E-Math notes · Chapter 10

Angles, Triangles and Polygons: O-Level E-Math Notes

Geometry is detective work: use one fact about angles to find a new angle, write the reason, and repeat. By the end of this chapter you can find unknown angles in lines, parallels, triangles, quadrilaterals and polygons, classify the special quadrilaterals from their properties, and construct figures from given data.

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Figure 1 · Square

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Angle types and basic angle facts

Three facts about angles that sit together.

Parallel lines

Parallel lines make equal pairs and straight-line pairs.

Triangles

The angles of a triangle add up to 180°.

By the end of this chapter

Recognise right, acute, obtuse and reflex angles, and use angles on a straight line, angles at a point and vertically opposite angles.
Use the angles made by parallel lines and a transversal: corresponding, alternate and interior angles.
Use the properties of triangles, special quadrilaterals and regular polygons, including their symmetry, and classify quadrilaterals.
Find the sum of interior and exterior angles of a convex polygon, and the angles of regular polygons.
Construct simple figures from given data with ruler, compasses, set squares and protractor.

Start here

Maths In Real Life

Three situations where angles fit together or are shared out. The pictures are made separately and the angles are drawn on top by script, so every number matches its problem.

$A bathroom floor tiled with regular hexagonal tiles, three tiles meeting at a point, with the three interior angles marked over the picture$

Maths In Real Life 1

Tiling a floor with hexagons

Tiles must meet at a point without gaps or overlaps, so their angles at the point have to add up to a full turn.

A floor is tiled with identical regular polygons.

(a)3 regular hexagon tiles meet at one point. Show that they fit together with no gap.
(b)3 regular pentagon tiles meet at one point. Do they fit together with no gap? Explain.

Show full solutionHide solution

(a)

\displaystyle \text{each interior angle} = 180 - 360 \div 6 = 120

interior angle of a regular polygon

\displaystyle 3 \times 120 = 360

angles at a point

(b)

\displaystyle \text{each interior angle} = 180 - 360 \div 5 = 108

interior angle of a regular polygon

\displaystyle 3 \times 108 = 324

angles at a point

\displaystyle 360 - 324 = 36

what is left of the full turn

Final answer

(a)

3 \times 120^{\circ} = 360^{\circ}

, a full turn, so there is no gap and no overlap.

(b)No:

3 \times 108^{\circ} = 324^{\circ}

leaves a gap of

36^{\circ}

$A courtyard paved with regular octagonal tiles and square tiles, with the angles at one meeting point marked over the picture$

Maths In Real Life 2

Octagons and squares

Some patterns mix two shapes: regular octagons leave square gaps, which can be filled with squares.

A designer uses regular octagonal tiles and square tiles.

Two regular octagons and one square meet at a point. Do they fit with no gap?

Show full solutionHide solution

\displaystyle \text{each octagon: } 180 - 360 \div 8 = 135

\displaystyle \text{the quadrilateral: } 180 - 360 \div 4 = 90

\displaystyle 135 + 135 + 90 = 360

angles at a point

Final answer

Yes, they fit with no gap:

135^{\circ} + 135^{\circ} + 90^{\circ} = 360^{\circ}.

$A step-ladder opened on a floor, with the two legs equal and the angle between each leg and the floor marked over the picture$

Maths In Real Life 3

The step-ladder

A step-ladder opens into an isosceles triangle, because its two legs are the same length.

A step-ladder has two legs of equal length.

A step-ladder is opened so that each leg makes an angle of 65° with the floor. Find the angle between the two legs at the top.

Show full solutionHide solution

The two legs and the floor form an isosceles triangle, because the legs have equal length.

equal legs, equal base angles

\displaystyle 65 + 65 + x = 180

angle sum of a triangle

\displaystyle x = 180 - 130 = 50

Final answer

50^{\circ}

Supports G1.1 · G1.2

Types of angle and the basic angle facts

An angle measures a turn, in degrees. Every angle in this chapter is described by how big it is, and by a few rules that connect angles that sit together.

Case table 1: the types of angle (right, acute, obtuse and reflex are the four named in the syllabus)
Type	Size	Picture
Acute	more than 0° and less than 90°	acute
Right	exactly 90°	right
Obtuse	more than 90° and less than 180°	obtuse
Straight	exactly 180°	a straight line
Reflex	more than 180° and less than 360°	reflex

The Core Idea

Three facts about angles that sit together.

Angles on a straight line add up to 180°. Angles at a point add up to 360°. Vertically opposite angles (opposite each other where two lines cross) are equal.

angles on a straight line

angles at a point

vertically opposite angles

SGSchoolKaki Tip

Write the reason beside every step, in standard words: "angles on a straight line", "angles at a point", "vertically opposite angles". A correct number with no reason loses the reasoning mark.

Think About It

Two angles on a straight line are equal. What must each one be?

Show the worked answerHide the worked answer

\displaystyle 2x = 180

angles on a straight line

\displaystyle x = 180 \div 2 = 90

Final answerEach is 90°, so they are right angles.

Example 10.1

Name the angle

Name each angle as acute, right, obtuse, straight or reflex.

(i)35°
(ii)90°
(iii)127°
(iv)200°
(v)180°

Show solutionHide solution

(i)

35° is more than 0° and less than 90°.

(ii)

90° is exactly 90°.

(iii)

127° is more than 90° and less than 180°.

(iv)

200° is more than 180° and less than 360°.

(v)

180° is exactly 180°.

Final answer

(i)an acute angle

(ii)a right angle

(iii)an obtuse angle

(iv)a reflex angle

(v)a straight angle

Diagram for Example 10.2

Example 10.2

Angles on a straight line

The diagram shows 3 angles on a straight line. Find $x$ .

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

\displaystyle 47 + x + 68 = 180

angles on a straight line

\displaystyle x = 180 - 47 - 68

subtract the known angles

\displaystyle x = 65

Final answer

x = 65^{\circ}

Diagram for Example 10.3

Example 10.3

Angles at a point

Four lines meet at a point, as shown. Find $x$ .

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

\displaystyle 105 + 88 + x + 73 = 360

angles at a point

\displaystyle x = 360 - 105 - 88 - 73 = 94

subtract the known angles

Final answer

x = 94^{\circ}

Diagram for Example 10.4

Example 10.4

Vertically opposite angles with algebra

Two straight lines AB and CD cross at O. Find $x$ and the angle marked $y$ .

(i)Find $x$ .
(ii)Find the angle AOD (the angle marked $y$ ).

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

\displaystyle 2x + 10 = 40

vertically opposite angles

\displaystyle 2x = 40 - 10 = 30

move the number across

\displaystyle x = 30 \div 2 = 15

divide by the coefficient of x

(ii)

\displaystyle y + 40 = 180

angles on a straight line

\displaystyle y = 180 - 40 = 140

Final answer

(i)

x = 15

(ii)

y = 140^{\circ}

Diagram for Example 10.5

Example 10.5

Algebra on a straight line

Three angles on a straight line are marked with expressions in $x$ . Find $x$ and the size of the largest of the three angles.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

\displaystyle x + 2x + (x + 20) = 180

angles on a straight line

\displaystyle 4x + 20 = 180

collect the x terms and the numbers

\displaystyle 4x = 180 - 20 = 160

move the number across

\displaystyle x = 160 \div 4 = 40

The largest angle is

(2x)

80

°.

put x back into each expression

Final answer

x = 40

; the largest angle is

80^{\circ}

Links To

The free overview of geometry and congruence covers the same ideas in short: free E-Math revision page: Geometry & Congruence

Supports G1.3 · G1.2

Parallel lines and a transversal

A line that crosses two other lines is a transversal. When the two lines are parallel (marked with matching arrows, written AB // CD), the eight angles it makes have only two sizes, and three named pairs tell you which are equal and which add up to a straight line.

Case table 2: the three pairs of angles when AB // CD
Name	Picture (the two marked angles)	Shape to spot	Relationship
Corresponding angles	in matching positions	F	equal
Alternate angles	between the lines, opposite sides	Z	equal
Interior angles	between the lines, same side	C	add up to 180°

The Core Idea

Parallel lines make equal pairs and straight-line pairs.

If AB // CD then corresponding angles are equal, alternate angles are equal, and interior angles add up to 180°. The reverse also works: equal corresponding or alternate angles show that the lines are parallel.

SGSchoolKaki Tip

Write "AB // CD" inside the reason, for example "alternate angles, AB // CD". The parallel statement is part of the reason.

SGSchoolKaki Tip

Do not use these rules unless the lines are parallel. If the question does not say so and you cannot show it, the rules do not apply.

Think About It

Two interior angles on the same side of a transversal add up to 180°. What does that tell you about the two lines?

Show the worked answerHide the worked answer

Colour keyfinal answer

If the lines were not parallel, the interior angles on one side would add up to more or less than 180°.

so the sum tells you about the direction of the lines

Final answerThe lines are parallel, because the interior angles add up to a straight line.

Diagram for Example 10.6

Example 10.6

Corresponding, alternate and interior angles

In the diagram AB // CD and the line EF crosses them at P and Q. Find $x$ , $y$ and $z$ , giving a reason for each.

(i) $x$
(ii) $y$
(iii) $z$

Show solutionHide solution

(i)

\displaystyle x = 72

corresponding angles, AB // CD

(ii)

\displaystyle y = 72

alternate angles, AB // CD

(iii)

\displaystyle z + 72 = 180

interior angles, AB // CD

\displaystyle z = 180 - 72 = 108

Final answer

(i)

72^{\circ}

(corresponding angles, AB // CD)

(ii)

72^{\circ}

(alternate angles, AB // CD)

(iii)

108^{\circ}

(interior angles, AB // CD)

Diagram for Example 10.7

Example 10.7

A chain of reasons

AB // CD. Use the given angle to find $a$ , $b$ and $c$ , with a reason at every step.

(i) $a$ (the angle QPA)
(ii) $b$ (the angle EPB)
(iii) $c$ (the angle APE)

Show solutionHide solution

(i)

\displaystyle a = 128

corresponding angles, AB // CD (FQC and QPA)

(ii)

\displaystyle b = a = 128

vertically opposite angles (QPA and EPB)

(iii)

\displaystyle c + 128 = 180

angles on a straight line

\displaystyle c = 180 - 128 = 52

Final answer

(i)

128^{\circ}

(ii)

128^{\circ}

(iii)

52^{\circ}

Diagram for Example 10.8

Example 10.8

Parallel lines with algebra

AB // CD. The angles CQP and BPQ are marked as shown. Find $x$ and then the angle $y$ (the angle PQD).

(i)Find $x$ .
(ii)Find the angle PQD.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplifyexpand, rewrite, substitute or solve

(i)

\displaystyle 3x + 10 = x + 50

alternate angles, AB // CD

\displaystyle 2x = 40

collect the x terms on one side

\displaystyle x = 20

(ii)

\displaystyle \angle\text{CQP} = 3x + 10 = 70

substitute x

\displaystyle \angle\text{PQD} + 70 = 180

angles on a straight line (CQP and PQD on the line CD)

\displaystyle \angle\text{PQD} = 180 - 70 = 110

Final answer

(i)

x = 20

(ii)

y =110^{\circ}

Your Turn 10.8

AB // CD. The angles CQP and BPQ are marked as shown. Find $x$ and then the angle $y$ (the angle PQD).

(i)Find $x$ .
(ii)Find the angle PQD.

Diagram for Your Turn 10.8

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplifyexpand, rewrite, substitute or solve

(i)

\displaystyle 2x - 5 = x + 25

alternate angles, AB // CD

\displaystyle x = 30

collect the x terms on one side

(ii)

\displaystyle \angle\text{CQP} = 2x - 5 = 55

substitute x

\displaystyle \angle\text{PQD} + 55 = 180

angles on a straight line (CQP and PQD on the line CD)

\displaystyle \angle\text{PQD} = 180 - 55 = 125

Final answer

(i)

x = 30

(ii)

y =125^{\circ}

Diagram for Example 10.9

Example 10.9

A bent line between parallels

PQ // RS. X is on PQ and Y is on RS, and T lies between the two lines. Find $x$ , the angle XTY.

Show solutionHide solution

Draw a line through T parallel to PQ and RS. It splits

x

into two parts.

an extra line often unlocks the problem

\displaystyle \text{upper part} = 35

alternate angles, PQ // the new line

\displaystyle \text{lower part} = 40

alternate angles, RS // the new line

\displaystyle x = 35 + 40 = 75

Final answer

x = 75^{\circ}

Diagram for Example 10.10

Example 10.10

Are the lines parallel?

For each diagram, state whether AB is parallel to CD, with a reason.

(i)the first diagram
(ii)the second diagram

Show solutionHide solution

(i)

The marked angles are corresponding angles: 68° and 68°.

look at which pair of angles is marked

Equal corresponding angles mean parallel lines; unequal ones do not.

the converse of the corresponding-angles rule

(ii)

The marked angles are corresponding angles: 65° and 71°.

Compare them: equal means parallel.

Final answer

(i)AB // CD, because the corresponding angles are equal.

(ii)AB is not parallel to CD, because the corresponding angles are not equal.

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Notes version 1.0 · Last updated 3 Nov 2026 · Chapter 10 of 19