Angles on lines and around points | Year 7

01 / LEARN

Understand the idea.

Choose the total that the diagram gives

Adjacent angles share a vertex and an arm without overlapping. Their total is $180^{\circ}$ only when their outer arms form a straight line. Identify that line before using the straight-angle rule.

Write the marked relationship first, then subtract the known part from the total. A right-angle marker supplies a total of $90^{\circ}$ instead, so touching angles do not automatically call for a total of $180^{\circ}$ .

a + b = 180^{\circ}

A complete turn has no gap

Non-overlapping angles that fill a complete turn around a point total $360^{\circ}$ . Include every region exactly once. Add all the known angles, then subtract their combined total to find the missing region.

Read which region is marked, especially if it is reflex. Give the reason with your calculation: angles on a straight line, in a right angle, or around a point. The diagram determines which total is justified.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Missing angle on a line

The two marked angles lie on a straight line. Find $x$ .

Worked solution

1
Use the marked straight line.
$x + 127^{\circ} = 180^{\circ}$
The two adjacent regions together fill a straight angle.
2
Subtract the known angle.
$x = 180^{\circ} - 127^{\circ} = 53^{\circ}$
The missing region is the part of the half-turn left after removing 127 degrees.
3
Check the combined angle.
$53^{\circ} + 127^{\circ} = 180^{\circ}$
The calculated pair fills the required straight angle.

Answer

$x = 53^{\circ}$ .

EXAMPLE 02

Three angles around one point

Three angles around a point are $96^{\circ}$ , $118^{\circ}$ and $y$ . Find $y$ .

Worked solution

1
Write the complete-turn total.
$96^{\circ} + 118^{\circ} + y = 360^{\circ}$
The three non-overlapping angles fill the turn around the point.
2
Combine the known angles.
$96^{\circ} + 118^{\circ} = 214^{\circ}$
These two regions use 214 degrees of the full turn.
3
Find and check the remaining angle.
$\begin{aligned} y & = 360^{\circ} - 214^{\circ} = 146^{\circ} \\ 214^{\circ} + 146^{\circ} & = 360^{\circ} \end{aligned}$
The remaining 146 degrees completes the turn with no gap or overlap.

Answer

$y = 146^{\circ}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

A right angle is split into $32^{\circ}$ and $x$ . Find $x$ .

Show answer & reasoning

$58^{\circ}$ .

The marker gives a right-angle total of $90^{\circ}$ . Subtract the known $32^{\circ}$ to get $x = 58^{\circ}$ ; the two parts then add back to $90^{\circ}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Two adjacent angles on a straight line are $148^{\circ}$ and $a$ . Find $a$ .
2 marks
Four angles around a point are $85^{\circ}$ , $110^{\circ}$ , $74^{\circ}$ and $b$ . Find $b$ .
2 marks
Three equal adjacent angles fill a straight angle. Find the size of each.
2 marks
An angle of $54^{\circ}$ and angle $c$ fill a right angle. Find $c$ .
2 marks
A robot turns $135^{\circ}$ clockwise, then continues clockwise until it has completed one full turn. How far is the second turn?
2 marks
Alex subtracts $72^{\circ}$ from $360^{\circ}$ to find the adjacent angle on a straight line. Explain the error and give the correct answer.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two adjacent angles on a straight line are $148^{\circ}$ and $a$ . Find $a$ .
2 marks
$a = 180^{\circ} - 148^{\circ} = 32^{\circ}$ .
$32^{\circ}$ .
Four angles around a point are $85^{\circ}$ , $110^{\circ}$ , $74^{\circ}$ and $b$ . Find $b$ .
2 marks
$85 + 110 + 74 = 269$ .
$b = 360^{\circ} - 269^{\circ} = 91^{\circ}$ .
$91^{\circ}$ .
Three equal adjacent angles fill a straight angle. Find the size of each.
2 marks
Their total is $180^{\circ}$ .
$180 \div 3 = 60$ .
$60^{\circ}$ .
An angle of $54^{\circ}$ and angle $c$ fill a right angle. Find $c$ .
2 marks
$c = 90^{\circ} - 54^{\circ} = 36^{\circ}$ .
$36^{\circ}$ .
A robot turns $135^{\circ}$ clockwise, then continues clockwise until it has completed one full turn. How far is the second turn?
2 marks
The full turn is $360^{\circ}$ .
$360 - 135 = 225$ . The second turn is reflex.
$225^{\circ}$ .
Alex subtracts $72^{\circ}$ from $360^{\circ}$ to find the adjacent angle on a straight line. Explain the error and give the correct answer.
3 marks
The outer arms form a straight line, not a full turn.
$180^{\circ} - 72^{\circ} = 108^{\circ}$ .
The total should be 180 degrees; the missing angle is $108^{\circ}$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Mathematics

Year 7 angle relationships, polygon properties and classification algorithms. Chapter grouping follows the approved Ascent outline.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose the total that the diagram gives

A complete turn has no gap

See the method in action.

Missing angle on a line

Use the marked straight line.

Subtract the known angle.

Check the combined angle.

Three angles around one point

Write the complete-turn total.

Combine the known angles.

Find and check the remaining angle.

Try it yourself.

Check your understanding

Keep practising