Triangles: Congruence

Two triangles are congruent if they have exactly the same shape and the same size, so that one can be placed on top of the other and cover it perfectly. When that happens, all three pairs of sides are equal and all three pairs of angles are equal. We write triangle ABC is congruent to triangle DEF, and the ORDER of the letters matters: it tells you A matches D, B matches E and C matches F. Congruence is the geometry of an exact copy.

You do not need all six pairs of parts to be sure two triangles are congruent. Just the right three are enough. SSS: three pairs of sides equal. SAS: two sides and the angle BETWEEN them equal. ASA: two angles and the side between them equal. AAS: two angles and a side that is not between them. RHS: for right-angled triangles, the right angle, the hypotenuse and one other side. Two famous-looking rules fail: AAA fixes only the shape (think of a small and a large equilateral triangle), and SSA can give two different triangles.

The real power of congruence is what comes after. Once you have proved two triangles congruent using one of the criteria, every other pair of corresponding parts must also be equal. This is CPCT: Corresponding Parts of Congruent Triangles are Equal. So to prove two awkward sides or angles are equal, you find a pair of triangles that contain them, prove those triangles congruent, and then quote CPCT. This single idea powers most proofs in the chapter.

Congruence proves two beautiful facts about an isosceles triangle (two equal sides). First: the angles opposite the equal sides are equal. Second (the converse): if two angles of a triangle are equal, the sides opposite them are equal. The chapter also covers inequalities: in any triangle the side opposite the larger angle is longer, and the sum of any two sides is greater than the third side (the triangle inequality).