Angles in Polygons and Parallel Lines

Mathematics revision notes · Cambridge (CIE) · 0580

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1. The basic angle facts

Four facts that start every question

Angles meeting at a point add to 360°
Angles on one side of a straight line add to 180°
Vertically opposite angles, formed where two straight lines cross, are equal

Two straight lines crossing, drawn twice: in the first the pair of opposite angles above and below the crossing point is shaded, and in the second the other opposite pair, left and right, is shaded, so each pair is equal — Source: Basic angle properties by Save My Exams

The angles inside a triangle add to 180°, and the angles inside a quadrilateral add to 360°
Naming uses three letters with the vertex in the middle, so angle ABC is the angle at B, and it means the angle smaller than 180° rather than the reflex one
Diagrams are usually marked "NOT TO SCALE", so work from the numbers and the markings rather than from how the picture looks

Three straight lines crossing at one point, marked not to scale, with angles of 25 degrees and 98 degrees given and two more, x and y, to be found from the facts that angles at a point add to 360 and angles on a line add to 180 — Source: Basic angle properties by Save My Exams

Filling in what you can

Mark every angle you can work out on the diagram, even ones the question has not asked for
One of them is almost always the step that unlocks the angle you actually want
A question worth more than one mark expects the intermediate angles to appear in the working

A triangle standing on a straight line with 60 degrees and 130 degrees marked outside it and x inside at the apex, and the two angles y and z between the triangle and the line filled in as the intermediate step — Source: Basic angle properties by Save My Exams

Worked example

Crossing lines and a straight line

Three straight lines meet at a point. One angle is 112°, and x is vertically opposite it. On the straight line through that point sit x, y and an angle of 41°. Find x and y.

Three straight lines crossing at a single point. Going anticlockwise from the right-hand horizontal ray, the first angle is marked 41 degrees, the next is labelled y, and the third, up to the left-hand horizontal ray, is labelled x. Below the horizontal line, the angle vertically opposite x is marked 112 degrees.

Solution:

x is vertically opposite the 112° angle, and vertically opposite angles are equal
So x = 112
The three angles x, y and 41° lie on a straight line, so they add to 180°
112 + y + 41 = 180
153 + y = 180
So y = 27
Check: 112 + 27 + 41 = 180

Exam tip

Label every angle you work out

Many angle questions give a mark for a correct angle found on the way, even when the final answer is wrong, as long as it is clear which angle it is. So write each angle on the diagram in its exact position, or name it with three letters in your working (angle PQR = 47°). A number standing alone in the working does not show which angle it is.

2. Angles in triangles and quadrilaterals

Using the shape's own properties

Any triangle's three angles total 180°, and knowing two gives the third by subtraction
An isosceles triangle has two equal angles, and they are the ones opposite the two equal sides
- With an apex angle of 34°, the two base angles are (180 − 34) ÷ 2 = 73° each
An equilateral triangle has three angles of 60°
Any quadrilateral's four angles total 360°

A quadrilateral with interior angles of 115, 85 and 97 degrees and one side extended, so the fourth interior angle works out at 63 degrees and the exterior angle y beside it is what the question wants — Source: Basic angle properties by Save My Exams

Symmetry supplies extra equal angles: a parallelogram and a rhombus have equal opposite angles, and a kite has one pair of equal opposite angles
A square or rectangle has four right angles

Which angles a quadrilateral's own shape makes equal: a square and a rectangle have four right angles, a parallelogram and a rhombus have equal opposite angles marked in pairs, and a kite has one pair of equal opposite angles — Source: Basic angle properties by Save My Exams

Worked example

The fourth angle of a quadrilateral

Three angles of a quadrilateral are 88°, 124° and 97°. Find the fourth angle.

Solution:

The four angles of any quadrilateral add to 360°
Add the three given: 88 + 124 + 97 = 309
Subtract from 360: 360 − 309 = 51
The fourth angle is 51°

3. Angles in parallel lines

The three relationships

When a straight line crosses a pair of parallel lines, three named relationships appear
Corresponding angles are equal. They sit in matching positions at the two crossings

Two parallel lines marked with arrows and cut by a third line, drawn twice: in the first a pair of corresponding angles sitting in matching positions at the two crossings is shaded, and in the second a pair of alternate angles on opposite sides of the crossing line between the parallels — Source: Angles in parallel lines by Save My Exams

Alternate angles are equal. They sit on opposite sides of the crossing line, between the parallel lines
Co-interior angles add to 180°. They sit on the same side of the crossing line, between the parallel lines
The syllabus also calls co-interior angles supplementary, which means they add to 180°

The same pair of parallel lines cut by a third, with two co-interior pairs shaded in turn: each pair sits on the same side of the crossing line between the parallels, and the two shaded angles in each pair add to 180 degrees — Source: Angles in parallel lines by Save My Exams

Vertically opposite angles and angles on a straight line still apply, and combining them with the three above solves almost everything

A line crossing two parallel lines with all eight angles filled in: every angle is either x or 180 − x, arranged so the matching ones at each crossing are equal and any two next to each other add to 180 — Source: Angles in parallel lines by Save My Exams

Arrows on a diagram are what tell you the lines are parallel, so look for them before using any of these

Two parallel lines marked with arrows and crossed by a sloping line, with one angle given as 64 degrees and two others labelled a and b to be found from it — Source: Angles in parallel lines by Save My Exams

Naming the reason

Questions ask for the angle and a geometrical reason, with a mark for each
Give the full name: "alternate angles are equal", "corresponding angles are equal", "co-interior angles add to 180°"
The F, Z and C shapes are useful for spotting the relationship but are not accepted as the reason on their own
Abbreviating the name loses the mark, so write it out in full

Worked example

Two angles with reasons

A straight line crosses two parallel lines. One of the angles it makes is 118°. Find the angle corresponding to it, and the angle co-interior with that one, giving a reason for each.

Two parallel horizontal lines, each marked with a single arrowhead to show they are parallel, crossed by a sloping transversal. At the upper crossing, the angle between the line to the right and the transversal going up to the left is marked 118 degrees. No other angle is marked.

Solution:

The corresponding angle sits in the matching position at the other crossing: above the lower line, to the right of the crossing line
Corresponding angles are equal, so that angle is 118°
It lies between the parallel lines, so it has a co-interior partner: the angle at the upper crossing on the same side of the crossing line, also between the parallel lines
Co-interior angles add to 180°, so that angle is 180 − 118 = 62°
Check: the 62° angle and the given 118° lie on a straight line at the upper crossing, and 118 + 62 = 180

4. Angles in polygons

Interior and exterior angles

An interior angle sits inside the polygon at a corner
An exterior angle sits between one side and the extension of the next side
At every corner the interior and exterior angles lie on a straight line, so they add to 180°
Whatever the polygon and however many sides it has, its exterior angles total 360°
For the inside angles the total depends on n: a polygon with n sides splits into n − 2 triangles, so they come to 180° × (n − 2)
- A triangle gives 180°, a quadrilateral 360°, a pentagon 540° and a hexagon 720°

An irregular hexagon with each of its six interior angles labelled a to f in blue, and each side extended with a dashed line so the exterior angle beside it, p to u, is marked in red, the interior and exterior angle at each corner lying on a straight line — Source: Angles in polygons by Save My Exams

Regular polygons

A regular polygon has all sides equal and all angles equal, so its angles can be found by dividing
Each exterior angle is 360° ÷ n, and this is usually the quicker route
Each interior angle is then 180° minus that exterior angle
The same result comes from 180(n − 2) ÷ n, so either method is acceptable
Working backwards, n = 360° ÷ exterior angle; when a question gives the interior angle, take it from 180° first

Regular polygon	Sides	Interior sum	Each interior	Each exterior
Equilateral triangle	3	180°	60°	120°
Square	4	360°	90°	90°
Pentagon	5	540°	108°	72°
Hexagon	6	720°	120°	60°
Octagon	8	1080°	135°	45°
Decagon	10	1440°	144°	36°

Worked example

Finding the number of sides

Each exterior angle of a regular polygon is 24°. Work out the number of sides, and the size of each interior angle.

Solution:

The exterior angles of any polygon add to 360°
For a regular polygon they are all equal, so the number of sides is 360 ÷ 24
The polygon has 15 sides
The interior and exterior angles at each corner lie on a straight line
Each interior angle is 180 − 24 = 156°
Check with the other method: 180 × (15 − 2) = 2340, and 2340 ÷ 15 = 156

Worked example

Number of sides from an interior angle

Each interior angle of a regular polygon is 168°. Find the number of sides.

Solution:

The interior and exterior angles at each corner lie on a straight line, so each exterior angle is 180 − 168 = 12°
The exterior angles add to 360° and are all equal, so the number of sides is 360 ÷ 12
The polygon has 30 sides
Check with the other method: 180(n − 2) ÷ n = 168 gives 180n − 360 = 168n, so 12n = 360 and n = 30

Dividing 360 by the interior angle itself gives 360 ÷ 168, which is not a whole number: a sign that the step to the exterior angle was missed.

Irregular polygons

The interior angle sum formula works for any polygon, so an irregular one with a missing angle is found by subtraction
Add every angle you are given, then take the total from 180° × (n − 2)
- Five angles of a hexagon are 100°, 145°, 98°, 130° and 122°, which total 595°
- A hexagon's interior angles total 720°, so the missing angle is 720 − 595 = 125°
The exterior angles still add to 360° whether the polygon is regular or not