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# Problems with quadrilaterals

Angles and shapes · Congruence and similarity

<!-- Speaker note: Students use the angle and diagonal properties of quadrilaterals to find unknown angles and lengths, with reasons (AC9M8SP02, node 2). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** 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°.

- 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.

<!-- Speaker note: 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). -->

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## Why it matters

> 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?

<!-- Speaker note: 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°. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A parallelogram has one angle of 70°. What are the other three?**

- 110°, 110°, 70°
- 110°, 70°, 110°
- 70°, 70°, 70°
- You cannot tell

<!-- Speaker note: What happens: 110°, 70° and 110°. The opposite angle is equal (70°), and each neighbour is 180° − 70° = 110°. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask which angle is opposite the 70° one. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fquadrilateral-problems%2Fstep-2&code=widget%3A+geometry%0Ashape%3A+polygon%0Asides%3A+4%0Acorners%3A+1%2C1%3B+8%2C2%3B+7%2C6%3B+2%2C5%0Aclassify%3A+on%0Asum%3A+off%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/quadrilateral-problems/?full=step-2#step-2)

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what stays the same as the parallelogram changes. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 110°, 70° and 110°. The opposite angle is equal (70°), and each neighbour is 180° − 70° = 110°.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Neighbours

Here is the routine.

$$\angle K = 72^\circ \text{ in parallelogram } KLMN$$

1) $\angle K + \angle L = 180^\circ$ — Co-interior angles: KN is parallel to LM.
2) $\angle L = 180^\circ - 72^\circ = 108^\circ$ — Work it out.
3) $\angle M = 72^\circ,\ \angle N = 108^\circ$ — Opposite angles are equal. Check: they add to 360°.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write K = 72^ in parallelogram KLMN and say "What is given?" Line 1: write K + L = 180^ and say "Co-interior angles: KN is parallel to LM." Line 2: write L = 180^ - 72^ = 108^ and say "Work it out." Line 3: write M = 72^, N = 108^ and say "Opposite angles are equal. Check: they add to 360°." -->

<!-- Speaker note: Teacher: ask Jo which pairs are co-interior. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Neighbours

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

* Answer: 115

<!-- Speaker note: P and Q are co-interior angles between the parallel sides PS and QR: 180 − 65 = 115°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

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

```
 1  ABCD is a parallelogram with angle A = 65°.
 2  Opposite angles add to 180°, so angle C = 180° − 65° = 115°.
 3  Angles A and B are co-interior, so angle B = 180° − 65° = 115°.
 4  So the angles are A = 65°, B = 115°, C = 115° and D = 65°.
```

* Answer: line 2

<!-- Speaker note: Line 2: opposite angles are equal, so C = 65°. Neighbouring angles add to 180°. Then D = 115°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A kite

**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 B
- Angle A: it is opposite
- Angle D: triangles ABC and ADC are congruent (SSS)
- None of them

* Answer: Angle D: triangles ABC and ADC are congruent (SSS)

<!-- Speaker note: AB = AD, CB = CD and AC is shared, so the triangles are congruent and angle B = angle D. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Other shapes



- **Rhombus:** 
- **Rectangle diagonals:** 
- **Kite:** 
- **Isosceles trapezium:** 

<!-- Speaker note: What this step is for: The same reasoning on a rhombus, a rectangle's diagonals, a kite and an isosceles trapezium. For students who need more support: The properties table from the lesson, printed with a column for 'reason words'. For early finishers: Prove the carpenter's check: a parallelogram with equal diagonals is a rectangle. 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'. -->

<!-- Speaker note: Teacher: ask for a check using 360°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Angles with x

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

* Answer: 60

<!-- Speaker note: x + 2x = 180, so 3x = 180 and x = 60. The angles are 60° and 120°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Rhombus sides

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

* Answer: 5

<!-- Speaker note: The diagonals cut each other in half at 90°, making right-angled triangles with legs 3 and 4: the side is 5 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Opposite angles

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

* Answer: 52

<!-- Speaker note: Opposite angles of a parallelogram are equal: 52°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: similar shapes

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

* Answer: 12

<!-- Speaker note: 3 × 4 = 12 cm. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: 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.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/quadrilateral-problems/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a kite with a given angle, then solving for the others, joins the board, the properties and the angle sum. Hints, in order: A kite has AB = AD and CB = CD. / Put B and D level, the same distance either side of A, so that angle A is 90°. / Try A at 5,1, B at 7,3 and D at 3,3, then move C up the middle. -->

<!-- Speaker note: Teacher: ask which angles of the kite must be equal, and why. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**In parallelogram ABCD, angle B = 3 × angle A. Find all four angles.**

* A + 3A = 180, so A = 45°. B = 135°, C = 45°, D = 135°.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M8SP02. Where it leads: congruence algorithms (next lesson), and circle geometry and trigonometry in Years 9 and 10. -->