Sec 1 · G3 (Express)

Angles and parallel lines

Practise angles and parallel lines with step-by-step help. Free account.

1. Types of angles

Angle	Size
acute	less than $90^\circ$
right	$90^\circ$
obtuse	between $90^\circ$ and $180^\circ$
reflex	between $180^\circ$ and $360^\circ$
complementary angles	add up to $90^\circ$
supplementary angles	add up to $180^\circ$

📖 Worked example

$\angle ABC$ and $\angle PQR$ are supplementary angles. Find $x$ .

Step 1 of 2

Supplementary angles: what do they add up to?

x^\circ + 78^\circ = 180^\circ

2. Angles at a point and on a line

Angles on a straight line, 130° and 50°, add to 180°. Angles at a point, 120°, 110° and 130°, add to 360°. Two crossing lines make vertically opposite angles of 85° and 85°.

Property	Reason to write
add up to $180^\circ$	$\angle$ s on a str. line
add up to $360^\circ$	$\angle$ s at a pt.
equal	vert. opp. $\angle$ s

Angles on a straight line: 3x + 2x = 180°

\displaystyle 3x + 2x = 180

e.g.

\displaystyle 3x + 2x = 180

\displaystyle 5x = 180

\displaystyle x = 36

reason for the first line: $\angle$ s on a str. line

📖 Worked example

In the diagram, the line segments $OA$ , $OB$ , $OC$ and $OD$ meet at the point $O$ . Find $x$ .

Step 1 of 2

Add up the angles round the point O

48^\circ + 72^\circ + 95^\circ + x^\circ = 360^\circ

📖 Worked example

In the diagram, the straight lines $AB$ and $CD$ meet at $O$ . Find $x$ .

Step 1 of 1

Find x and give the reason

x = 106

(vert. opp. ∠s)

Practise this →

3. Parallel lines

Two parallel lines cut by a transversal at 62°. Corresponding angles are both 62°; alternate angles are both 62°; interior angles are 62° and 118°, adding to 180°.

Angles	Shape	Rule	Reason to write
corresponding	F	equal	corr. $\angle$ s, $AB \parallel CD$
alternate	Z	equal	alt. $\angle$ s, $AB \parallel CD$
interior	C	add up to $180^\circ$	int. $\angle$ s, $AB \parallel CD$

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ , and the straight line $EF$ cuts them at $P$ and $Q$ . Find $x$ .

Step 1 of 1

Find x and give the reason

x = 66

(alt. ∠s,

AB \parallel CD

)

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ , and the straight line $EF$ cuts them at $P$ and $Q$ . Find $x$ .

Step 1 of 2

Which angle fact links x° and the given angle?

x^\circ + 66^\circ = 180^\circ

(int. ∠s,

AB \parallel CD

)

Figure	Use
a bend with a reflex angle	line through the bend parallel to both: inside angle $=$ sum of the two alternate angles; reflex angle $= 360^\circ -$ inside angle
a triangle between parallel lines	alternate angles give its angles; equal angles face equal sides; the longest side faces the largest angle

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ . The reflex angle $BED$ is $(3x + 15)^\circ$ . Find $x$ .

Step 1 of 5

Draw a line through E parallel to AB and CD

Draw the line

EF

through

E

, parallel to

AB

and

CD

, pointing towards

B

and

D

. It splits the angle

BED

inside the bend into

\angle BEF

and

\angle FED

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ . $QR$ bisects angle $PQD$ . Which two sides of triangle $PQR$ are equal? Choose the answer.

Step 1 of 4

Find ∠PQD

\angle PQD = 180^\circ - 114^\circ

\phantom{\angle PQD} = 66^\circ

(int. ∠s,

AB \parallel CD

)

Practise this →

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Sec 1 · G3 (Express)

Angles and parallel lines

Practise angles and parallel lines with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Types of angles

Angle	Size
acute	less than $90^\circ$
right	$90^\circ$
obtuse	between $90^\circ$ and $180^\circ$
reflex	between $180^\circ$ and $360^\circ$
complementary angles	add up to $90^\circ$
supplementary angles	add up to $180^\circ$

📖 Worked example

$\angle ABC$ and $\angle PQR$ are supplementary angles. Find $x$ .

Step 1 of 2

Supplementary angles: what do they add up to?

x^\circ + 78^\circ = 180^\circ

2. Angles at a point and on a line

Property	Reason to write
add up to $180^\circ$	$\angle$ s on a str. line
add up to $360^\circ$	$\angle$ s at a pt.
equal	vert. opp. $\angle$ s

Angles on a straight line: 3x + 2x = 180°

\displaystyle 3x + 2x = 180

e.g.

\displaystyle 3x + 2x = 180

\displaystyle 5x = 180

\displaystyle x = 36

reason for the first line: $\angle$ s on a str. line

📖 Worked example

In the diagram, the line segments $OA$ , $OB$ , $OC$ and $OD$ meet at the point $O$ . Find $x$ .

Step 1 of 2

Add up the angles round the point O

48^\circ + 72^\circ + 95^\circ + x^\circ = 360^\circ

📖 Worked example

In the diagram, the straight lines $AB$ and $CD$ meet at $O$ . Find $x$ .

Step 1 of 1

Find x and give the reason

x = 106

(vert. opp. ∠s)

Practise this →

3. Parallel lines

Angles	Shape	Rule	Reason to write
corresponding	F	equal	corr. $\angle$ s, $AB \parallel CD$
alternate	Z	equal	alt. $\angle$ s, $AB \parallel CD$
interior	C	add up to $180^\circ$	int. $\angle$ s, $AB \parallel CD$

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ , and the straight line $EF$ cuts them at $P$ and $Q$ . Find $x$ .

Step 1 of 1

Find x and give the reason

x = 66

(alt. ∠s,

AB \parallel CD

)

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ , and the straight line $EF$ cuts them at $P$ and $Q$ . Find $x$ .

Step 1 of 2

Which angle fact links x° and the given angle?

x^\circ + 66^\circ = 180^\circ

(int. ∠s,

AB \parallel CD

)

Figure	Use
a bend with a reflex angle	line through the bend parallel to both: inside angle $=$ sum of the two alternate angles; reflex angle $= 360^\circ -$ inside angle
a triangle between parallel lines	alternate angles give its angles; equal angles face equal sides; the longest side faces the largest angle

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ . The reflex angle $BED$ is $(3x + 15)^\circ$ . Find $x$ .

Step 1 of 5

Draw a line through E parallel to AB and CD

Draw the line

EF

through

E

, parallel to

AB

and

CD

, pointing towards

B

and

D

. It splits the angle

BED

inside the bend into

\angle BEF

and

\angle FED

📖 Worked example

In the diagram, $AB$ is parallel to $CD$ . $QR$ bisects angle $PQD$ . Which two sides of triangle $PQR$ are equal? Choose the answer.

Step 1 of 4

Find ∠PQD

\angle PQD = 180^\circ - 114^\circ

\phantom{\angle PQD} = 66^\circ

(int. ∠s,

AB \parallel CD

)

Practise this →

Try a question: Angles and parallel lines

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise angles and parallel lines with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

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