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# Congruent triangles

Angles and shapes · Congruence and similarity

<!-- Speaker note: Students build triangles from given parts and see that SSS, SAS, ASA, AAS and RHS fix one triangle while SSA can give two (AC9M8SP01, node 1). About 40 minutes (Predict and Run 10 min · Investigate 10 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** congruent triangles are the same shape and size. You don't need all six parts to check: some sets of three parts fix the triangle (the congruence tests: SSS, SAS, ASA or AAS, RHS), and one set of three can fit two different triangles.

- I can name the four congruence tests (SSS, SAS, ASA or AAS, RHS).
- I can choose the test that shows two triangles are congruent.
- I can explain why SSA is not a congruence test.

<!-- Speaker note: Before class: Do the Run board yourself first: try SSA with a short BC. Students need triangle classification and the angle sum (Year 7). The one idea: a test is a set of parts that fixes the triangle, so any two triangles with those parts are congruent. The common mistake is treating SSA as SAS (Tom's mistake). The board draws every triangle the parts allow: none, one, or two (only for SSA). Wider: show two triangles formed by a rectangle's diagonal are congruent (it leads to the quadrilateral proofs in two lessons). -->

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## Why it matters

> A builder cuts a hundred identical roof trusses. To check two are the same, does she need to measure every side and every angle, or are three measurements enough?

<!-- Speaker note: Two shapes are **congruent** when one fits exactly on the other: the same shape and the same size. For triangles, matching sides and matching angles are equal. Write $\triangle ABC \equiv \triangle PQR$, with the corners in matching order. Do you need all six measurements to be sure? -->

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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

**Two triangles both have sides of 8 cm and 6 cm and a 40° angle. Must they be congruent?**

- Yes: two sides and an angle are enough
- No: you always need all three sides
- Not always: it depends where the angle is

<!-- Speaker note: What happens: Not always. With the angle between the two sides (SAS) the triangle is fixed. With the angle somewhere else (SSA), two different triangles can fit. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask where the 40° angle could be. -->

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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%2Fcongruent-triangles%2Fstep-2&code=widget%3A+congruence%0Atest%3A+choose%0Aslider+q%3A+1%2C+12%2C+7%2C+0.5%0A&fresh=1}}

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

<!-- Speaker note: Press a test to choose which parts are given, then change them with the sliders. -->

<!-- Speaker note: Teacher: ask what changed between one triangle and two. -->

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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?

* Not always. With the angle between the two sides (SAS) the triangle is fixed. With the angle somewhere else (SSA), two different triangles can fit.

<!-- 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>

## Which test

Here is the routine.

$$\text{AB} = 6,\ \text{BC} = 4,\ \angle B = 70^\circ$$

1) $\angle B \text{ is between AB and BC}$ — Where is the angle?
2) $\text{SAS}$ — Two sides and the included angle.
3) $\triangle ABC \equiv \triangle PQR$ — Write the congruence with matching corners in order.

<!-- 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 AB = 6,BC = 4, B = 70^ and say "List the equal parts." Line 1: write B is between AB and BC and say "Where is the angle?" Line 2: write SAS and say "Two sides and the included angle." Line 3: write ABC PQR and say "Write the congruence with matching corners in order." -->

<!-- Speaker note: Teacher: ask Tom which sides meet at the 40° angle. -->

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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>

## Which test

**In both triangles, AB = 5 cm, AC = 7 cm and angle A = 40°. Which test shows they are congruent?**

- SSS
- ASA
- They might not be congruent
- SAS: angle A is between AB and AC

* Answer: SAS: angle A is between AB and AC

<!-- Speaker note: Angle A is where AB and AC meet: two sides and the included angle. -->

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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

**Tom decides two triangles are congruent. Click the line where his reasoning goes wrong.**

```
 1  Two triangles each have a 6 cm side, a 5 cm side and a 40° angle.
 2  In both, the 40° angle is opposite the 5 cm side, not between the two sides.
 3  Two sides and an angle match, so they are congruent by SAS.
```

* Answer: line 3

<!-- Speaker note: Line 3: SAS needs the angle between the two sides. Here it is not (SSA), and SSA can give two different triangles. -->

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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 only

**Two triangles both have angles of 50°, 60° and 70°. Are they congruent?**

- Yes, by ASA
- Yes: all three angles match
- Not necessarily: they are the same shape but can be different sizes
- No, never

* Answer: Not necessarily: they are the same shape but can be different sizes

<!-- Speaker note: Equal angles make them similar (same shape). A matching side is needed for congruence. -->

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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>

## Tests that fail



- **No triangle:** 
- **RHS:** 
- **Sides that fail:** 
- **Matching order:** 

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

<!-- Speaker note: What this step is for: Finding parts that make no triangle, checking RHS, and reading congruence statements in order. Hints, in order: What must the two angles add to for the two lines to meet? / The hypotenuse is the longest side. For students who need more support: A card with the four tests drawn, the equal parts marked. For early finishers: Find the range of BC (with AB = 6 and angle A = 40°) that gives no triangle, two triangles and one triangle. What is special at the boundaries? The hard part, on purpose: Where the angle is. Ask students to point to the angle and say which sides meet there before they name a test. When students get stuck: A student calls SSA 'SAS'. Ask: Is the angle between the two sides? Fix: SAS needs the included angle. / A student says AAA proves congruence. Ask: Could a bigger triangle have the same angles? Fix: Equal angles mean similar; congruence needs a side too. / A student matches the wrong sides. Ask: Which corners match? Read the letters in order. Fix: △ABC ≡ △PQR: A with P, B with Q, C with R. -->

<!-- Speaker note: Teacher: ask why the angles must add to less than 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>

## SSS

**Both triangles have sides 4, 6 and 7 cm. Which test?**

- SSS
- SAS
- RHS
- Not enough to tell

* Answer: SSS

<!-- Speaker note: Three equal sides: SSS. -->

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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>

## AAS

**Both triangles have angles 30° and 80°, and the side opposite the 80° angle is 9 cm. Which test?**

- SSS
- AAS
- SAS
- Not enough to tell

* Answer: AAS

<!-- Speaker note: Two angles and a side that is not between them: AAS. -->

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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>

## SSA

**Both triangles have sides 9 cm and 6 cm, and a 35° angle opposite the 6 cm side (not between the two sides). Must they be congruent?**

- Yes, by RHS
- Yes, by SSS
- No: SSA is not a congruence test
- Yes, by SAS

* Answer: No: SSA is not a congruence test

<!-- Speaker note: Two sides and a non-included angle can make two different triangles. -->

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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 of 47° and 68°. What is the third angle, in degrees?**

* Answer: 65

<!-- Speaker note: 180 − 47 − 68 = 65°. -->

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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: exactly one

Then explain why SSA still is not a congruence test.

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

<!-- Speaker note: What this step is for: Finding SSA parts that give exactly one triangle shows when the ambiguous case disappears. Hints, in order: Too short a BC gives none; a little longer gives two. / Try BC as long as AB, or longer. -->

<!-- Speaker note: Teacher: ask when SSA gives one triangle, and why that is not enough. -->

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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

**Two triangles have AB = PQ = 7 cm, BC = QR = 5 cm and angle B = angle Q = 60°. Are they congruent? Which test?**

* Yes, by SAS: angle B is between AB and BC.

<!-- 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 starts AC9M8SP01. Where it leads: similar shapes (next lesson), proving quadrilateral properties, and congruence algorithms. -->