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# Classifying triangles

Angles and shapes · Classifying shapes

<!-- Speaker note: Students drag triangles on a dot grid and see them named by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse), combine the two names, and reason about impossible pairs (AC9M7SP02, 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** name a triangle twice: by its sides (scalene, isosceles, equilateral) and by its biggest angle (acute, right, obtuse). Equal sides sit opposite equal angles, so some pairs are impossible: no right-angled equilateral triangle.

- I can name a triangle by its sides and by its angles.
- I can explain why some triangles are impossible, like a right-angled equilateral triangle.
- I can use equal sides and equal angles to find missing angles.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the triangle angle sum (Angles in a triangle) and the words acute, right and obtuse for angles. The one idea: every triangle has a sides name and an angles name, and equal sides sit opposite equal angles. The common mistake is naming by any acute angle (Zara's mistake): every triangle has at least two. A square grid cannot make an exact equilateral triangle; the board counts sides within 2% as equal and says so. -->

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## Why it matters

> Roof trusses, bridges and set squares are all triangles, but not the same kind. Builders name them by their sides and their angles, because that tells you how they fit together.

<!-- Speaker note: Every triangle gets **two** names. By its **sides**: - **scalene**: no sides equal; - **isosceles**: two sides equal (marked with the same tick); - **equilateral**: all three sides equal. By its **angles**: - **acute**: every angle is less than 90°; - **right-angled**: one angle is 90° (marked ∟); - **obtuse**: one angle is more than 90°. Equal sides sit opposite equal angles: an isosceles triangle has two equal angles, and an equilateral triangle has three, each 60°. -->

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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

**Can a triangle be right-angled **and** isosceles?**

- No: a right-angled triangle can't have equal sides
- Yes
- Only if it is equilateral

<!-- Speaker note: What happens: Yes: angles 90°, 45° and 45°, with the two sides at the right angle equal (half a square, cut along its diagonal). Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; ask a 'yes' student to sketch 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%2Fclassifying-triangles%2Fstep-2&code=widget%3A+geometry%0Ashape%3A+triangle%0Acorners%3A+1%2C1%3B+9%2C1%3B+6%2C6%0Aclassify%3A+on%0Asum%3A+off%0A&fresh=1}}

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

<!-- Speaker note: Equal sides get matching ticks; a right angle gets a small square. -->

<!-- Speaker note: Teacher: ask what the ticks and the small square mean. -->

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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?

* Yes: angles 90°, 45° and 45°, with the two sides at the right angle equal (half a square, cut along its diagonal).

<!-- 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>

## Sides, then angles

Here is the routine for naming a triangle.

Now try one: sides 5, 5 and 8.

$$90^\circ, 45^\circ, 45^\circ$$

1) $\text{isosceles}$ — Sides name: two equal sides.
2) $\text{right-angled}$ — Angles name: the biggest angle is 90°. Give both: isosceles, right-angled.
3) $60^\circ, 60^\circ, 60^\circ$ — Equilateral means equal angles too, so an equilateral triangle is always acute.

<!-- 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 90^, 45^, 45^ and say "A triangle with these angles: two are equal, so the sides opposite them are equal." Line 1: write isosceles and say "Sides name: two equal sides." Line 2: write right-angled and say "Angles name: the biggest angle is 90°. Give both: isosceles, right-angled." Line 3: write 60^, 60^, 60^ and say "Equilateral means equal angles too, so an equilateral triangle is always acute." -->

<!-- Speaker note: Teacher: for the impossible pair, ask what angles an equilateral triangle must have. -->

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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>

## Sides, then angles

**A triangle has sides 5 cm, 5 cm and 8 cm. Name it by its sides.**

- Right-angled
- Isosceles
- Scalene
- Equilateral

* Answer: Isosceles

<!-- Speaker note: Two equal sides: isosceles. -->

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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

**Zara names a triangle by its angles. Click the line where her reasoning goes wrong.**

```
 1  A triangle has angles 110°, 40° and 30°.
 2  It has two acute angles, 40° and 30°.
 3  So it is an acute triangle.
```

* Answer: line 3

<!-- Speaker note: Line 3: every triangle has at least two acute angles. The biggest angle decides: 110° is more than 90°, so it is obtuse. -->

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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>

## Impossible pairs

**Which of these triangles is impossible?**

- An obtuse isosceles triangle
- An acute scalene triangle
- A right-angled scalene triangle
- A right-angled equilateral triangle

* Answer: A right-angled equilateral triangle

<!-- Speaker note: An equilateral triangle has three equal angles, 60° each: none can be 90°. -->

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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



- **Make it scalene:** Each task changes one thing.
- **Obtuse and scalene:** 
- **Nearly equilateral:** 
- **Two right angles?:** 

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

<!-- Speaker note: What this step is for: One change at a time on the board: from isosceles to scalene, then an obtuse scalene triangle, then nearly equilateral on half squares, then the edge case of two right angles. Hints, in order: Move one corner so that no two sides match: the ticks disappear. / Pull the top far to one side, past the end of the bottom side. / Corners land on half squares here. / Try a bottom side of 4, with the top 3.5 up, half way along. For students who need more support: Keep a two-column table on the board: sides names and angles names with a picture of each. For early finishers: List all nine pairs (3 side names × 3 angle names). Which are possible? Sketch each possible one. The hard part, on purpose: Making a triangle with two names at once on the board. Ask students to say which name they will fix first and what has to stay true while they fix the second, rather than moving corners for them. When students get stuck: A student calls a triangle acute because it has an acute angle (Zara's mistake). Ask: Which is the biggest angle? Fix: The biggest angle decides: under 90° acute, 90° right, over 90° obtuse. / A student gives only one name. Ask: What do the sides say, and what do the angles say? Fix: Name the sides first, then the biggest angle. / A student thinks a right-angled triangle can't be isosceles. Ask: What happens if you cut a square along its diagonal? Fix: Two equal sides at the right angle: 45°, 45°, 90°. -->

<!-- Speaker note: Teacher: for nearly equilateral, ask why the grid cannot do it exactly (the height is not a whole number of squares). -->

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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>

## Equal angles

**A triangle has angles 50°, 50° and 80°. Name it by its sides and its angles.**

- Isosceles and obtuse
- Scalene and acute
- Isosceles and acute
- Equilateral and acute

* Answer: Isosceles and acute

<!-- Speaker note: Two equal angles sit opposite two equal sides: isosceles. The biggest angle, 80°, is under 90°: acute. -->

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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 right angle

**A triangle has angles 25°, 65° and 90°. Name it.**

- Scalene and right-angled
- Isosceles and right-angled
- Scalene and obtuse
- Scalene and acute

* Answer: Scalene and right-angled

<!-- Speaker note: No equal angles, so no equal sides: scalene. One 90° angle: right-angled. -->

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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>

## Base angles

**An isosceles triangle has one angle of 40° between its two equal sides. What is each of the other two angles, in degrees?**

* Answer: 70

<!-- Speaker note: 180 − 40 = 140, shared equally: 70° each. -->

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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: angle sum

**Review: a triangle has angles 35° and 85°. What is the third, in degrees?**

* Answer: 60

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

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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: obtuse and isosceles

Then explain why the obtuse angle has to be the one between the equal sides.

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

<!-- Speaker note: What this step is for: Building an obtuse isosceles triangle combines both names on purpose: the equal sides meet at the obtuse angle. Hints, in order: The obtuse angle has to be between the two equal sides. / Try a long flat bottom with the top just above its middle. -->

<!-- Speaker note: Teacher: ask which angle is obtuse 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

**A triangle has angles 30°, 30° and 120°. Name it by its sides and its angles.**

* Isosceles (two equal angles, so two equal sides) and obtuse (120°).

<!-- 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: The quadrilateral family (node 2). Where it leads: Pythagoras' theorem (Year 8) for right-angled triangles, and congruent and similar triangles (Years 8 and 9). -->