Lesson 9 ~25 min Unit 3 · Geometry +85 XP

Rhombuses, Squares, Kites and Trapeziums

Four more quadrilaterals, each with their own defining feature. Rhombuses have four equal sides and perpendicular diagonals; squares inherit from everyone; kites have two pairs of adjacent equal sides; trapeziums have exactly one pair of parallel sides.

Today's hook: What makes a diamond shape "special enough" to get its own name?

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Think First

warm-up

A kite flies through the air. Draw what a paper kite looks like from the front. Then draw a "diamond" (rhombus). Compare them: how many sides are equal? Are any sides parallel? Are the diagonals perpendicular? Now sketch a square. Which features appear in ALL three?

Record your answer in your workbook.

The Rhombus — "Equal Sides Diamond"

+5 XP

A rhombus is a parallelogram with all four sides equal in length. Because it's a parallelogram, it inherits the 5 parallelogram properties. It also gets THREE extras: (i) all four sides equal, (ii) diagonals are perpendicular (cross at $90^{\circ}$), and (iii) diagonals bisect the vertex angles.

In rhombus $ABCD$: $AB = BC = CD = DA$, $AC \perp BD$ (diagonals perpendicular), and each diagonal cuts the corner angles into two equal halves. The diagonals STILL bisect each other (parallelogram property).

Rhombus = parallelogram + 4 equal sides ⇒ perpendicular diagonals.

4 equal sides

Mark with single tick on every side — tells you "all four equal".

Diagonals at $90^{\circ}$

$AC \perp BD$ — mark the small square at the centre.

Bisects vertex angles

Diagonal $AC$ splits $\angle A$ and $\angle C$ in half.

What You'll Master

objectives

Know

Defining feature of rhombus, square, kite, trapezium
The extra properties each one inherits
The reason "co-int. angles" applies to trapeziums

Understand

Why a square inherits from BOTH rectangle and rhombus
Why a kite's diagonals are perpendicular but only one bisects the other
Why the angle pair at parallel-side ends is supplementary

Can Do

Find missing sides/angles in a rhombus or square
Solve kite problems using the symmetry diagonal
Find an unknown angle in a trapezium using co-interior

Words You Need

vocabulary

RhombusParallelogram with four equal sides.

SquareParallelogram with four equal sides AND four right angles.

KiteTwo pairs of ADJACENT equal sides; NOT a parallelogram in general.

TrapeziumQuadrilateral with EXACTLY one pair of parallel sides.

Perpendicular ($\perp$)Meeting at right angles ($90^{\circ}$).

Adjacent sidesSides that meet at a common vertex.

Axis of symmetryA line where the shape folds onto itself.

Co-interiorTwo same-side angles between parallel lines, sum $= 180^{\circ}$.

The Square — The Royal Quadrilateral

+5 XP

A square is a rectangle AND a rhombus at the same time. It has every parallelogram property, plus every rectangle property (4 right angles, equal diagonals), plus every rhombus property (4 equal sides, perpendicular diagonals, diagonals bisect angles). It's the most "loaded" of all the special quadrilaterals.

Square $WXYZ$ checklist: $WX = XY = YZ = ZW$ (4 equal sides), $\angle W = \angle X = \angle Y = \angle Z = 90^{\circ}$, diagonals equal (from rectangle), diagonals perpendicular (from rhombus), and diagonals bisect the $90^{\circ}$ corner angles into two $45^{\circ}$ pieces.

Square = rectangle ∩ rhombus (the ∩ means "both at once").

Has it all

Square wins every property test — useful in proofs.

Diagonals make $45^{\circ}$

Each $90^{\circ}$ angle is split into two $45^{\circ}$ halves.

4 axes of symmetry

Two through midpoints of sides, two along diagonals.

Book notes · Card 4

Square = rectangle + rhombus — gets every property of both.
Diagonals are equal AND perpendicular AND bisect $90^{\circ}$ corners into $45^{\circ}$ halves.
Has 4 axes of symmetry (most of any quadrilateral).

Quick check+1 coin

True or False: In a square, each diagonal cuts the $90^{\circ}$ corner into two $45^{\circ}$ angles.

The Kite — The Outsider

+5 XP

A kite has two pairs of adjacent (touching) equal sides — e.g. $AB = AD$ AND $CB = CD$. A kite is NOT a parallelogram (no parallel sides usually). Its properties: (i) ONE pair of opposite angles equal (the ones between unequal sides), (ii) diagonals perpendicular, (iii) the axis-of-symmetry diagonal bisects the other diagonal (but the other diagonal does NOT bisect the symmetry diagonal in general).

Kite $ABCD$: $AB = AD$ and $CB = CD$. The diagonal $AC$ is the axis of symmetry — it bisects $\angle A$ and $\angle C$, and is perpendicular to $BD$. $\angle B = \angle D$ (the two "side" angles between the unequal-length sides).

Kite: $AB = AD$, $CB = CD$ ⇒ diagonals perpendicular, $\angle B = \angle D$.

2 pairs adjacent

Equal sides MUST share a vertex (top pair / bottom pair).

One axis of symmetry

The "long" diagonal — bisects the other one at $90^{\circ}$.

Only one pair of equal angles

$\angle B = \angle D$ — the angles BETWEEN the unequal sides.

Book notes · Card 5

Kite has 2 pairs of ADJACENT equal sides (not opposite).
Diagonals meet at $90^{\circ}$; symmetry diagonal bisects the other diagonal.
One pair of opposite angles is equal — the angles between the unequal-length sides.

Quick check+1 coin

Which property is TRUE of every kite?

The Trapezium — "Exactly One Pair Parallel"

+5 XP

A trapezium (NSW Stage 4 definition) has exactly one pair of parallel sides. The two parallel sides are called the "parallel sides" and the other two are the "legs". Because two sides are parallel, the two angles at each leg are co-interior angles — so they sum to $180^{\circ}$.

Trapezium $ABCD$ with $AB \parallel DC$: $\angle A + \angle D = 180^{\circ}$ and $\angle B + \angle C = 180^{\circ}$ (co-int. angles, $AB \parallel DC$). The angles still sum to $360^{\circ}$ overall, which matches $180 + 180$.

Trapezium: $AB \parallel DC$ ⇒ $\angle A + \angle D = 180^{\circ}$, $\angle B + \angle C = 180^{\circ}$.

Exactly one pair

If both pairs are parallel, it's a parallelogram, not a trapezium (NSW).

Co-int. at each leg

The two angles at the same leg sum to $180^{\circ}$.

Isosceles trapezium

Special case: legs equal ⇒ two pairs of equal angles AND equal diagonals.

Book notes · Card 6

NSW trapezium: EXACTLY one pair of parallel sides (not "at least one").
At each leg: two co-interior angles summing to $180^{\circ}$.
Isosceles trapezium: legs equal ⇒ equal base angles, equal diagonals.

Fill the blank+1 coin

Trapezium $ABCD$ has $AB \parallel DC$ and $\angle A = 120^{\circ}$. Then $\angle D = $ $^{\circ}$ because angles at the same leg are co-interior.

Watch Me Solve It · 3 examples

Watch Me Solve It · Rhombus angles

+15 XP per step

PROBLEM

Rhombus $PQRS$ has $\angle P = 110^{\circ}$. Find the other three angles.

1
Opposite angles equal (rhombus is parallelogram)
$\angle R = \angle P = 110^{\circ}$ (opp. angles of parallelogram)
2
Adjacent angles co-int.
$\angle Q = 180 - 110 = 70^{\circ}$ (co-int. angles, $PQ \parallel SR$)
3
Last angle
$\angle S = \angle Q = 70^{\circ}$ (opp. angles)
Check $110 + 70 + 110 + 70 = 360^{\circ}$ ✓

Answer$\angle Q = 70^{\circ}, \angle R = 110^{\circ}, \angle S = 70^{\circ}$.

Watch Me Solve It · Kite angles

+15 XP per step

PROBLEM

Kite $ABCD$ has $AB = AD$ and $CB = CD$. $\angle A = 90^{\circ}$ and $\angle C = 30^{\circ}$. Find $\angle B$ and $\angle D$.

1
Use angle sum of quadrilateral
$\angle A + \angle B + \angle C + \angle D = 360^{\circ}$.
2
Kite property: $\angle B = \angle D$
$90 + \angle B + 30 + \angle B = 360 \Rightarrow 2\angle B = 240$.
3
Solve
$\angle B = 120^{\circ}$, so $\angle D = 120^{\circ}$.
Check $90 + 120 + 30 + 120 = 360^{\circ}$ ✓

Answer$\angle B = \angle D = 120^{\circ}$.

Watch Me Solve It · Trapezium leg angles

+15 XP per step

PROBLEM

Trapezium $ABCD$ has $AB \parallel DC$. $\angle A = 115^{\circ}$ and $\angle B = 75^{\circ}$. Find $\angle C$ and $\angle D$.

1
Co-int. on leg $AD$
$\angle A + \angle D = 180^{\circ}$ (co-int. angles, $AB \parallel DC$).
2
Find $\angle D$
$\angle D = 180 - 115 = 65^{\circ}$.
3
Co-int. on leg $BC$
$\angle C = 180 - 75 = 105^{\circ}$.
Check $115 + 75 + 105 + 65 = 360^{\circ}$ ✓

Answer$\angle C = 105^{\circ}, \angle D = 65^{\circ}$.

Common Pitfalls

heads-up

Calling a kite a rhombus

A rhombus needs ALL four sides equal. A kite has two pairs of adjacent equal sides — the pairs are DIFFERENT lengths.

Fix: 4 equal ⇒ rhombus. 2 + 2 of different lengths ⇒ kite.

"Both pairs of angles equal" in a kite

In a kite, only ONE pair of opposite angles is equal — the angles between the unequal-length sides.

Fix: Set up $\angle B = \angle D$, then use $\angle A + \angle B + \angle C + \angle D = 360^{\circ}$.

Trapezium = "at least one pair parallel"

NSW Stage 4 uses EXACTLY one pair. If both pairs are parallel, it's a parallelogram.

Fix: Trapezium ⇒ one pair parallel, other pair NOT parallel.

Copy Into Your Books

Rhombus

4 equal sides
Diagonals perpendicular
Diagonals bisect angles
(All parallelogram properties)

Square

4 equal sides + 4 right angles
Diagonals equal AND perpendicular
Diagonals bisect corners into $45^{\circ}$
(All rectangle & rhombus properties)

Kite

2 pairs adjacent equal sides
Diagonals perpendicular
One pair opp. angles equal
Symmetry diagonal bisects the other

Trapezium

Exactly 1 pair parallel sides
Co-int. at each leg sum $180^{\circ}$
Isosceles: legs equal ⇒ symmetric

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Special Quadrilaterals

4 problems

Four quick drills. Solve, then reveal.

1 Rhombus $ABCD$ has $\angle A = 80^{\circ}$. Find $\angle B$.

Co-int. with $\angle A$: $180 - 80$.$\angle B = 100^{\circ}$
2 Square has diagonals meeting at $90^{\circ}$. What angle does the diagonal make with each side?

Diagonal bisects $90^{\circ}$ corner.$45^{\circ}$
3 Kite $ABCD$ has $\angle A = 100^{\circ}$, $\angle C = 60^{\circ}$ (axis through $AC$). Find $\angle B$.

$\angle B = \angle D$, sum $= 360 - 100 - 60 = 200$, each is $100$.$\angle B = 100^{\circ}$
4 Trapezium $ABCD$, $AB \parallel DC$, $\angle A = 95^{\circ}$. Find $\angle D$.

Co-int. on leg $AD$: $180 - 95$.$\angle D = 85^{\circ}$

Complete in your workbook.

Quick Check · 5 questions

Which quadrilateral has FOUR equal sides and perpendicular diagonals but is NOT necessarily a square?

+10 XP

A diagonal of a square makes what angle with each side?

+10 XP

The defining feature of a kite is:

+10 XP

Trapezium $ABCD$ has $AB \parallel DC$ and $\angle A = 130^{\circ}$. What is $\angle D$?

+10 XP

A quadrilateral has 2 pairs of adjacent equal sides AND perpendicular diagonals, but its sides are NOT all equal. The shape is a:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Rhombus $PQRS$ has $\angle P = 124^{\circ}$ and $PQ = 7$ cm.
(a) Find $\angle Q$, with a reason.
(b) Find $\angle R$, with a reason.
(c) Find $QR$, with a reason.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. Kite $ABCD$ has $AB = AD$, $CB = CD$, $\angle A = 84^{\circ}$ and $\angle C = 116^{\circ}$.
(a) Use the angle sum of a quadrilateral to set up an equation for $\angle B + \angle D$.
(b) Use the kite property to write $\angle B = \angle D$ and solve.
(c) State $\angle B$ and $\angle D$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. Trapezium $WXYZ$ has $WX \parallel ZY$. Its angles are $\angle W = (3x)^{\circ}$, $\angle Z = (2x + 30)^{\circ}$.
(a) Write an equation using co-int. angles on leg $WZ$.
(b) Solve for $x$.
(c) State $\angle W$ and $\angle Z$.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — Rhombus (4 equal sides, perpendicular diagonals).

2. C — $45^{\circ}$ (bisection of $90^{\circ}$ corner).

3. B — Two pairs of adjacent equal sides.

4. D — $\angle D = 180 - 130 = 50^{\circ}$ (co-int.).

5. B — Kite.

Show Your Working Model Answers

Q6 (3 marks): (a) $\angle Q = 180 - 124 = 56^{\circ}$ (co-int. angles, $PQ \parallel SR$) [1]. (b) $\angle R = \angle P = 124^{\circ}$ (opp. angles of parallelogram, rhombus is a parallelogram) [1]. (c) $QR = PQ = 7$ cm (all sides of rhombus equal) [1].

Q7 (3 marks): (a) $\angle A + \angle B + \angle C + \angle D = 360^{\circ}$ so $\angle B + \angle D = 360 - 84 - 116 = 160^{\circ}$ [1]. (b) $\angle B = \angle D$ (one pair of equal opp. angles in kite), so $2\angle B = 160 \Rightarrow \angle B = 80^{\circ}$ [1]. (c) $\angle B = \angle D = 80^{\circ}$ [1].

Q8 (3 marks): (a) Co-int. angles on leg $WZ$: $3x + (2x + 30) = 180$ [1]. (b) $5x + 30 = 180 \Rightarrow 5x = 150 \Rightarrow x = 30$ [1]. (c) $\angle W = 90^{\circ}, \angle Z = 90^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

The Symmetry Detective

A quadrilateral has four equal sides AND perpendicular diagonals AND equal diagonals. (a) What is its most specific name? Justify by listing every property condition you've used. (b) Could a NON-square rhombus ever have equal diagonals? Explain. (c) A kite has its symmetry diagonal $AC = 12$ cm, and $BD = 6$ cm. If $AC$ bisects $BD$ at $M$, find $BM$ and explain why $AM = MC$ is not guaranteed.

Reveal solution

(a) Square. 4 equal sides + perpendicular diagonals ⇒ rhombus. Equal diagonals in a rhombus only happens when angles are $90^{\circ}$ ⇒ square. (b) NO — a rhombus has equal diagonals only when it's a square; otherwise one diagonal is "long" and the other is "short". (c) $BM = 3$ cm (symmetry diagonal bisects the other). However in a kite, the diagonal $BD$ does NOT generally bisect $AC$ — $AM$ may not equal $MC$ unless the kite is also a rhombus.

Quick Review

Rhombus

4 equal sides + perpendicular diagonals.

Square

Rhombus + 4 right angles (or rectangle + 4 equal sides).

Kite

2 pairs adjacent equal sides; diagonals $\perp$.

Trapezium

Exactly 1 pair parallel; co-int. at each leg.

Diagonals $\perp$

Rhombus, square, kite — all have $\perp$ diagonals.

Square wins

4 axes of symmetry, equal & $\perp$ diagonals.

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