Two-column proofs (or three-column if you count the diagram column) work well for many geometric proofs, however, they are not the only type you can use. The proof in the following example is written as a paragraph. The type of reasoning is very similar to a two-column proof, but is displayed differently. Which style do you prefer to read? You will convert the proof in Example 3 into a two-column proof in the next Coach's Corner.

Example 3

Prove that angle ABC is equal to angle ACB if triangle ABF is congruent to triangle ACD.

Proof:

It can be seen that triangle BCD is congruent to triangle CBF by SAS:
BD = CF is given, angle CFB and angle BDC are corresponding angles in congruent triangles and both triangles share side BC. Angle DBC must equal angle FCB because they are corresponding angles in congruent triangles DBC and FCB. This means their supplements, angle ABC and angle ACB must also be equal.

For further information about congruent triangles see pp. 107–111 of Principles of Mathematics 11.

Showing that two triangles are congruent can be done using four different congruence relationships: SSS, SAS, ASA, and SAA. There are two cases where three pieces of information do not allow you to conclude that triangles are congruent: SSA and AAA. In the next lesson you will continue your study of triangles by exploring trigonometry using triangles that do not include a right angle.