Classifying triangles/quadrilaterals | Year 7

01 / LEARN

Understand the idea.

A triangle can have two descriptions

Describe a triangle by its sides and by its angles separately. Equilateral means three equal sides; isosceles means at least two equal sides; scalene means no equal sides. With this inclusive definition, an equilateral triangle is also isosceles, but equilateral is more specific.

An acute triangle has three acute angles, a right triangle has one right angle, and an obtuse triangle has one obtuse angle. Equal angles face equal sides, so the angles $40^{\circ}, 40^{\circ}, 100^{\circ}$ identify an obtuse isosceles triangle.

Families are based on properties, not appearance

Classify quadrilaterals from their properties. A parallelogram has two pairs of opposite parallel sides; a rectangle has four right angles; a rhombus has four equal sides. A square satisfies both the rectangle and rhombus definitions, so it also belongs to the parallelogram family.

In this chapter a trapezium has exactly one pair of opposite parallel sides, and a kite has two pairs of adjacent equal sides. Use the definition supplied in the question: trapezium conventions vary. Rotation and an unmeasured sketch cannot change or establish these properties.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Two kinds of triangle name

A triangle has angles $40^{\circ}$ , $40^{\circ}$ and $100^{\circ}$ . Describe it by sides and by angles.

Worked solution

1
Use the two equal angles.
$40^{\circ} = 40^{\circ}$
Sides opposite equal angles are equal, so the triangle is isosceles.
2
Classify its largest angle.
$90^{\circ} < 100^{\circ} < 180^{\circ}$
An angle of 100 degrees is obtuse, so this is an obtuse triangle.
3
Combine the independent descriptions.
The side and angle classifications both apply: it is an obtuse isosceles triangle.

Answer

An obtuse isosceles triangle.

EXAMPLE 02

Most specific quadrilateral

A quadrilateral has four right angles. Two adjacent side lengths are 8 cm and 5 cm. Give its most specific name and one wider family.

Worked solution

1
Apply the right-angle property.
A quadrilateral with four right angles is a rectangle.
2
Check whether the name can be more specific.
$8 cm \neq 5 cm$
The adjacent sides have unequal lengths, so it cannot be a square.
3
Identify a wider family.
A rectangle has two pairs of opposite parallel sides, so it is also a parallelogram.

Answer

A rectangle; also a parallelogram.

03 / YOUR TURN

Try it yourself.

Check your understanding

A square is turned so one corner points upward. Is it still a square?

Show answer & reasoning

Yes.

Yes. Turning the square preserves its side lengths and angle sizes. It still has four equal sides and four right angles, so it remains a square.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

A triangle has side lengths 6 cm, 6 cm and 6 cm. Give its most specific side-based name and each angle size.
2 marks
A triangle has angles $90^{\circ}$ , $35^{\circ}$ and $55^{\circ}$ . Classify it by angles and by sides.
2 marks
A simple quadrilateral has four sides each measuring 5 cm. No angle sizes are supplied. What family is guaranteed? Is it necessarily a square?
2 marks
A quadrilateral has two pairs of opposite parallel sides. State its guaranteed family. What angle property would establish that it is a rectangle?
3 marks
Using this chapter’s definition, name a quadrilateral with exactly one pair of opposite parallel sides.
1 mark
Explain why a square belongs to both the rectangle and rhombus families.
2 marks

Read the worked solutions

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

A triangle has side lengths 6 cm, 6 cm and 6 cm. Give its most specific side-based name and each angle size.
2 marks
All three sides are equal.
Its three equal angles total 180 degrees, so each is $180 \div 3 = 60$ degrees.
Equilateral; each angle is $60^{\circ}$ .
A triangle has angles $90^{\circ}$ , $35^{\circ}$ and $55^{\circ}$ . Classify it by angles and by sides.
2 marks
It contains one right angle.
All angles differ, so their opposite side lengths differ.
Right-angled and scalene.
A simple quadrilateral has four sides each measuring 5 cm. No angle sizes are supplied. What family is guaranteed? Is it necessarily a square?
2 marks
Four equal sides define a rhombus.
Four right angles have not been established.
It is a rhombus. It need not be a square.
A quadrilateral has two pairs of opposite parallel sides. State its guaranteed family. What angle property would establish that it is a rectangle?
3 marks
Two pairs of opposite parallel sides define a parallelogram.
A right angle forces the adjacent supplementary angles and opposite angles to be right angles too.
Parallelogram; four right angles (or one right angle together with the parallelogram properties) establish a rectangle.
Using this chapter’s definition, name a quadrilateral with exactly one pair of opposite parallel sides.
1 mark
The definition specifies exactly one parallel pair.
Trapezium.
Explain why a square belongs to both the rectangle and rhombus families.
2 marks
The four right angles satisfy the rectangle definition.
The four equal sides satisfy the rhombus definition.
It has four right angles and four equal sides.

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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.

A triangle can have two descriptions

Families are based on properties, not appearance

See the method in action.

Two kinds of triangle name

Use the two equal angles.

Classify its largest angle.

Combine the independent descriptions.

Most specific quadrilateral

Apply the right-angle property.

Check whether the name can be more specific.

Identify a wider family.

Try it yourself.

Check your understanding

Keep practising