Congruent triangles

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.

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?

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?

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Two sides and an angle

You do not need all six. These congruence tests each fix a triangle, so two triangles that share them are congruent. The angle between two sides is the included angle.

Test	Equal parts
SSS	all three sides
SAS	two sides and the included angle (the angle between them)
ASA / AAS	two angles and a matching side
RHS	a right angle, the hypotenuse and one other side

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 enoughNo: you always need all three sidesNot always: it depends where the angle is

TryTry it on the board and compare the result with your prediction.

Build from parts

Press a test to choose which parts are given, then change them with the sliders.

Try it yourselfYour edits are saved in this browser.

Try SSS, SAS and ASA, then SSA. Find parts that make two different triangles. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Which test

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\text{AB} = 6,\ \text{BC} = 4,\ \angle B = 70^\circ$ List the equal parts.
$\angle B \text{ is between AB and BC}$ Where is the angle?
$\text{SAS}$ Two sides and the included angle.
$\triangle ABC \equiv \triangle PQR$ Write the congruence with matching corners in order.

Read Tom's working. Where is the 40° angle?

In both triangles, AB = 5 cm, AC = 7 cm and angle A = 40°. Which test shows they are congruent?SSSASAThey might not be congruentSAS: angle A is between AB and AC

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

Two triangles both have angles of 50°, 60° and 70°. Are they congruent?Yes, by ASAYes: all three angles matchNot necessarily: they are the same shape but can be different sizesNo, never

ModifyChange the problem on purpose and see what happens.

Tests that fail

Need a hint?

What must the two angles add to for the two lines to meet?

Need a hint?

The hypotenuse is the longest side.

Try it yourselfYour edits are saved in this browser.

Change the two angles so that no triangle can be made.

Use the board. This ticks when you get there.

With a right angle at A, set AB = 8 and the hypotenuse BC = 10. How many triangles fit?

Use the board. This ticks when you get there.

Find three sides that do not make a triangle.

Use the board. This ticks when you get there.

△ABC ≡ △PQR. Which side matches BC?QRPQPRYou cannot tell

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Exactly one

Then explain why SSA still is not a congruence test.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Too short a BC gives none; a little longer gives two.

Another hint

Try BC as long as AB, or longer.

With AB = 6 and angle A = 40°, find a length of BC that gives exactly one triangle.

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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).

Running the lesson

Predict: ask where the 40° angle could be.
Run: ask what changed between one triangle and two.
Investigate: ask Tom which sides meet at the 40° angle.
Modify: ask why the angles must add to less than 180°.
Make: ask when SSA gives one triangle, and why that is not enough.

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.

Stretch and support

Stretch: 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?

Support: A card with the four tests drawn, the equal parts marked.

Where it leads

This starts AC9M8SP01.

Where it leads: similar shapes (next lesson), proving quadrilateral properties, and congruence algorithms.

What each step is for

two sides and an angle: Committing to whether two sides and any angle fix a triangle shows whether they know the angle's position matters.
build from parts: Trying each set of parts on the board shows which sets fix one triangle and which (SSA) can give two.
which test: Name the test from the parts and where the angle is; SSA is not a test (Tom treats it as SAS); equal angles alone give the same shape but not the same size.
tests that fail: Finding parts that make no triangle, checking RHS, and reading congruence statements in order.
congruence tests: Naming tests from given parts, mixed; treating SSA as a test is the planted slip; the angle sum review keeps triangles fresh.
exactly one: Finding SSA parts that give exactly one triangle shows when the ambiguous case disappears.

Now you can

Name the four congruence tests (SSS, SAS, ASA or AAS, RHS).
Choose the test that shows two triangles are congruent.
Explain why SSA is not a congruence test.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson17. Similar shapesSimilar shapes are the same shape but can be different sizes, like a photo and its enlargement. Find the scale factor, use it to work out missing sides, and learn the conditions that make two triangles similar.

Source: content/maths/angles/congruent-triangles/lesson.md

Two sides and an angle

Build from parts

Which test

Tests that fail

Congruence tests

Exactly one

Students will be able to

Before class

Running the lesson

The hard part (on purpose)

When students get stuck

Stretch and support

Where it leads

What each step is for