KS3 · Maths

Criteria for congruent triangles (SSS, SAS, ASA, RHS)

Your friend describes a triangle using a few measurements. You draw it — and yours looks different. Who made the mistake? Possibly neither of you.

Maths · Congruent triangles

Could these measurements make two different triangles?

AB 6.0 cm. AC 7.0 cm. BC 5.6 cm. Angle A 50°. Angle C 56°. Classification: SSA: angle A = 50°, AB = 6 cm, and BC = 5 cm opposite the angle. Relationship: Choose AC first and BC is settled: one triangle (SAS, because angle A sits between AB and AC). Choose BC = 5.0 cm first and AC can be about 1.9 cm or about 5.8 cm: two triangles (SSA). Where BC is shortest, angle C reads 90° and only one place fits — AB is then the hypotenuse, the side opposite the right angle.

AB6.0 cmAC7.0 cmBC5.6 cmAngle A50°Angle C56°Drag C up and down the ray

SSA: angle A = 50°, AB = 6 cm, and BC = 5 cm opposite the angle

Choose AC first and BC is settled: one triangle (SAS, because angle A sits between AB and AC). Choose BC = 5.0 cm first and AC can be about 1.9 cm or about 5.8 cm: two triangles (SSA). Where BC is shortest, angle C reads 90° and only one place fits — AB is then the hypotenuse, the side opposite the right angle.

Triangle ABC has a 50° angle at A and AB = 6 cm. Picture BC as a ladder of fixed length, with its foot at B and its top on the ray from A. Drag C until BC reads 5.0 cm — then hunt for a second place where it does too.

Maths · Congruent triangles

What do you think congruent means?

You are deciding whether two shapes are congruent.

Which of these is closest to what you think right now?

If two triangles have the same three angles, they must be congruent.Two shapes are congruent if one can be moved to fit exactly on top of the other — turning, flipping or sliding are all allowed.If two quadrilaterals have the same four edge lengths, they must be congruent.If one triangle is turned round or flipped over compared with the other, they can't be congruent.

How sure are you?

I'm sureI think soI'm guessing

If two triangles have the same three angles, they must be congruent.

This is a common misconception

Same angles only makes the triangles similar — the same shape, not necessarily the same size. Every equilateral triangle has three 60° angles, but one with 6.5 cm edges is not congruent to one with 6.3 cm edges. An enlargement keeps every angle and changes the size, so angles on their own can never prove congruence.

Two shapes are congruent if one can be moved to fit exactly on top of the other — turning, flipping or sliding are all allowed.

This matches the evidence

Yes. If a shape can fit exactly on top of another using rotation, reflection or translation, the two are congruent. Corresponding angles and edges must be the same size and in the same relative position.

If two quadrilaterals have the same four edge lengths, they must be congruent.

This is a common misconception

Not for quadrilaterals. The edges must also be in the same order, and the angles must match too. A rectangle and a parallelogram can both have edges of 3 cm, 6 cm, 3 cm, 6 cm in that order and still have different angles. Triangles are different: wherever you put one side it has to join the other two, so three edges can only go together in one order.

If one triangle is turned round or flipped over compared with the other, they can't be congruent.

This is a common misconception

Orientation doesn't matter. You are allowed to rotate, reflect or slide a shape to make it fit on top of its partner, so a turned or flipped triangle can still be congruent. What matters is that the matching angles and edges are the same size and in matching positions.

Congruent means an exact match in size and shape, whichever way round it sits. Matching angles alone only gets you as far as similar.

Constructing a triangle from two angles and a side

Problem

Construct triangle MNO with MN = 7 cm, angle MON = 70° and angle ONM = 45°.

Maths · Congruent triangles

Which piece of information does the job?

For each pair of triangles, pick the criterion the information proves congruence with — or decide it is not enough.

Still to sort

SSS (0)

All three sides match.

Where the line is: Three pairs of sides, not two. No angles needed.

SAS (0)

Two sides and the angle between them.

Where the line is: The angle must sit between the two sides. If it is not between them, that is SSA, which is not enough.

ASA / AAS (0)

Two angles and one side.

Where the line is: The side must be a corresponding side. It can be between the two angles (ASA) or not between them (AAS).

RHS (0)

Right angle, hypotenuse and one other side.

Where the line is: Right-angled triangles only, and the right angle must be marked, given or calculated — not just look right.

Not enough information (0)

No criterion applies.

Where the line is: The triangles might still be congruent. There is just not enough to prove it.

11 of 11 still to sort.

Maths · Congruent triangles

Your turn to supply the reasons

PQRS is a parallelogram. T is a point on PS and U is a point on RQ, and PT = RU. Prove that TQ = SU.

We want to show TQ = SU. These are sides of triangles PTQ and RUS, so we prove those two triangles congruent first.
PT = RU, because we are told so in the question.
missing step

Which line is step 3?

PQ = SR, because opposite sides of a parallelogram are equal.PQ = SR, because they are both marked with hash marks.PQ = SR, because they are both hypotenuses.

Maths · Congruent triangles

Write two short congruence proofs

(a) E is the midpoint of AC and also the midpoint of BD, so the lines AC and BD cross at E. Prove that triangle AEB is congruent to triangle CED. (b) A new diagram: ABCD and DEFG are two squares that share the corner D. Going round D, the edges meet in the order DC, DA, DG, DE. Prove that triangle CDG is congruent to triangle ADE. Write each proof as statements with reasons. [6 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

(a) AE = EC and DE = EB, because E is the midpoint of both AC and BD. (1 mark)
(a) Angle AEB = angle CED, because vertically opposite angles are equal. (1 mark)
(a) Two pairs of equal sides with the equal angle between them, so △AEB ≅ △CED by SAS. (1 mark)
(b) AD = DC and GD = DE, because each pair are edges of the same square. (1 mark)
(b) Angle CDG = 90° + angle ADG and angle ADE = 90° + angle ADG, so the two angles are equal. (1 mark)
(b) Two pairs of equal sides with the equal angle between them, so △CDG ≅ △ADE by SAS. (1 mark)

Model answer

(a) AE = EC and DE = EB (E is the midpoint of AC and of BD). Angle AEB = angle CED (vertically opposite angles). Two pairs of equal sides with the angle between them equal, so △AEB ≅ △CED by SAS. (b) AD = DC and GD = DE (edges of the same square). Angle CDG = 90° + angle ADG and angle ADE = 90° + angle ADG, so the angles are equal. Two pairs of equal sides with the angle between them equal, so △CDG ≅ △ADE by SAS.

Good proofs use the same moves each time: find equal sides and equal angles from the shape's properties, say which criterion they add up to, then write the triangles in matching order. In (b) the congruence was hiding until you spotted the angles were each made of 90° plus angle ADG.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find the pieces of information that allow only one triangle — and the ones that leave a second.

What you need to know

Congruent shapes match exactly: one fits on top of the other after turning, flipping or sliding, with matching angles and edges in matching places.
Four sets of information prove two triangles congruent, because each allows only one triangle: SSS, SAS, ASA/AAS and RHS.
Equal angles only (AAA) proves similar, not congruent. Two sides and an angle not between them (SSA) can fit two different triangles. Neither proves congruence.
You can work out missing facts instead of measuring: the third angle (angles in a triangle add up to 180°), or what shape properties tell you.
Once two triangles are congruent, all their corresponding sides and angles are equal, so congruence can prove that two lengths or angles are equal.

The big picture

Two triangles are congruent when one fits exactly on top of the other. You don't have to measure every part to prove it: SSS, SAS, ASA/AAS and RHS each give just enough information to allow only one triangle. AAA and SSA do not, because they leave room for a different triangle.

Key points

1SSS: all three pairs of sides are equal.

2SAS: two pairs of equal sides, with the equal angle between them.

3ASA / AAS: two pairs of equal angles and one pair of equal corresponding sides.

4RHS: right-angled triangles only — equal hypotenuses and one other pair of equal sides, with the right angle marked, given or calculated.

5AAA and SSA are not criteria. If no criterion fits, the honest answer is "not enough information to prove it".

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Worked example

Problem

ABCD is a rectangle with the diagonal AC drawn. Show that triangle ABC is congruent to triangle ADC.

⚠ Watch out

Using equal parts from the wrong place. For SAS, ASA and AAS the equal parts must sit in the same position in both triangles: between the same sides or angles, or opposite the same angle.

🧠

Memory hook

Can it wobble? If the information leaves the triangle room to wobble into a second shape, it proves nothing. SSS, SAS, ASA/AAS and RHS hold it still; AAA and SSA do not.

✓

Check yourself

Two triangles share angles of 40° and 60° and a 5 cm side. What must you check before saying they are congruent? Answer: that the 5 cm side is in the same place in both.

Flashcards

(14)

What does it mean for two shapes to be congruent?

They are exactly the same size and shape: one fits exactly on top of the other after turning, flipping or sliding, with matching angles and edges in matching positions.

What does SSS stand for, and what does it need?

Side, side, side: all three sides of one triangle equal the three corresponding sides of the other. Three side lengths fix the angles too, so only one triangle fits.

SAS

Side, angle, side. Two pairs of equal sides and the equal angle between them. The angle must sit between the two sides.

ASA and AAS

Two pairs of equal angles and one pair of equal corresponding sides. ASA if the side is between the angles, AAS if it is not. They are equivalent because the third angle can always be worked out.

RHS

Right angle, hypotenuse, side. For right-angled triangles only: equal hypotenuses and one other equal side. The right angle must be marked, given or calculated.

What is the hypotenuse?

The side of a right-angled triangle opposite the right angle. It is the longest side. A triangle with no right angle has no hypotenuse.

Why does AAA not prove congruence?

Three equal angles only prove the triangles are similar. An enlargement keeps all the angles but changes the size.

Why does SSA not prove congruence?

With two sides and an angle that is not between them, two different triangles can sometimes be formed.

What if the information fits none of the criteria?

You cannot be sure the triangles are congruent. They might be, but there is not enough information to prove it.

What do hash marks and the sign ≅ mean?

Hash marks show edges of equal length. △ABC ≅ △ADC means triangle ABC is congruent to triangle ADC.

What follows once two triangles are proved congruent?

All their corresponding sides and angles are equal, so congruence can be used to prove that two lengths or angles are equal.

How can you show sides or angles are equal without measuring?

Use properties: opposite sides and angles of a parallelogram, edges of the same square, vertically opposite angles, a midpoint, a shared side — or work out a missing angle from the 180° total.

What should you check before comparing two lengths?

Check the units match. 35 cm is 0.35 m and 12 cm is 120 mm, so those pairs of sides are equal.

What are corresponding sides?

Sides in the same position in each triangle, for example both opposite the same angle. For ASA or AAS the equal side must be a corresponding side.

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 do you think congruent means?

Which piece of information does the job?

Your turn to supply the reasons

Write two short congruence proofs

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