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O Level · Mathematics D 4024

Angles

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Cambridge O Level Mathematics (Syllabus D) 4024

Angles

The big idea: An angle answer is a chain of facts, and each link needs its reason written in the syllabus words: "co-interior angles sum to 180°", not "C-angles".

Summary — What This Chapter Covers

Angles at a point add up to 360°; angles on a straight line add up to 180°; vertically opposite angles are equal.
Angles in a triangle add up to 180°; angles in a quadrilateral add up to 360°.
With parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles add up to 180°.
Angle sum of a polygon = (n − 2) × 180°; exterior angles of any polygon add up to 360°.
Use three-letter notation (angle ABC) and give a geometric reason for every step.

1. Angles at a Point and on a Line

Syllabus 4.6 · Paper 1 & 2

The first three facts

Angle ABC means the angle at B, between BA and BC. When a question says "Give a reason", quote the fact in words.

Reasons Angles at a point sum to 360° · Angles on a straight line sum to 180° · Vertically opposite angles are equal

Worked Example

Two straight lines cross. One of the four angles is 3x° and the angle next to it is (x + 40)°. Find all four angles.

2. Triangles and Quadrilaterals

Syllabus 4.6 · Paper 1 & 2

180° and 360°

A diagonal splits a quadrilateral into two triangles, which is why its angles sum to 2 × 180° = 360°. Isosceles triangles add one more fact: base angles are equal.

Reasons Angle sum of a triangle = 180° · Angle sum of a quadrilateral = 360° · Base angles of an isosceles triangle are equal

Practice Question 1

Triangle PQR is isosceles with PQ = PR. Angle QPR = 38°. Find angle PQR, giving reasons.

Practice Question 2

The angles of a quadrilateral are x°, 2x°, (x + 30)° and 110°. Find x.

3. Angles With Parallel Lines

Syllabus 4.6 · Paper 1 & 2

Corresponding, alternate, co-interior

A line crossing two parallel lines is a transversal. It makes three pairs of related angles:

Reasons Corresponding angles are equal (F shape) · Alternate angles are equal (Z shape) · Co-interior angles sum to 180°, they are supplementary (C shape)

Worked Example

AB and CD are parallel. A transversal meets AB at P and CD at Q, with B and D on the same side of the transversal. Angle BPQ = (2x + 15)° and angle PQD = (3x − 5)°. Find x, giving reasons.

Write the real words

Mark schemes accept "alternate angles", "corresponding angles", "co-interior angles". "Z-angles", "F-angles" and "C-angles" describe the shape and are not accepted as reasons.

4. Angles in Polygons

Syllabus 4.6 · Paper 1 & 2

Regular and irregular

A polygon with n sides splits into (n − 2) triangles from one vertex, so its interior angles sum to (n − 2) × 180°. Walking round any polygon you turn through one full turn, so the exterior angles sum to 360°. In a regular polygon all exterior angles are equal.

Polygon angles (not given) Sum of interior angles = (n − 2) × 180° Sum of exterior angles = 360°
Regular: exterior angle = 360° ÷ n interior angle = 180° − exterior angle

Worked Example

Practice Question 3

Practice Question 4

Memorise

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One fact, one reason, per line

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Angles

Angles

The first three facts

180° and 360°

Corresponding, alternate, co-interior

Regular and irregular

Past papers that test Angles

More Mathematics D topics