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# Unknown angles in shapes

Angles and shapes · Angle sums

<!-- Speaker note: Students find unknown angles in triangles, quadrilaterals and pentagons in steps, with a reason for each (angle sum, straight line, isosceles), and explain why an exterior angle equals the sum of the two interior opposite angles; working is their own, the answer and a key reason are marked (AC9M7M05, node 3). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** find unknown angles in triangles and polygons in steps, with a reason for each: the angle sum of the shape, angles on a straight line, equal angles in an isosceles triangle. An exterior angle of a triangle equals the sum of the two interior opposite angles.

- I can find an unknown angle in a triangle or polygon using its angle sum.
- I can combine angle sums with straight-line and parallel-line angles, giving a reason for each step.
- I can explain why an exterior angle of a triangle equals the sum of the two interior opposite angles.

<!-- Speaker note: Before class: Do the first reasoning task yourself, then compare with its model. Students need Angles in a triangle, Angles in a polygon and Reasoning with angles. The one idea: choose the right angle sum for the shape, then chain steps with reasons. The common mistake is using 180° for every shape; it is planted in Zoe's working, where a negative angle is the clue. As in Reasoning with angles, the free working is not marked: students write it their way. The answer is marked, and at the key step they choose the reason from a list. Then they compare with a model and tick a self-check. The isosceles task is wider than the minimum: the equal angles are stated in the question. -->

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## Why it matters

> A roof truss, a kite, a bridge brace: builders work out the angle they cannot measure from the ones they can. Every step needs a reason.

<!-- Speaker note: You know the angle sums: a triangle's angles add to $180^\circ$, and a polygon with $n$ sides has $(n - 2) \times 180^\circ$. With straight-line angles and the parallel-line rules, you can find angles you cannot measure. As in Reasoning with angles, each step needs a reason. Write your working your own way; the checker marks your answer and a reason at the key step, then shows a model to compare. -->

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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 triangle has $a = 50^\circ$ and $b = 60^\circ$. The side through $C$ is extended past $C$, making an outside angle $x$ with the third side. What is $x$?**

- $70^\circ$
- $110^\circ$
- $120^\circ$
- $130^\circ$

<!-- Speaker note: What happens: 110°. c = 180° − 50° − 60° = 70° (angles in a triangle), then x = 180° − 70° = 110° (a straight line). Notice: 110° = 50° + 60°. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands, then ask students who chose 110° how they got it. Collect both routes (two steps, or a + b). -->

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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%2Funknown-angles%2Fstep-2&code=widget%3A+geometry%0Acorners%3A+1%2C1%3B+9%2C1%3B+4%2C6%0Aexterior%3A+C%0A&fresh=1}}

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

<!-- Speaker note: Was your prediction right? Is $x$ always $a + b$? Why must it be? -->

<!-- Speaker note: Teacher: ask why x = a + b must be true (a + b + c = 180° and x + c = 180°). -->

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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°. c = 180° − 50° − 60° = 70° (angles in a triangle), then x = 180° − 70° = 110° (a straight line). Notice: 110° = 50° + 60°.

<!-- 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>

## Triangle, then a straight line

Here is the routine: know, choose the rule, say why.

Now find $x$ yourself, your own way.

$$a = 59^\circ,\; b = 45^\circ$$

1) $c = 180^\circ - 59^\circ - 45^\circ = 76^\circ$ — Choose: $a$, $b$, $c$ are a triangle's angles. Why: angles in a triangle add to $180^\circ$.
2) $x = 180^\circ - 76^\circ = 104^\circ$ — Choose: $c$ and $x$ sit on one straight line. Why: angles on a straight line add to $180^\circ$.
3) $x = 59^\circ + 45^\circ$ — A shortcut you can now explain: an exterior angle equals the sum of the two interior opposite angles.

<!-- 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 a = 59^, b = 45^ and say "Know: the given angles." Line 1: write c = 180^ - 59^ - 45^ = 76^ and say "Choose: a, b, c are a triangle's angles. Why: angles in a triangle add to 180^." Line 2: write x = 180^ - 76^ = 104^ and say "Choose: c and x sit on one straight line. Why: angles on a straight line add to 180^." Line 3: write x = 59^ + 45^ and say "A shortcut you can now explain: an exterior angle equals the sum of the two interior opposite angles." -->

<!-- Speaker note: Teacher: model know, choose, say why for the triangle. Ask: what does a negative answer tell you (Zoe)? -->

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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>

## Triangle, then a straight line

**Find $x$. Write each step with its reason.**

* Answer: $c = 180^\circ - 59^\circ - 45^\circ = 76^\circ$ (angles in a triangle add to $180^\circ$); $x = 180^\circ - 76^\circ = 104^\circ$ ($c$ and $x$ are on a straight line)

<!-- Speaker note: c = 76° (angles in a triangle), x = 104° (a straight line with c). Check: 59° + 45° = 104°. -->

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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

**Zoe finds the fourth angle of a quadrilateral. Click the line where her working goes wrong.**

```
 1  a = 76°, b = 68°, c = 102° (given).
 2  The angles of a shape add to 180°.
 3  d = 180° − 76° − 68° − 102°.
 4  d = −66°.
```

* Answer: line 2

<!-- Speaker note: Line 2: a quadrilateral's angles add to (4 − 2) × 180° = 360°, not 180°. d = 360° − 76° − 68° − 102° = 114°. A negative angle on line 4 is the clue that something earlier went wrong. -->

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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 quadrilateral

**Find $d$, with your reason.**

* Answer: $a + b + c + d = 360^\circ$ (angles in a quadrilateral: $(4 - 2) \times 180^\circ$); $d = 360^\circ - 76^\circ - 68^\circ - 102^\circ = 114^\circ$ (subtract the angles you know)

<!-- Speaker note: (4 − 2) × 180° = 360°, so d = 360° − 246° = 114°. -->

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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>

## One change at a time



- **Isosceles:** Each task changes one thing.
- **A pentagon:** 
- **Exterior at another corner:** 
- **A right angle:** 
- **Impossible triangles:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/unknown-angles/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: an isosceles triangle, a pentagon's sum, the exterior angle at another corner, a right angle in a quadrilateral, then the edge case of a triangle that cannot exist. For students who need more support: Use the Run board to check: set a triangle, read its angles, and compare with your working. For early finishers: Prove that the exterior angles of any convex polygon add to 360°, starting from a triangle (each exterior + interior = 180°). When students get stuck: A student uses 180° for a quadrilateral (Zoe's mistake). Ask: How many triangles does a quadrilateral split into? Fix: (n − 2) × 180°: two triangles, 360°. / A student stops at the interior angle when asked for the exterior one. Ask: Which angle is x? Is it inside the triangle? Fix: Mark x on the diagram, then find what links it to an angle you have (a straight line). / A student writes numbers with no reasons. Ask: Which fact did you use for that line? Fix: One reason per line, in short words: 'angle sum of triangle', 'straight line'. -->

<!-- Speaker note: Teacher: before the pentagon, ask students for its angle sum and how they know. -->

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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>

## Triangle

**A triangle has angles of $35^\circ$ and $85^\circ$. Find the third angle.**

* Answer: 60

<!-- Speaker note: 180° − 35° − 85° = 60°. -->

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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>

## Quadrilateral

**Three angles of a quadrilateral are $90^\circ$, $85^\circ$ and $120^\circ$. Find the fourth.**

* Answer: 65

<!-- Speaker note: 360° − 295° = 65°. -->

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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>

## Exterior angle backwards

**A triangle has an exterior angle of $130^\circ$ at $C$, and $a = 70^\circ$. Find $b$, with your reasons.**

* Answer: $c = 180^\circ - 130^\circ = 50^\circ$ ($c$ and the exterior angle are on a straight line); $b = 180^\circ - 70^\circ - 50^\circ = 60^\circ$ (angles in a triangle add to $180^\circ$)

<!-- Speaker note: c = 50°, b = 60°. Or: 130° = a + b, so b = 60°. -->

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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: parallel lines

**Review: parallel lines, $c = 70^\circ$. Which is a correct first step to find $e$?**

- $e = 70^\circ$: alternate to $c$
- $e = 70^\circ$: corresponding to $c$
- $e = 110^\circ$: co-interior with $c$

* Answer: $e = 110^\circ$: co-interior with $c$

<!-- Speaker note: Co-interior angles add to 180°: e = 110°. -->

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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 given exterior angle

First work out what $c$ must be. Then find a triangle that has it.

Stretch: design an angle puzzle for a partner with two unknown angles, and write the model reasons.

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

<!-- Speaker note: What this step is for: Making a triangle with a given exterior angle reverses the reasoning: x = 135° needs c = 45°, from a + b = 135°. Hints, in order: x = 135° means c = 45°. / A right angle at A and equal legs give 45° at C: try A (1, 1), B (7, 1), C (1, 7). -->

<!-- Speaker note: Teacher: ask students what c must be before they move anything. -->

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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

**A triangle has angles of 48° and 67°. Find the third angle, and the exterior angle next to it, with reasons.**

* 180° − 48° − 67° = 65° (angles in a triangle). Exterior angle 180° − 65° = 115° (a straight line). Check: 48° + 67° = 115°.

<!-- 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 the Year 7 angle work: parallel lines (AC9M7M04) and angle sums (AC9M7M05). Where it leads: congruence and similarity (Years 8 and 9) and geometric proof, where chains of reasons are the whole point. -->