Congruence: Same Shape, Same Size

Congruent, similar, and the difference in one line

Congruent shapes are identical twins; similar shapes are photocopies at different settings. Similar shapes have equal angles and sides in the same ratio, so a scale factor is involved. Congruent shapes have the same angles and the same lengths, so the scale factor is 1.

Position and orientation are irrelevant. A triangle rotated, slid across the page, or reflected in a mirror line is still congruent to the original, because none of those movements changes any length or any angle. Only resizing does.

The symbol is ≅, and the order of the letters is part of the statement: triangle ABC ≅ triangle DEF says that A matches D, B matches E and C matches F, so a correspondence is being claimed as well as a congruence.

The four conditions for triangles

You never have to check all six measurements. Three of the right kind are enough to force everything else, and there are four combinations that do it:

AAS — two angles and a side that is not between them — is sometimes listed as a fifth. It is really ASA in disguise, since knowing two angles gives you the third, and the side can then be repositioned.

• SSS — all three sides equal
• SAS — two sides and the angle between them equal
• ASA — two angles and the side between them equal
• RHS — right angle, hypotenuse and one other side equal, for right-angled triangles only

Why AAA is not on the list

Three matching angles fix the shape but say nothing about the size. A triangle with sides 3, 4 and 5 has exactly the same angles as one with sides 6, 8 and 10, and the second is twice as large. Equal angles prove similarity, never congruence.

Two sides and an angle outside them, SSA, fails for a different reason: the information can often be satisfied by two different triangles, one with an acute angle and one obtuse. RHS is the special case where that ambiguity disappears, which is why right-angled triangles get a rule of their own.