Problems with quadrilaterals

Use the properties of parallelograms, rhombuses, rectangles, kites and trapeziums to find unknown angles and lengths, with a reason for each step, and check with the angle sum of 360°.

A gate is braced with a diagonal so it stays a rectangle. A carpenter checks it by measuring both diagonals. Why does that work, and what does it tell her?

Problems with quadrilaterals use their properties, with a reason for each step. Start with what you already know: every quadrilateral's angles add to 360°, and co-interior angles between parallel lines add to 180°.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

One angle

Here are the properties to use:

Shape	Angles	Diagonals
any quadrilateral	add to 360°
parallelogram	opposite angles equal; neighbours add to 180°	bisect each other
rhombus	as a parallelogram	bisect each other at 90°
rectangle	four right angles	equal, and bisect each other
kite	the angles between the unequal sides are equal	cross at 90°
isosceles trapezium	base angles equal (in pairs)	equal

Neighbouring angles of a parallelogram add to 180° because they are co-interior angles between parallel sides.

A parallelogram has one angle of 70°. What are the other three?110°, 110°, 70°110°, 70°, 110°70°, 70°, 70°You cannot tell

TryTry it on the board and compare the result with your prediction.

Make a parallelogram

Try it yourselfYour edits are saved in this browser.

Drag the corners to make a parallelogram that is not a rectangle. Check the angles: were you right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Find the angles

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\angle K = 72^\circ \text{ in parallelogram } KLMN$ What is given?
$\angle K + \angle L = 180^\circ$ Co-interior angles: KN is parallel to LM.
$\angle L = 180^\circ - 72^\circ = 108^\circ$ Work it out.
$\angle M = 72^\circ,\ \angle N = 108^\circ$ Opposite angles are equal. Check: they add to 360°.

Read Jo's working. Which angle is opposite A?

In parallelogram PQRS, angle P = 65°. What is angle Q, in degrees?

Jo works out the angles of a parallelogram. Click the line where she goes wrong.

Kite ABCD has AB = AD and CB = CD, and angle B = 100°. Which other angle must be 100°, and why?Angle C: it is next to BAngle A: it is oppositeAngle D: triangles ABC and ADC are congruent (SSS)None of them

ModifyChange the problem on purpose and see what happens.

Other shapes

A rhombus has an angle of 70°. What is the size of its larger angles, in degrees?

Rectangle ABCD has diagonals that cross at M. AC = 10 cm. How long is BM, in cm?

Kite ABCD has AB = AD and CB = CD. Angle A = 70° and angle C = 90°. What is angle B, in degrees?

An isosceles trapezium has two base angles of 70°. What is each of the other two angles, in degrees?

PractiseA few on your own, to check you have it.

Quadrilateral problems

Two neighbouring angles of a parallelogram are x° and 2x°. What is x?

A rhombus has diagonals of 6 cm and 8 cm. How long is each side, in cm?

In parallelogram ABCD, angle A = 52°. What is angle C, in degrees?

Review: a 2 cm by 3 cm rectangle is enlarged with scale factor 4. What is the longer side now, in cm?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A right-angled kite

Then read angle C on the board and work out angles B and D, with a reason for each. The board rounds each angle to a whole degree, so its B and D can differ by 1°: your answer can be a half.

Try it yourselfYour edits are saved in this browser.

Need a hint?

A kite has AB = AD and CB = CD.

Another hint

Put B and D level, the same distance either side of A, so that angle A is 90°.

Another hint

Try A at 5,1, B at 7,3 and D at 3,3, then move C up the middle.

Make a kite (not a square) with a right angle at A, where AB = AD.

Use the board. This ticks when you get there.

For teachers

Students use the angle and diagonal properties of quadrilaterals to find unknown angles and lengths, with reasons (AC9M8SP02, node 2). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can find unknown angles in parallelograms, rhombuses, kites and trapeziums.
I can use diagonal properties to find lengths.
I can give a reason for each step.

Before class

Do the Run board yourself first. Students need the quadrilateral family, co-interior angles and the angle sum of a quadrilateral (Year 7), and the proofs from the last lesson.

The one idea: each property is a rule with a reason, so a problem is a chain of rules. The common mistake is saying opposite angles of a parallelogram add to 180° (Jo's mistake); neighbouring angles do.

Reasons students should use: co-interior angles, opposite angles of a parallelogram, angle sum of a quadrilateral (360°), properties of the diagonals, congruent triangles.

Wider: the carpenter's check in the hook (equal diagonals mean a parallelogram frame is a rectangle).

Running the lesson

Predict: ask which angle is opposite the 70° one.
Run: ask what stays the same as the parallelogram changes.
Investigate: ask Jo which pairs are co-interior.
Modify: ask for a check using 360°.
Make: ask which angles of the kite must be equal, and why.

The hard part (on purpose)

Telling opposite from neighbouring angles. Ask students to trace the parallel sides with a finger before they decide which rule applies.

When students get stuck

A student subtracts from 180 for the opposite angle. Ask: Are those two angles between the same pair of parallel sides? Fix: Opposite angles are equal; neighbours add to 180°.
A student uses the whole diagonal for half. Ask: Where do the diagonals cross? Fix: They cut each other in half: each part is half the diagonal.
A student gives answers with no reasons. Ask: Which property did you use? Fix: Write the reason after each line, such as 'co-interior angles'.

Stretch and support

Stretch: Prove the carpenter's check: a parallelogram with equal diagonals is a rectangle.

Support: The properties table from the lesson, printed with a column for 'reason words'.

Where it leads

This completes AC9M8SP02.

Where it leads: congruence algorithms (next lesson), and circle geometry and trigonometry in Years 9 and 10.

What each step is for

one angle: Committing to all four angles of a parallelogram from one shows whether they use opposite-equal and co-interior.
make a parallelogram: Making parallelograms on the board shows opposite angles equal and neighbours adding to 180° every time.
find the angles: Use co-interior angles for neighbours and equality for opposites, with reasons; opposite angles adding to 180° is the slip (Jo); a kite's equal angles come from congruent triangles.
other shapes: The same reasoning on a rhombus, a rectangle's diagonals, a kite and an isosceles trapezium.
quadrilateral problems: Angles with algebra, diagonals and Pythagoras, mixed; opposite angles as supplementary is the planted slip again; similar shapes review.
a right-angled kite: Building a kite with a given angle, then solving for the others, joins the board, the properties and the angle sum.

Now you can

Find unknown angles in parallelograms, rhombuses, kites and trapeziums.
Use diagonal properties to find lengths.
Give a reason for each step.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson20. Algorithms for congruenceAn algorithm can decide whether two triangles are congruent, similar or neither by asking about their equal parts in a careful order: the congruence tests first, then the similarity conditions, and 'not enough to tell' when nothing fits. You test it on one pair of every kind, find what it misses, and fix it.

Source: content/maths/angles/quadrilateral-problems/lesson.md