GCSE · Maths · AQA · Spec 8300 · Foundation

Conjecture and prove with congruence, similarity, Pythagoras

Drag a triangle into a hundred different shapes and the same thing happens every time. Is that enough to be sure it will always happen?

Conjecture

Drag A. What stays the same?

AB 10.8. AC 10.8. angle ABC 56°. angle ACB 56°. Classification: Isosceles: A stays above M, the midpoint of BC, so AB = AC. Relationship: Conjecture: if AB = AC, then angle ABC = angle ACB

AB10.8AC10.8angle ABC56°angle ACB56°drag A up and down

Isosceles: A stays above M, the midpoint of BC, so AB = AC

Conjecture: if AB = AC, then angle ABC = angle ACB

However many positions you try, you have only tested the triangles you drew, so this is a conjecture, not yet a proof.

Conjecture or proof?

What did the dragging actually show?

You dragged A to lots of positions and angle ABC matched angle ACB every time. A friend says: "So that's proved."

Which is closest to what you think right now?

It worked in every position I tried, so it's proved.The readouts show the same number, so the angles are equal.You can't prove it for every triangle. There are too many to check.It's a conjecture. To prove it I need steps, each with a reason from a known result, that work for any isosceles triangle.

How sure are you?

I'm sureI think soI'm guessing

It worked in every position I tried, so it's proved.

This is a common misconception

Every position you tried is still only the triangles you tried. A statement can hold in many cases and fail in one nobody tested, so checking cases gives you a conjecture: something you have good reason to believe but haven't shown for every case. A proof has to cover every isosceles triangle at once.

The readouts show the same number, so the angles are equal.

This is a common misconception

A measurement describes one drawing, to the accuracy of the measurement. These angle readouts are rounded to the nearest degree, so two angles 0.3° apart would show the same number. Measuring and looking can suggest a result, but they can't prove it.

You can't prove it for every triangle. There are too many to check.

This is a common misconception

You don't check them one by one. A proof works with a general isosceles triangle: AB = AC, with no particular lengths or angles. It uses results already known to be true for every triangle, such as the tests for congruent triangles. Each step holds whatever the triangle looks like, so the conclusion does too.

It's a conjecture. To prove it I need steps, each with a reason from a known result, that work for any isosceles triangle.

This matches the evidence

Yes. The dragging gave you strong evidence and a clear statement to prove. The proof below uses congruent triangles, so every step has a reason and the conclusion covers every isosceles triangle, including ones nobody has drawn.

Examples and measurements suggest a conjecture. A proof uses known results to show it is true in every case.

Proof: base angles of an isosceles triangle

Problem

Triangle ABC has AB = AC. Prove that angle ABC = angle ACB.

Your turn

Prove a property of a parallelogram

ABCD is a parallelogram, so AB is parallel to DC and AD is parallel to BC. The diagonal AC is drawn. Prove that angle ABC = angle CDA.

Look at triangles ABC and CDA. The diagonal AC is a side of both.
missing step

Which line is step 2?

Angle BAC = angle DCA (alternate angles, because AB is parallel to DC).Angle BAC = angle DCA (corresponding angles, because AB is parallel to DC).Angle BAC = angle DCA (base angles of an isosceles triangle).

Deriving Pythagoras' theorem

Why a² + b² = c² in every right-angled triangle

View: 1. The conjecture. Showing 1 layer: One triangle

View

Build squares on the sides of a 3, 4, 5 right-angled triangle: 9 + 16 = 25. For 5, 12, 13: 25 + 144 = 169. Cases like these suggest the conjecture a² + b² = c², where c is the longest side, opposite the right angle. Here is any right-angled triangle, with acute angles x and y.

Step through the four views in order. Each one adds a reason.

Check the proof

Similar is not the same as congruent

The straight lines AE and BD cross at C. Angle BAC = angle DEC, and AC = EC. Prove that AB = DE.

A student's proof — which line goes wrong?

Angle ACB = angle ECD (vertically opposite angles).
Angle BAC = angle DEC (given).
Angle ABC = angle EDC (the third angles, since each triangle's angles add to 180°).
So triangle ABC is congruent to triangle EDC (AAA: all three pairs of angles are equal).
So AB = DE, because they are corresponding sides of congruent triangles.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Spotting a pattern tells you what might be true. A chain of reasons shows that it is.

What you need to know

A conjecture is a statement you believe because of the cases you have checked or measured. A proof shows it is true in every case.
A proof is a chain of statements, and each one is justified by a known result: an angle fact, a congruence test, a similarity fact or a property of a quadrilateral.
Two results you can prove this way are that the base angles of an isosceles triangle are equal, and that a² + b² = c² in any right-angled triangle (Pythagoras' theorem).

The big picture

A conjecture is a statement suggested by examples or measurements, such as noticing that the base angles of an isosceles triangle always match. A proof shows it is true in every case, using a chain of statements each justified by a known result: an angle fact, a congruence test, a similarity fact or a property of a quadrilateral. Congruent triangles are the main tool, because all their matching sides and angles are equal. Equal angles alone prove only that triangles are similar. Pythagoras' theorem can be proved by fitting four copies of a right-angled triangle in a square and finding its area in two ways.

Key points

1To prove two angles or sides are equal, find two triangles that contain them and show that the triangles are congruent.

2Name the congruence test you use: SSS, SAS, ASA or RHS. Three pairs of equal angles only proves that the triangles are similar.

3Write a reason next to every statement, for example 'given', 'alternate angles', 'common side' or 'corresponding angles of congruent triangles'.

4Never use the result you are proving as one of your reasons, and never use 'it looks equal' as a reason.

5Pythagoras' theorem follows from finding the area of a square of side a + b in two ways: (a + b)² = 4 × ½ab + c².

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Worked example

Problem

Prove that the angles in any triangle ABC add up to 180°.

⚠ Watch out

Using what a diagram looks like as a reason. 'Angle x = angle y because they look the same' is not a step in a proof: every equal pair needs a known result behind it, such as 'alternate angles' or 'corresponding angles of congruent triangles'.

🧠

Memory hook

A pattern earns a guess; reasons earn a proof. No reason, no step.

✓

Check yourself

Kite ABCD has AB = AD and CB = CD. Prove that angle ABC = angle ADC. Check: triangles ABC and ADC are congruent (SSS, with AC common), so these corresponding angles are equal.

Flashcards

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What is a conjecture?

A statement you think is true because of the cases you have checked or measured, but haven't yet proved for every case.

What makes an argument a proof?

Every step is a statement justified by a known result, and the chain works for every case, not just the ones you drew.

What do three pairs of equal angles prove about two triangles?

Only that they are similar: the same shape, possibly different sizes. It doesn't prove they are congruent.

Triangle ABC has AB = AC. Which two angles are equal?

The base angles: angle ABC = angle ACB, the angles opposite the equal sides.

How do you prove the base angles of an isosceles triangle are equal?

Join the apex to the midpoint M of the base. The two triangles have three pairs of equal sides (SSS), so they are congruent, and the base angles are corresponding angles.

Once two triangles are congruent, what reason proves a pair of their angles are equal?

'Corresponding angles of congruent triangles are equal.' The same reason works for corresponding sides.

State Pythagoras' theorem.

In a right-angled triangle with shorter sides a and b and longest side c (the hypotenuse), a² + b² = c².

In the four-triangle proof of Pythagoras' theorem, which two expressions for the big square's area are set equal?

(a + b)² and 4 × ½ab + c². Expanding gives a² + 2ab + b² = 2ab + c², so a² + b² = c².

Why is the middle shape in the four-triangle proof a square?

Its four sides are all c, and each corner is 180° − 90° = 90°, because the two acute angles of a right-angled triangle add to 90°.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What did the dragging actually show?

Prove a property of a parallelogram

Similar is not the same as congruent

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