Properties of quadrilaterals established using triangles | Year 8

01 / LEARN

The idea, step by step

A diagonal creates two triangles

Drawing a diagonal often turns a quadrilateral problem into a triangle problem. In parallelogram ABCD, opposite sides are parallel by definition. Diagonal AC gives two pairs of equal alternate angles and a shared side.

Those facts establish congruence of triangles ABC and CDA by ASA. Corresponding parts are then equal, so opposite sides and opposite angles of the parallelogram are equal. Notice that we did not assume the side equality we were trying to prove.

Special quadrilaterals add information

A rectangle is a parallelogram with right angles. Its diagonals are equal as well as bisecting each other. A rhombus has all four sides equal; its diagonals bisect each other at right angles. A square satisfies both sets of conditions.

State which given fact or established theorem justifies each step. One example drawn accurately may suggest a property, but it does not prove that the property holds for every quadrilateral of that type.

A common trap

Using a property as a premise while trying to prove that same property.

Instead: Begin with the shape’s given definition or previously proved results, then justify the new conclusion.

02 / WORKED EXAMPLES

See the thinking

Example 1

Explain opposite-side equality

ABCD is a parallelogram with diagonal AC. Explain why AB = CD.

Since AB is parallel to CD, angle BAC equals angle DCA.
Since BC is parallel to AD, angle BCA equals angle DAC.
AC is common to both triangles, so triangles ABC and CDA are congruent by ASA.
Corresponding sides AB and CD are equal.

AB = CD by corresponding parts of congruent triangles.

Example 2

Use a rectangle theorem

The diagonals of rectangle ABCD are AC = 17 cm and BD = $(2 x + 3)$ cm. Find $x$ .

In a rectangle, the two right triangles using adjacent equal opposite sides show that its diagonals are equal by SAS.
Set $2 x + 3 = 17$ .
Subtract 3, then divide by 2: $x = 7$ .

x = 7

Check your understanding

In a parallelogram, one diagonal is split by the other into lengths 5 cm and $x$ cm. Find $x$ .

Show answer

5 cm.

The diagonals of a parallelogram bisect each other.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Parallelogram ABCD has AB = 9 cm and BC = 5 cm. Find CD and DA.
2 marks
One angle of a parallelogram is 65°. Find an adjacent angle and its opposite angle.
3 marks
Parallelogram diagonals AC and BD meet at O. AO = 4 cm and BO = 6 cm. Find OC and OD.
2 marks
A rectangle has one diagonal 13 cm long. Find the length of its other diagonal and explain.
2 marks
A rhombus has one side 6 cm. Find its perimeter.
2 marks
A parallelogram ABCD is split by diagonal AC. Which equal angles follow from AB parallel to CD? Explain the transversal used.
3 marks

Read the worked solutions

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

Parallelogram ABCD has AB = 9 cm and BC = 5 cm. Find CD and DA.
2 marks
Opposite sides of a parallelogram are equal.
CD = 9 cm; DA = 5 cm.
One angle of a parallelogram is 65°. Find an adjacent angle and its opposite angle.
3 marks
Adjacent angles sum to 180°, giving $180 - 65 = 115$ °.
Opposite angles are equal.
Adjacent angle 115°; opposite angle 65°.
Parallelogram diagonals AC and BD meet at O. AO = 4 cm and BO = 6 cm. Find OC and OD.
2 marks
The diagonals bisect each other, so AO = OC and BO = OD.
OC = 4 cm; OD = 6 cm.
A rectangle has one diagonal 13 cm long. Find the length of its other diagonal and explain.
2 marks
A rectangle’s diagonals are equal.
13 cm.
A rhombus has one side 6 cm. Find its perimeter.
2 marks
All four sides are equal, so $P = 4 (6) = 24$ cm.
24 cm.
A parallelogram ABCD is split by diagonal AC. Which equal angles follow from AB parallel to CD? Explain the transversal used.
3 marks
AC crosses both parallel lines.
These angles are an alternate-angle pair.
Angle BAC equals angle DCA.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.

Official curriculum source ↗

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