Proving quadrilateral properties

Classroom resources

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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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 13 min

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

1. Predict: ask what would convince someone who has never seen a rectangle.
2. Run: ask whether any rectangle breaks the rule.
3. Investigate: ask why each pair of parts is equal.
4. Modify: ask which diagonal of the kite gets cut in half.
5. 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

Answers to the checks

1. 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?
   No. Examples suggest the rule, but a proof gives reasons that work for every rectangle, such as two congruent triangles.
2. test many shapes: Drag the corners to make a rectangle. Are its diagonals equal? Try a few different rectangles. Does that prove it, as you predicted?
   
   Every rectangle you make has equal diagonals. That is strong evidence, but only a proof covers every rectangle.
3. The proof: Put the proof that a rectangle's diagonals are equal in order.
   see the program below
   Find two triangles, give the equal parts with reasons, name the test, then use matching sides.

   ABCD is a rectangle. Look at triangles ABC and BAD, made by the diagonals AC and BD.
   Side AB is in both, BC = AD (opposite sides of a rectangle), and angle B = angle A = 90°.
   Two sides and the angle between them match, so △ABC ≡ △BAD (SAS).
   Matching sides of congruent triangles are equal, so AC = BD: the diagonals are equal.
   

4. Find the mistake: Leo tries to prove a property of kites. Click the line where his reasoning goes wrong.
   line 3
   Line 3: one measured kite is an example, not a proof. A proof gives reasons that hold for every kite.
5. A kite: Kite ABCD has AB = AD and CB = CD. The diagonal AC splits it into triangles ABC and ADC. Which test shows they are congruent?
   SSS: AB = AD, CB = CD and AC is shared
   AC is in both triangles, so all three sides match: SSS. Then angle BAC = angle DAC, so AC cuts angle A in half.
6. A kite: Make a kite that is not a rhombus. Do its diagonals cross at 90°? Do they cut each other in half?
   
   They cross at 90°, and AC cuts BD in half, but BD does not cut AC in half.
7. A rhombus: Make a rhombus that is not a square. What do you notice about its diagonals?
   
   They cut each other in half at 90°, but they are not equal.
8. A parallelogram: Make a parallelogram that is not a rectangle. Do its diagonals cut each other in half?
   
   Yes: every parallelogram's diagonals bisect each other. The two triangles at each end are congruent (ASA, with alternate angles).
9. A square: A square is a rectangle and a rhombus. Which properties must its diagonals have?
   Equal, cutting each other in half, at 90°
   It has the rectangle's diagonal properties and the rhombus's.
10. Parallelogram: In parallelogram ABCD, the diagonal AC makes triangles ABC and CDA. AB = CD and BC = DA (opposite sides). Which test shows they are congruent?
    SSS: AC is shared
    AB = CD, BC = DA and AC is in both: SSS. So angle B = angle D: opposite angles are equal.
11. Rhombus: Which is a proof that a rhombus's diagonals cross at 90°?
    The four small triangles are congruent by SSS, so the four angles at the centre are equal and add to 360°
    Four equal angles that make 360° are 90° each.
12. Which shape: Which quadrilateral's diagonals always cut each other in half, but are not always equal?
    A parallelogram
    Every parallelogram's diagonals bisect each other; they are equal only when it is a rectangle.
13. Review: parallel lines: Review: two parallel lines are cut by a transversal. One co-interior angle is 115°. What is the other, in degrees?
    65
    Co-interior angles add to 180°: 180 − 115 = 65°.
14. equal, not halved: Make a quadrilateral with one pair of parallel sides whose diagonals are equal but do not cut each other in half.
    
    An isosceles trapezium, such as corners 1,1; 9,1; 7,5; 3,5.

Solutions

One possible solution for each task. Other correct programs are fine.

more properties › A kite

Open task: any program that does what the step asks.

more properties › A rhombus

Open task: any program that does what the step asks.

more properties › A parallelogram

Open task: any program that does what the step asks.

more properties › A square

Open task: any program that does what the step asks.

equal, not halved

Open task: any program that does what the step asks.

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 __ = __.'

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Proving quadrilateral properties".

The idea: Establishing quadrilateral properties via congruent triangles: I can Students use diagonal decomposition into triangles and congruence arguments to establish properties of specific quadrilaterals.

What understanding looks like:
- Uses a congruence argument to establish at least one quadrilateral property.
- States each step with a geometric reason.
- Does not assert properties without justification.

Mistakes students often make here:
- A student says 'they look equal'.
- A student picks two triangles that don't share parts.
- A student names the test but stops.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Proving quadrilateral properties".

Exit ticket

Kite ABCD has AB = AD and CB = CD. Prove that angle B = angle D.

Triangles ABC and ADC: AB = AD, CB = CD, AC shared, so they are congruent (SSS). Angle B and angle D match, so they are equal.

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.

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).