Geometry
Congruent triangles
Decide when two triangles are congruent using SSS, SAS, ASA and RHS.
Two triangles are congruent when one could be picked up and laid exactly on top of the other — same shape and same size, turned or flipped as much as you like. Checking all six pairs of parts would settle it, but that is far more than anyone needs. Three well-chosen pairs are always enough, and knowing which three is the whole of this skill.
The four rules
SSS: three pairs of sides. SAS: two sides and the angle between them. ASA: two angles and the side between them. RHS: in right-angled triangles, the hypotenuse and one other side. In SAS the word between is doing real work — move the angle away from between the sides and the rule stops holding. ASA is more forgiving: once two angles are known the third is forced, so the side may sit against either angle — that form has its own name, AAS.
Worked example
Why do the two halves of an isosceles triangle come out congruent?
Triangle ABC has AB = AC. Mark D, the mid-point of BC, and join AD.
One extra line turns one triangle into two, which is what lets a congruence rule get a grip.
Try it together
Triangle ABC has AB = 5, BC = 7 and CA = 9. Triangle PQR has PQ = 5, QR = 7 and RP = 9. Are they congruent?
Compare the parts you have been given, then name the rule they add up to.
1.How many pairs of equal sides are there between the two triangles?
Watch out
AAA is not a congruence rule. Three equal angles make two triangles the same shape, but one can be a scaled-up copy of the other. SSA is no good either: two sides and an angle that is not between them can sometimes be arranged into two genuinely different triangles.
Have a go
Have a go on your own: two triangles have two pairs of equal angles and the pair of sides between those angles is equal too. Which rule proves them congruent?
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.