Reasoning about unknowns | Year 6

01 / LEARN

Understand the idea.

Choose the whole or the pair

Choose an angle fact from the relationship shown. Adjacent parts of a straight angle total $180^{\circ}$ ; parts filling a full turn total $360^{\circ}$ ; vertically opposite angles at a crossing are equal. The positions of the arms decide the rule.

Use a letter for the missing angle and write an equation describing that relationship. Add known parts before subtracting them from the whole. Work with the labelled values even when a drawing is not exactly to scale.

Show a chain of reasons

In a linked problem, an angle found in one step becomes known information for the next. Keep each calculation beside its reason, such as opposite-angle equality or a straight-line total. This makes the argument easy to follow and check.

Test the final values against the original relationships. A negative missing angle or parts adding beyond the stated whole signals that a value, region or rule was misread. Check the geometry rather than changing the answer to match a visual guess.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Two known parts

A straight angle contains adjacent parts 42°, 65° and x. Find x and explain.

Worked solution

1
State the straight-line relationship.
$42^{\circ} + 65^{\circ} + x = 180^{\circ}$
The three adjacent parts fill the straight angle.
2
Combine the two known parts.
$42^{\circ} + 65^{\circ} = 107^{\circ}$
Their sum is the amount of the half-turn already occupied.
3
Subtract that sum from the whole.
$x = 180^{\circ} - 107^{\circ} = 73^{\circ}$
The missing angle is the remaining part of the straight angle.
4
Check the completed half-turn.
$42^{\circ} + 65^{\circ} + 73^{\circ} = 180^{\circ}$
The calculated value satisfies the original straight-line condition.

Answer

73°

EXAMPLE 02

Use an opposite angle, then a neighbour

At a crossing, a given angle is 116°. Angle a is vertically opposite it. Angle b is adjacent to a. Find a and b.

Worked solution

1
Use the vertically opposite pair.
$a = 116^{\circ}$
Angle a is directly opposite the given angle at the crossing.
2
Write the relationship for the neighbour.
$a + b = 180^{\circ}$
Adjacent angles formed by the two crossing straight lines fill a straight angle.
3
Substitute and solve for the neighbour.
$b = 180^{\circ} - 116^{\circ} = 64^{\circ}$
Subtracting angle a leaves the other part of the half-turn.
4
Check the two values together.
$116^{\circ} + 64^{\circ} = 180^{\circ}$
The values agree with both opposite-angle equality and the adjacent straight-angle condition.

Answer

a = 116° and b = 64°.

03 / YOUR TURN

Try it yourself.

Check your understanding

A straight angle has parts 25°, 80° and x. Find x and name the rule.

Show answer & reasoning

75°; adjacent angles on a straight line total 180°.

The known parts total $25^{\circ} + 80^{\circ} = 105^{\circ}$ . Since adjacent angles on a straight line total $180^{\circ}$ , $x = 180^{\circ} - 105^{\circ} = 75^{\circ}$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

A straight angle is divided into $45^{\circ}$ , $68^{\circ}$ and $x$ . Find $x$ .
2 marks
Four angles around a point are $50^{\circ}$ , $75^{\circ}$ , $95^{\circ}$ and $x$ . Find $x$ .
2 marks
Two lines cross. One angle is $132^{\circ}$ . Find its opposite angle and an adjacent angle.
2 marks
A right angle is divided into $29^{\circ}$ and $x$ . Find $x$ .
1 mark
A student says $115^{\circ}$ and $75^{\circ}$ are adjacent angles on a straight line. Explain the error and correct the second angle.
2 marks
A full turn contains one angle of $80^{\circ}$ and two equal angles. Find the equal angles and check your result.
3 marks

Read the worked solutions

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

A straight angle is divided into $45^{\circ}$ , $68^{\circ}$ and $x$ . Find $x$ .
2 marks
$45 + 68 = 113$ .
$180 - 113 = 67$ .
$67^{\circ}$
Four angles around a point are $50^{\circ}$ , $75^{\circ}$ , $95^{\circ}$ and $x$ . Find $x$ .
2 marks
$50 + 75 + 95 = 220$ .
$360 - 220 = 140$ .
$140^{\circ}$
Two lines cross. One angle is $132^{\circ}$ . Find its opposite angle and an adjacent angle.
2 marks
Opposite angles are equal.
$180^{\circ} - 132^{\circ} = 48^{\circ}$ .
Opposite $132^{\circ}$ ; adjacent $48^{\circ}$ .
A right angle is divided into $29^{\circ}$ and $x$ . Find $x$ .
1 mark
A right angle is $90^{\circ}$ .
$90^{\circ} - 29^{\circ} = 61^{\circ}$ .
$61^{\circ}$
A student says $115^{\circ}$ and $75^{\circ}$ are adjacent angles on a straight line. Explain the error and correct the second angle.
2 marks
$115^{\circ} + 75^{\circ} = 190^{\circ}$ , not $180^{\circ}$ .
$180^{\circ} - 115^{\circ} = 65^{\circ}$ .
Their total is $190^{\circ}$ ; the second angle should be $65^{\circ}$ .
A full turn contains one angle of $80^{\circ}$ and two equal angles. Find the equal angles and check your result.
3 marks
$360^{\circ} - 80^{\circ} = 280^{\circ}$ .
$280^{\circ} \div 2 = 140^{\circ}$ .
$80^{\circ} + 140^{\circ} + 140^{\circ} = 360^{\circ}$ .
$140^{\circ}$ each

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 6 angle relationships: straight-line angles, angles around a point, vertically opposite angles and connected reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose the whole or the pair

Show a chain of reasons

See the method in action.

Two known parts

State the straight-line relationship.

Combine the two known parts.

Subtract that sum from the whole.

Check the completed half-turn.

Use an opposite angle, then a neighbour

Use the vertically opposite pair.

Write the relationship for the neighbour.

Substitute and solve for the neighbour.

Check the two values together.

Try it yourself.

Check your understanding

Keep practising