GCSE · Maths · OCR · Spec J560
Congruent triangles
Two triangles can look identical and still not be. Today you'll learn the test that settles it, and it starts with you trying your hardest to cheat.
Maths · Congruent triangles
Two sides plus the angle between them, or all three sides, leave only one triangle. Two sides plus an angle that is not between them can leave two.
Two sides are locked: AB = 6.0 cm and AC = 4.5 cm. Drag C round A and add ONE more fact. Try three challenges. (1) Make angle A read 70°. (2) Make BC read 5.0 cm. (3) Make angle B read 40°. Each time, how many different triangles can you make? A flipped copy under line AB counts as the same triangle.
Maths · Which criterion?
Sort the evidence
For each pair of triangles, what do the facts prove? Pick a card, then pick its box.
Still to sort
SSS (0)
Three pairs of edges equal
Where the line is: Only edges are needed here. No angles.
SAS (0)
Two sides and the angle between them
Where the line is: The angle has to sit between the two sides, where they meet.
ASA / AAS (0)
Two angles and a corresponding side
Where the line is: The side must be in the same position in both triangles. Find the third angle first if you need to.
RHS (0)
Right angle, hypotenuse, one other side
Where the line is: Right-angled triangles only, and the right angle must be given or marked.
Not proven (0)
Not one of the four
Where the line is: It looks convincing, but the facts leave room for a different triangle, or the facts are not in the right places.
Some of these prove congruence. Some only look as if they do.
A full proof, line by line
Problem
Triangles ABD and CBD share the edge BD. Angle BAD = angle BCD = 100°, and angle ADB = angle CDB = 30°. The diagram is not drawn accurately. Prove that the triangles are congruent, then show that AB = CB.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
What you need to know
- Congruent shapes fit exactly on top of each other, using rotation, reflection or translation as necessary. Angles and edges match, in the same positions.
- Equal angles aren't enough. All equilateral triangles have three 60° angles, but they're congruent only if their edge lengths match too.
Have a goJess has two equilateral triangles. Both have three 60° angles, so she announces: 'Congruent!' What has she forgotten to check?
Whether the edge lengths match.
Equal angles alone don't make triangles congruent. Equilateral triangles are congruent only if their edge lengths are also the same.
- A congruence criterion is information that leaves room for only one triangle. The four are SSS, SAS, ASA and RHS.
- SSS: three edges the same. Three known edges allow only one triangle, and that fixes the angles too.
- SAS: two sides and the angle between them. The angle has to sit between the two sides.
- Two sides and an angle that isn't between them (SSA) don't prove congruence, because two different triangles can be built.
Have a goSam and Ali each draw a triangle with sides of 9 cm and 6 cm and a 35° angle that is not between them. Must their triangles match?
No. They could end up with two different triangles.
Two sides and a non-included angle (SSA) can be built into two different triangles, so it doesn't prove congruence.
- ASA: two corresponding angles and a corresponding side. AAS is the same idea, because the third angle can be calculated from the 180° sum.
Have a goA triangle has angles of 55° and 65°. What is the third angle, and what fact lets you find it?
60°, because angles in a triangle add up to 180°.
180° − 55° − 65° = 60°. Finding the third angle is what turns AAS into ASA.
- RHS works only for right-angled triangles: the right angle, the hypotenuse and one other side. The right angle must be given or marked.
- A proof gives a reason for every statement, such as given information, a shared edge or a defined property. Then it names the criterion.
- Once triangles are proved congruent, corresponding edges and angles are equal. That lets you prove two edges or two angles equal.
The big picture
Two triangles are congruent when they fit exactly on top of each other. You don't have to check everything: SSS, SAS, ASA (or AAS) and RHS are sets of facts that leave room for only one triangle. Two sides with a non-included angle, or angles alone, do not. Once congruence is proved, corresponding edges and angles are equal, and a proof needs a reason for every line.
Key points
Worked example
Problem
Triangle ABC is congruent to triangle PQR, with A matching P, B matching Q and C matching R. In triangle ABC, AB = 7 cm, BC = 5 cm, angle A = 48° and angle B = 70°. Find PQ, QR and angle R.
⚠ Watch out
Treating a likeness as a proof: 'they look the same', 'those lines look square', or 'two sides and an angle match'. Each time, check that the facts are one of SSS, SAS, ASA (or AAS) or RHS, in matching positions, with any right angle given or marked.
Memory hook
Only one triangle fits? Congruent. Room for a second? Not proven. SSS, SAS, ASA and RHS leave no room. SSA and angles-only are the two that fool you.
Check yourself
Pick any one criterion and explain to a friend, without looking back, why it leaves room for only one triangle. Then name the two situations that look like proof but fail.
Flashcards
(13)What does it mean for two shapes to be congruent?
Which transformations always give a congruent image?
Do equal angles make two triangles congruent?
SSS: what is known, and why does it work?
SAS: where must the angle be?
Why does SSA not prove congruence?
What is the difference between ASA and AAS?
RHS: what do you need, and for which triangles?
Which edge is the hypotenuse?
Can you assume a right angle because two lines look perpendicular?
How can you check by hand that two shapes drawn to the same scale are congruent?
What counts as a justification in a congruence proof?
How does a congruence proof end, and what does it let you do?
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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