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# Angles in a triangle

Angles and shapes · Angle sums

<!-- Speaker note: Students drag triangles of every shape on a dot grid, see that the three angles always add to 180°, and explain why by tearing off the corners to make a straight line (AC9M7M05, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** however you change a triangle, its three angles add to 180°. Tear off the corners and they make a straight line.

- I can show that the angles of a triangle add to 180°, by measuring and by tearing off the corners.
- I can explain why this is true for every triangle, whatever its shape or size.
- I can tell whether three angles could be the angles of a triangle.

<!-- Speaker note: Before class: Do the Run board and the tear-off task yourself first. Students should know that angles on a straight line add to 180° (Year 6). The board measures to the nearest degree and shares out the rounding so the shown angles add to the shown total. With a real protractor, totals of 178° to 182° are normal: say so when students measure by hand (Practise has a question on it). If you have time, do the paper version too: every student draws a different triangle, tears off the corners and lines them up. Seeing 25 different triangles all make a straight line is the strongest evidence. The formal proof with parallel lines comes with AC9M7M04. Finding a missing angle from the other two is the next step in class (node 3), not this lesson. -->

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## Why it matters

> A roof truss, a bike frame, a folded paper plane: triangles of every shape. One thing about their angles never changes.

<!-- Speaker note: An **angle** measures a turn, in degrees ($°$). A full turn is $360°$, and the angles on a straight line add to $180°$. A triangle has three corners, so three angles inside it. In this lesson you find out what happens to their total when you change the triangle's shape. -->

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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

**You drag one corner of a triangle so one angle gets much bigger. What happens to the total of the three angles?**

- It gets bigger
- It gets smaller
- It stays the same
- It depends on how long the sides are

<!-- Speaker note: What happens: It stays the same: 180°. As one angle grows, the other two shrink by exactly as much. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 'it gets bigger' are thinking of the big angle only. Do not correct it yet. -->

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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%2Ftriangle-angles%2Fstep-2&code=widget%3A+geometry%0Ashape%3A+triangle%0Acorners%3A+2%2C1%3B+9%2C1%3B+5%2C6%0A&fresh=1}}

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

<!-- Speaker note: Drag any corner to a new dot. The board measures each angle and adds them up under the triangle. Make tall, wide and flat triangles. Was your prediction right? -->

<!-- Speaker note: Teacher: ask students to make the tallest, the widest and the flattest triangle they can, and to read the total each time. -->

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## What happened?

* It stays the same: 180°. As one angle grows, the other two shrink by exactly as much.

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

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## Why 180°?

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Ftriangle-angles%2Fstep-3&code=widget%3A+geometry%0Ashape%3A+triangle%0Acorners%3A+1%2C1%3B+8%2C1%3B+3%2C6%0Atear%3A+on%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/triangle-angles/?full=step-3#step-3)

<!-- Speaker note: What this step is for: Tearing off the corners and putting them side by side shows why the total is fixed: the three angles fill a straight line, which is 180°. Kai's mistake ties the angles to the side lengths. -->

<!-- Speaker note: Teacher: model the worked example live with a paper triangle: mark the three angles, tear them off at writing pace, and put the points together on a ruler's edge, saying each step. Then show the board's torn pieces changing as a corner moves. -->

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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

**Kai compares a small triangle with one twice as big. Click the line where his reasoning goes wrong.**

```
 1  Kai draws a small triangle, then one with every side twice as long.
 2  Small triangle: 50° + 60° + 70° = 180°.
 3  The big triangle has longer sides, so its angles are bigger.
 4  So the big triangle's angles add to more than 180°.
```

* Answer: line 3

<!-- Speaker note: Line 3: making every side twice as long does not change the angles, only the size. The big triangle's angles are still 50°, 60° and 70°, adding to 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>

## Why a straight line?

**Why do the three torn corners always make a straight line?**

- Because every triangle has a $90°$ angle
- Because the angles add to $180°$, and a straight line is $180°$
- Because the corners are all the same size

* Answer: Because the angles add to $180°$, and a straight line is $180°$

<!-- Speaker note: Side by side at one point, the three angles fill a straight line exactly. Angles on a straight line add to 180°, so the triangle's angles do too. -->

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## One change at a time



- **A right angle:** Each task changes one thing. First: make a square corner.
- **Very flat:** Now make one angle very big.
- **Three equal angles:** 
- **Two right angles?:** 

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

<!-- Speaker note: What this step is for: One change at a time: a right angle (the other two must make 90°), a very flat triangle (the other two are tiny), then two edge cases that the 180° rule decides. Hints, in order: A right angle is a square corner. Use the dots: one side across, one side straight up. / Put C straight above A. / Pull one corner until the triangle is nearly a line. Tear off the corners to see the pieces. For students who need more support: Use the Run board throughout, and the preset buttons are on the explain clip's board. Give students paper triangles to tear. For early finishers: Can a triangle have one angle that is twice the other two added together? (Yes: 120°, since 120 = 2 × 60.) What about three times? (135°.) When students get stuck: A student thinks a bigger triangle has bigger angles (Kai's mistake). Ask: If you make a photocopy twice as big, do the corners get pointier or blunter? Fix: Enlarge a triangle on the board by doubling each corner's coordinates: the angles stay the same. / A student adds the angles and gets 179° or 181° with a protractor. Ask: How accurately can you read a protractor? Fix: Each reading can be a degree out, so three readings can be two or three degrees out. The true total is 180°. / A student says the rule only works for the triangles they tried. Ask: Which triangle could you try that would break it? Fix: Let them try their 'breaking' triangle on the board or on paper. Then point to the torn corners: they always fill a straight line, whatever the shape. -->

<!-- Speaker note: Teacher: before the right-angle task ask: if one angle is 90°, what must the other two add to? Then check on the board. -->

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## Possible?

**Which three angles could be the angles of a triangle?**

- $50°, 60°, 80°$
- $90°, 90°, 10°$
- $40°, 60°, 80°$

* Answer: $40°, 60°, 80°$

<!-- Speaker note: 40° + 60° + 80° = 180°. -->

<!-- Speaker note: Teacher: Lia's measurement question is a chance to talk about measuring error, which science lessons need too. -->

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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>

## Measuring by hand

**Lia measures a paper triangle with a protractor: $57°, 68°, 56°$. That adds to $181°$. What is the best explanation?**

- She has measured the wrong angles
- Her triangle is a special one that adds to $181°$
- Her measurements are a little out: a protractor reads to about a degree

* Answer: Her measurements are a little out: a protractor reads to about a degree

<!-- Speaker note: Real measurements are rounded. 181° is within a degree or two of 180°, which is what measuring three angles by hand gives. -->

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## Bigger triangle

**Triangle P has sides 3 cm, 4 cm and 5 cm. Triangle Q has sides 30 cm, 40 cm and 50 cm. How do their angle sums compare?**

- Q's is 10 times bigger
- They are the same: $180°$
- Q's is bigger, but not 10 times

* Answer: They are the same: $180°$

<!-- Speaker note: Every triangle's angles add to 180°. Q is P made 10 times bigger, with exactly the same angles. -->

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## Review

**Review: two angles sit side by side on a straight line. One is $130°$. How big is the other?**

* Answer: 50

<!-- Speaker note: 180° − 130° = 50°. -->

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## Make: one angle equals the other two

Think first: if one angle is the other two added together, what fraction of $180°$ is it? Then build it.

Stretch: can a triangle have one angle that is exactly twice the sum of the other two? How big is it?

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

<!-- Speaker note: What this step is for: Building a triangle where one angle equals the other two together reverses the lesson: from the 180° rule to a shape it forces, a right angle. Hints, in order: If one angle equals the other two added together, it is half of the total. / Half of 180° is 90°: make a square corner. -->

<!-- Speaker note: Teacher: ask a student to explain why the angle must be 90° before anyone builds it. -->

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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 student says the angles of a very large triangle add to more than 180°. Explain why they are wrong.**

* Every triangle's angles add to 180°: tear off the corners and they fill a straight line. Size changes the sides, not the angles.

<!-- 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: Next: angles in any polygon, by splitting it into triangles. Where it leads: finding unknown angles with reasons (node 3), the proof with parallel lines (AC9M7M04), and why regular hexagons tessellate. -->