Classifying triangles/quadrilaterals | Year 7

01 / LEARN

The idea, step by step

A triangle can have two descriptions

Classify sides and angles separately. An equilateral triangle has three equal sides; an isosceles triangle has at least two equal sides; a scalene triangle has no equal sides. Under this inclusive definition, an equilateral triangle is also a special isosceles triangle. Use the most specific name when asked.

An acute triangle has three acute angles. A right triangle has one right angle, and an obtuse triangle has one obtuse angle. A triangle with angles 40, 40 and 100 degrees is both isosceles and obtuse: the equal angles face equal sides.

Families are based on properties, not appearance

A quadrilateral has four straight sides. A parallelogram has two pairs of opposite parallel sides; a rectangle has four right angles; a rhombus has four equal sides; a square has both four right angles and four equal sides. Consequently every square is a rectangle, a rhombus and a parallelogram. Turning the drawing does not change these facts.

For this chapter, trapezium means a quadrilateral with exactly one pair of opposite parallel sides. A kite has two pairs of adjacent equal sides. Definitions of trapezium differ between books, so use the definition supplied in a question. Do not assume a shape is a square just because its sides look similar: use measurements or markings.

A common trap

A square cannot be a rectangle because their drawings usually look different.

Instead: A rectangle is any quadrilateral with four right angles. A square satisfies that definition and adds the extra condition of four equal sides.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two kinds of triangle name

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

Two angles are equal, so the sides opposite those angles are equal: it is isosceles.
The 100-degree angle is obtuse, so it is an obtuse triangle.
The descriptions give different information and can both be true.

An obtuse isosceles triangle.

Example 2

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.

Four right angles make it a rectangle.
The adjacent lengths differ, so its four sides are not all equal; it is not a square.
Its opposite sides are parallel, so it also belongs to the parallelogram family.

A rectangle; also a parallelogram.

Check your understanding

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

Show answer

Yes.

A rotation preserves equal lengths and right angles. Its orientation is irrelevant to its classification.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 ↗

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