GCSE · Maths · OCR · Spec J560
Applying congruent triangles
Two sides of 8 cm and 5 cm, plus a 30° angle: build me a triangle. Now give those exact facts to a friend. Will they build the same one?
Maths · Congruent triangles
Neither is enough to prove congruence
Two sides and an angle that is not between them can fit two different triangles. Three equal angles can fit triangles of different sizes.
Bottom figure: AB is 8 cm and the angle at A is 30°, and neither ever changes. Slide C along the faint line and find every spot where BC reads 5.0 cm. Top figure: two triangles with matching angles. Compare their readouts.
Maths · Congruent triangles
Which rule is it?
Pick a card, then pick the rule it proves. If it can't prove congruence, put it in Not enough.
Still to sort
SSS (0)
Three pairs of equal sides.
Where the line is: No angles are needed: three sides already leave only one triangle.
SAS (0)
Two pairs of equal sides and the angle between them.
Where the line is: The angle must be between the two sides. An angle outside them is a different case.
ASA (0)
Two pairs of equal angles and the side between them.
Where the line is: The side sits between the two angles. If it doesn't, it is AAS.
AAS (0)
Two pairs of equal angles and a side that is not between them.
Where the line is: The side is in the same place on both triangles, outside the two angles.
RHS (0)
A right angle, the hypotenuse and one shorter side.
Where the line is: The hypotenuse is the longest side, opposite the right angle. The other equal side is shorter.
Not enough (0)
The facts leave room for more than one triangle.
Where the line is: Includes two sides and an angle not between them, three angles, and a right angle with a hypotenuse only.
Each card lists what you know is equal on a pair of triangles. Decide if it is enough, and if so, which rule it is.
Worked proof
Problem
ABCD is a parallelogram: A bottom-left, B bottom-right, C top-right, D top-left. The diagonal AC is drawn. Prove that triangle ABC is congruent to triangle CDA.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Which facts pin a triangle down, and how do you prove it in writing?
What you need to know
- Congruent triangles are exactly the same shape and size: one fits on the other, even if it's turned or flipped over.
- To prove it, you don't need all six facts. You just need enough to leave only one possible triangle.
- SSS is three pairs of equal sides. SAS is two pairs of equal sides and the angle between them.
Have a goZara says: 'My triangle has sides 6, 6 and 4 cm. Ben's has 6, 6 and 4 cm too, but his is pointing the other way, so they can't be congruent.' Is Zara right?
No. All three pairs of sides are equal (SSS), so the triangles are congruent.
Congruent triangles can be turned or flipped, and three equal sides fix the triangle.
- ASA: two angles and the side between them. AAS: two angles and a side not between them, in the same place on both.
- RHS needs a right angle, equal hypotenuses and one more equal side. A right angle and hypotenuse alone aren't enough.
- Two sides and an angle that isn't between them (SSA) can make two different triangles, so it isn't a proof.
Have a goA friend says: 'Two sides and an angle are equal, so the triangles are congruent by SAS.' What is the first thing you'd ask them?
Is the angle between the two sides?
Only an angle between the two sides gives SAS. An angle outside them is SSA, which can make two different triangles.
- Three equal angles (AAA) give the same shape but not necessarily the same size, so that isn't enough either.
- A congruence proof is a list of statements, each with a reason, ending by naming the triangles and the rule.
- Once the triangles are congruent, matching sides and angles are equal. That's how you prove something new about a shape.
Have a goTriangle ABC is congruent to triangle PQR, with A matching P, B matching Q and C matching R. Which side of PQR must be equal to BC?
QR
B matches Q and C matches R, so the side joining B and C matches the side joining Q and R.
- Never write a right angle just because it looks like one. Use only what's given or already proved.
The big picture
Two triangles are congruent when one fits exactly on the other. To prove it you only need enough facts to leave one possible triangle: SSS, SAS, ASA, AAS or RHS. Two sides with an angle outside them, or three angles, are not enough. A proof gives a reason for every statement, names the triangles and the rule, and then lets you state that matching sides or angles are equal.
Key points
Worked example
Problem
Triangle PQR has PQ = 7 cm, QR = 5 cm and angle PQR = 35°. Triangle XYZ has XY = 7 cm, YZ = 5 cm and angle XYZ = 35°. Are the triangles congruent? If so, give the rule and say what else is equal.
⚠ Watch out
Using SSA or AAA as a reason. They feel like rules because three facts match, but two sides with an angle outside them can build two different triangles, and three equal angles fix only the shape, not the size.
Memory hook
Think sandwiches. In SAS the angle is the filling between two sides, and in ASA the side is the filling between two angles. Move the angle outside the sides and you get SSA, which doesn't work.
Check yourself
Without looking back, explain to a friend why 'two sides and an angle are equal' isn't automatically enough to prove congruence. Which one word decides whether it is?
Flashcards
(13)What does 'congruent' mean for two triangles?
Name the five rules that prove two triangles congruent.
What is an 'included' angle?
ASA or AAS: how do you tell them apart?
What three things does RHS need?
What is the hypotenuse?
Three equal angles: are the triangles congruent?
Can you write an angle as 90° because it looks like one in the diagram?
What goes on every line of a congruence proof?
Which reason covers a side that both triangles share?
Which reason covers equal angles in a Z shape between parallel lines?
What does the letter order in 'triangle ABC is congruent to triangle PQR' tell you?
Once you have proved two triangles congruent, what can you now say?
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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