Angle reasoning consolidation | Year 9

01 / LEARN

Understand the idea.

Choose the relationship

Choose an angle fact from the given information, rather than from the appearance of the sketch. Angles on a straight line total $180^{\circ}$ ; angles around a point total $360^{\circ}$ ; triangle interior angles total $180^{\circ}$ .

When lines are known to be parallel, corresponding and alternate angles are equal, while co-interior angles total $180^{\circ}$ . Identify the transversal and the positions of the two angles before naming a relationship.

Build a chain of reasons

An unknown angle may require two linked steps: first transfer a known angle through parallel lines, then use a triangle sum. Write the equation for each relationship and attach its reason so the argument can be checked.

A simple polygon with $n$ sides can be divided into $n - 2$ non-overlapping triangles. Its interior sum is $(n - 2) 180^{\circ}$ . Divide by $n$ to find an individual angle only when all interior angles are equal.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Triangle sum

A triangle has angles 48° and 67°. Find the third angle.

Worked solution

1
Write the triangle angle equation.
$x + 48^{\circ} + 67^{\circ} = 180^{\circ}$
The interior angles of a triangle total $180^{\circ}$ .
2
Add the known angles.
$48^{\circ} + 67^{\circ} = 115^{\circ}$
These are the two given interior angles.
3
Find the remaining angle.
$x = 180^{\circ} - 115^{\circ} = 65^{\circ}$
The unknown angle fills the part of the triangle sum not yet used.

Answer

65°

EXAMPLE 02

Adjacent angles

Find the acute angle x next to 122° at the upper intersection.

Worked solution

1
Identify the straight-line pair.
$x + 122^{\circ} = 180^{\circ}$
The adjacent angles at the upper intersection together form a straight angle.
2
Subtract the known angle.
$x = 180^{\circ} - 122^{\circ} = 58^{\circ}$
Subtracting $122^{\circ}$ from both sides isolates $x$ .
3
Check the requested angle type.
$0^{\circ} < 58^{\circ} < 90^{\circ}$
The calculated angle is acute, as specified; the second parallel line is not needed for this pair.

Answer

58°

03 / YOUR TURN

Try it yourself.

Check your understanding

A triangle has angles 35° and 82°. Find the third.

Show answer & reasoning

63°

The given angles total $35^{\circ} + 82^{\circ} = 117^{\circ}$ . The triangle sum leaves $180^{\circ} - 117^{\circ} = 63^{\circ}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Two adjacent angles form a straight line. One is $136^{\circ}$ . Find the other.
2 marks
Angles around a point are $85^{\circ}$ , $140^{\circ}$ and $x$ . Find $x$ .
2 marks
A triangle has angles $72^{\circ}$ and $49^{\circ}$ . Find its third angle.
2 marks
Two parallel lines are crossed by a transversal. An angle at the first crossing is $71^{\circ}$ . Find its corresponding angle at the second crossing.
2 marks
A quadrilateral has angles $88^{\circ}$ , $97^{\circ}$ , $104^{\circ}$ and $x$ . Find $x$ .
2 marks
Find one interior angle of a regular hexagon. Explain why division by 6 is valid.
3 marks

Read the worked solutions

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

Two adjacent angles form a straight line. One is $136^{\circ}$ . Find the other.
2 marks
Adjacent angles on a line total $180^{\circ}$ .
$180^{\circ} - 136^{\circ} = 44^{\circ}$ .
$44^{\circ}$
Angles around a point are $85^{\circ}$ , $140^{\circ}$ and $x$ . Find $x$ .
2 marks
Angles around a point total $360^{\circ}$ .
$360^{\circ} - 85^{\circ} - 140^{\circ} = 135^{\circ}$ .
$135^{\circ}$
A triangle has angles $72^{\circ}$ and $49^{\circ}$ . Find its third angle.
2 marks
Triangle sum is $180^{\circ}$ .
$180^{\circ} - 72^{\circ} - 49^{\circ} = 59^{\circ}$ .
$59^{\circ}$
Two parallel lines are crossed by a transversal. An angle at the first crossing is $71^{\circ}$ . Find its corresponding angle at the second crossing.
2 marks
Corresponding angles in parallel lines are equal.
$71^{\circ}$
A quadrilateral has angles $88^{\circ}$ , $97^{\circ}$ , $104^{\circ}$ and $x$ . Find $x$ .
2 marks
Its interior angles total $360^{\circ}$ .
$360^{\circ} - 88^{\circ} - 97^{\circ} - 104^{\circ} = 71^{\circ}$ .
$71^{\circ}$
Find one interior angle of a regular hexagon. Explain why division by 6 is valid.
3 marks
The angle sum is $(6 - 2) \times 180^{\circ} = 720^{\circ}$ .
Regular means all six angles are equal.
$720^{\circ} \div 6 = 120^{\circ}$ .
$120^{\circ}$

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose the relationship

Build a chain of reasons

See the method in action.

Triangle sum

Write the triangle angle equation.

Add the known angles.

Find the remaining angle.

Adjacent angles

Identify the straight-line pair.

Subtract the known angle.

Check the requested angle type.

Try it yourself.

Check your understanding

Keep practising