Proving quadrilateral properties

A diagonal cuts a quadrilateral into two triangles. When those triangles are congruent, their matching sides and angles are equal, and that proves a property for every shape of the kind: a rectangle's diagonals are equal, a parallelogram's cut each other in half, a kite's cross at 90°.

You measure the diagonals of one rectangle: both 10 cm. Is that true for every rectangle in the world? Measuring can't check them all, but a proof can.

A proof shows that something is true for every shape of a kind, with a reason for each step.

For quadrilaterals, the main tool is congruent triangles. Draw a diagonal (or both), find two triangles, and show they are congruent with a test (SSS, SAS, ASA, AAS or RHS). Then matching parts of congruent triangles are equal.

Shape	Its diagonals
parallelogram	cut each other in half (bisect each other)
rectangle	equal, and bisect each other
rhombus	bisect each other at 90°
square	equal, and bisect each other at 90°
kite	cross at 90°; one cuts the other in half

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.

Does one drawing prove it?

Ana draws one rectangle and measures both diagonals: 10 cm each. Has she proved that every rectangle's diagonals are equal?Yes, if she draws two moreYes: she measured carefullyNo: one example is not a proof

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

Test many shapes

The dashed lines are the diagonals AC and BD. The panel gives their lengths, in two parts from where they cross.

Try it yourselfYour edits are saved in this browser.

Drag the corners to make a rectangle. Are its diagonals equal? Try a few different rectangles. Does that prove it, as you predicted?

Use the board. This ticks when you get there.

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

Write the proof

Here is a proof that a parallelogram's diagonals cut each other in half. Every proof in this lesson has the same plan.

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

$\text{Parallelogram } ABCD,\ \text{diagonals cross at } M$ Look at triangles ABM and CDM.
$AB = CD$ Opposite sides of a parallelogram are equal.
$\angle BAM = \angle DCM,\ \angle ABM = \angle CDM$ Alternate angles: AB is parallel to DC.
$\triangle ABM \equiv \triangle CDM\ (\text{ASA})$ Name the test.
$AM = CM,\ BM = DM$ Matching sides are equal: the diagonals cut each other in half.

Read Leo's working. How many kites did he check?

Put the proof that a rectangle's diagonals are equal in order.

Pieces (click to add)

Your program

Leo tries to prove a property of kites. Click the line where his reasoning goes wrong.

Kite ABCD has AB = AD and CB = CD. The diagonal AC splits it into triangles ABC and ADC. Which test shows they are congruent?They are not congruentSASSSS: AB = AD, CB = CD and AC is sharedRHS

ModifyChange the problem on purpose and see what happens.

More properties

Need a hint?

A kite has two pairs of equal sides next to each other.

Another hint

Put A and C on one vertical line, with B and D the same distance either side of it.

Another hint

Such as corners 5,1; 7,3; 5,7; 3,3.

Need a hint?

Four equal sides: try steps of 3 across and 4 up, which are 5 long.

Try it yourselfYour edits are saved in this browser.

Make a kite that is not a rhombus. Do its diagonals cross at 90°? Do they cut each other in half?

Use the board. This ticks when you get there.

Make a rhombus that is not a square. What do you notice about its diagonals?

Use the board. This ticks when you get there.

Make a parallelogram that is not a rectangle. Do its diagonals cut each other in half?

Use the board. This ticks when you get there.

A square is a rectangle and a rhombus. Which properties must its diagonals have?At 90° onlyEqual onlyEqual, cutting each other in half, at 90°None of these

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

Equal, not halved

Then write a proof (or the plan of one) that its diagonals are equal. Which two triangles would you use?

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Need a hint?

Rectangles have equal diagonals that also cut each other in half. Change one thing.

Another hint

Keep one pair of sides parallel, but make the other two sides slope in by the same amount.

Another hint

Try a trapezium with equal slanted sides.

Make a quadrilateral with one pair of parallel sides whose diagonals are equal but do not cut each other in half.

Use the board. This ticks when you get there.

For teachers

Students test diagonal properties of quadrilaterals on the board, then prove them with congruent triangles (AC9M8SP02, node 1). About 45 minutes.

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

Students will be able to

I can find two triangles in a quadrilateral and say which test makes them congruent.
I can use matching parts of congruent triangles to prove a property.
I can explain why measuring one shape is not a proof.

Before class

Do the Run board yourself first. Students need the congruence tests (Congruent triangles), the quadrilateral family (Year 7) and angles at parallel lines.

The one idea: two congruent triangles inside the shape give equal sides and angles, and that holds for every shape of the kind. The common mistake is treating one careful drawing as a proof (Leo's mistake).

The board measures diagonals: l is equal, u is cut in half, y is at 90°. Use the board for evidence and conjectures; the proofs are written. Year 8 proofs can be a plan with reasons, not a two-column format.

Wider: prove a parallelogram's diagonals bisect each other (ASA with alternate angles), or prove the converse: a quadrilateral whose diagonals bisect each other is a parallelogram.

Running the lesson

Predict: ask what would convince someone who has never seen a rectangle.
Run: ask whether any rectangle breaks the rule.
Investigate: ask why each pair of parts is equal.
Modify: ask which diagonal of the kite gets cut in half.
Make: ask which two triangles the proof would use.

The hard part (on purpose)

Giving a reason for every equal part. Ask 'how do you know?' after each line until the answer is a property, a test or a shared side.

When students get stuck

A student says 'they look equal'. Ask: How do you know, for every shape like this? Fix: Give a property, a shared side or a test for each equal part.
A student picks two triangles that don't share parts. Ask: Which side do the two triangles have in common? Fix: Choose triangles on either side of a diagonal, or the two triangles that share a side.
A student names the test but stops. Ask: Now what does congruence give you? Fix: Matching parts of congruent triangles are equal: that is the property.

Stretch and support

Stretch: Prove that a parallelogram's opposite angles are equal, then that its diagonals cut each other in half.

Support: A proof frame: 'Look at triangles __ and __. __ = __ because __ (three times). So they are congruent by __. So __ = __.'

Where it leads

This covers AC9M8SP02 node 1.

Where it leads: problems with quadrilateral properties (next lesson), and proof in Years 9 and 10 (circle geometry, similar triangles).

What each step is for

does one drawing prove it?: Committing to whether one measured rectangle proves a rule shows whether they know what a proof needs.
test many shapes: Dragging a parallelogram into many rectangles shows the diagonals always equal: evidence, which the proof then explains.
write the proof: A proof finds two triangles, names the test, then uses matching parts; one measured picture is not a proof (Leo); a kite's triangles are congruent by SSS.
more properties: Making a kite, a rhombus and a parallelogram on the board and reading off their diagonal properties.
reasons: Naming the test and the property it proves, mixed; one example as proof is the planted slip; the angle review keeps parallel lines fresh.
equal, not halved: Building a shape from a diagonal property, not a name, shows they can reason from properties to shapes.

Now you can

Find two triangles in a quadrilateral and say which test makes them congruent.
Use matching parts of congruent triangles to prove a property.
Explain why measuring one shape is not a proof.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson19. Problems with quadrilateralsUse 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°.

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

Does one drawing prove it?

Test many shapes

Write the proof

More properties

Reasons

Equal, not halved

Students will be able to

Before class

Running the lesson

The hard part (on purpose)

When students get stuck

Stretch and support

Where it leads

What each step is for