m + 20)°. Find m.
3. Find the interior angle sum of a nonagon (9 sides).
4. A regular polygon has exterior angle 40°. How many sides has it?
5. Find the interior angle of a regular polygon with 18 sides.
6. In an isosceles triangle, one base angle is 52°. Find the apex angle.
7. A parallelogram has one angle of 65°. Find the other three angles.
8. A rhombus has diagonals that cross. State the angle between them.
9. State the number of lines of symmetry and the order of rotation symmetry of a regular hexagon.
10. State the shape: a quadrilateral with rotation symmetry of order 2 and no lines of symmetry.
11. Triangle angles are (x + 10)°, (2x)° and (3x − 4)°. Find x.
12. AB = AC in a triangle, D is on BC, and AD = BD. Angle ABC = 35°. Find angle ADC, with reasons.

Answers

1. 360 − 90 − 135 = 135°
2. 5m + 30 = 180, m = 30 (angles 70° and 110°)
3. (9 − 2) × 180 = 1260°
4. 360 ÷ 40 = 9 sides
5. Exterior 20°, interior 160°
6. 180 − 2 × 52 = 76°
7. 65°, 115°, 115°
8. 90°
9. 6 lines; order 6
10. Parallelogram
11. 6x + 6 = 180, x = 29 (angles 39°, 58°, 83°)
12. Angle BAD = 35° (AD = BD, base angles), so
    angle ADC = 35 + 35 = 70° (exterior angle of triangle ABD)

Where marks are usually lost

• Writing an angle value with no reason, or a vague reason such as “Z angles” or “parallel”.
• Mixing up alternate and corresponding angles, or using a rule when the lines are not parallel.
• Using 180° for angles at a point, or 360° for angles on a line.
• Finding the exterior angle and forgetting to subtract from 180° for the interior angle.
• Using (n − 2) × 180 with the wrong n, or using it on a regular polygon without dividing by n.
• Assuming a quadrilateral is a rectangle or square because it looks like one in the diagram.
• Giving “1” line of symmetry when the shape has none (a parallelogram), or writing “order 0” when you mean order 1.
• Leaving out the final statement in a proof, or proving the result for one example only.

Official syllabus

OCR, GCSE (9-1) Mathematics J560, specification Version 2.1 (August 2026), for first assessment in 2017: sections 8.03 and 8.04.