Properties of quadrilaterals established using triangles | Year 8

01 / LEARN

Understand the idea.

A diagonal creates two triangles

A diagonal can turn a quadrilateral proof into two triangle proofs. In parallelogram $A B C D$ , begin with the defining parallel sides: $A B ∥ C D$ and $B C ∥ A D$ . Diagonal $A C$ forms two pairs of equal alternate angles.

The triangles also share side $A C$ , so ASA establishes $△ A B C ≅ △ C D A$ . Corresponding parts then give equal opposite sides. For opposite angles at $A$ and $C$ , combine the matching smaller angles formed by the diagonal.

Special quadrilaterals add information

Special quadrilaterals provide extra facts. A rectangle has right angles, and its diagonals are equal and bisect each other. A rhombus has four equal sides, and its diagonals bisect each other at right angles. A square has both sets of properties.

A proof needs a reason for every equality: a given fact, an established theorem or a valid deduction. Do not assume the property being proved. Measurements in one drawing can suggest a claim, but cannot establish it for every shape of that type.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Explain opposite-side equality

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

Worked solution

1
Use the first pair of parallel sides.
$A B ∥ C D ⟹ ∠ B A C = ∠ D C A$
AC is a transversal, so alternate interior angles are equal.
2
Use the second pair of parallel sides.
$B C ∥ A D ⟹ ∠ B C A = ∠ D A C$
The same diagonal gives a second pair of equal alternate interior angles.
3
Identify the shared included side.
$A C = C A$
Both triangles contain the same diagonal between their matched angles.
4
Conclude triangle congruence.
$△ A B C ≅ △ C D A by ASA$
Two angle pairs and the included side agree in this vertex order.
5
Transfer the matching side equality.
$A B = C D$
Corresponding parts of congruent triangles are equal.

Answer

AB = CD by corresponding parts of congruent triangles.

EXAMPLE 02

Use a rectangle theorem

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

Worked solution

1
Justify the equal-diagonal property.
$\begin{array}{c} A B = D C, B C = C B, \\ ∠ A B C = ∠ D C B = 90^{\circ} \\ △ A B C ≅ △ D C B by SAS \\ A C = D B \end{array}$
A rectangle has equal opposite sides and right angles; SAS makes its diagonals corresponding equal sides.
2
Substitute the two diagonal expressions.
$2 x + 3 = 17$
BD and AC must have equal lengths.
3
Subtract 3 from both sides.
$2 x = 14$
Subtracting equally preserves the equation.
4
Divide both sides by 2 and check.
$x = 7, 2 (7) + 3 = 17$
The value 7 makes the given diagonal lengths equal.

Answer

$x = 7$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

5 cm.

The diagonals of a parallelogram bisect each other, so each diagonal is cut into two equal lengths. One half is $5 cm$ , hence $x = 5 cm$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Parallelogram $A B C D$ has $A B = 9$ cm and $B C = 5$ cm. Find $C D$ and $D A$ .
2 marks
One angle of a parallelogram is 65°. Find an adjacent angle and its opposite angle.
3 marks
Parallelogram diagonals $A C$ and $B D$ meet at $O$ . $A O = 4$ cm and $B O = 6$ cm. Find $O C$ and $O D$ .
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 $A B C D$ is split by diagonal $A C$ . Which equal angles follow from $A B$ parallel to $C D$ ? Explain the transversal used.
3 marks

Read the worked solutions

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

Parallelogram $A B C D$ has $A B = 9$ cm and $B C = 5$ cm. Find $C D$ and $D A$ .
2 marks
Opposite sides of a parallelogram are equal.
$C D = 9$ cm; $D A = 5$ 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 $A C$ and $B D$ meet at $O$ . $A O = 4$ cm and $B O = 6$ cm. Find $O C$ and $O D$ .
2 marks
The diagonals bisect each other, so $A O = O C$ and $B O = O D$ .
$O C = 4$ cm; $O D = 6$ 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 $A B C D$ is split by diagonal $A C$ . Which equal angles follow from $A B$ parallel to $C D$ ? Explain the transversal used.
3 marks
$A C$ crosses both parallel lines.
These angles are an alternate-angle pair.
Angle $B A C$ equals angle $D C A$ .

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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 ↗

In this lesson

Understand the idea.

A diagonal creates two triangles

Special quadrilaterals add information

See the method in action.

Explain opposite-side equality

Use the first pair of parallel sides.

Use the second pair of parallel sides.

Identify the shared included side.

Conclude triangle congruence.

Transfer the matching side equality.

Use a rectangle theorem

Justify the equal-diagonal property.

Substitute the two diagonal expressions.

Subtract 3 from both sides.

Divide both sides by 2 and check.

Try it yourself.

Check your understanding

Keep practising