B. Proofs

In the previous lesson, you made conjectures and checked them. However, it was difficult to conclude definitively that a conjecture was true. Mathematical proofs help alleviate this difficulty. A mathematical proof is an argument that uses deductive reasoning to show a conjecture is true for all cases.

Two-Column Proofs

Proofs can take on many forms. A two-column proof organizes reasoning by showing a series of statements and how they are justified.

Example 1

Prove the following conjecture using a two-column proof and the fact that the area of a rectangle can be determined using the formula A = bh, where b represents the base length and h represents the height of the rectangle.

  • The area of a right triangle is , where b represents the base length and h represents the height of the triangle.
StatementJustification
The acute angles in a right triangle are complementary (complementary angles add to 90°). A right triangle contains a 90° angle and the sum of the interior angles in a triangle is 180°, so the two remaining acute angles must add to 180° − 90° = 90°.

A rectangle can be formed using any pair of identical right triangles.

 Placing the complementary angles together forms a quadrilateral with four 90° angles, which is a rectangle.
The area of a rectangle is This information was given.
The area of one triangle is
The rectangle was made from two identical right triangles, so each will have an area equal to half that of the rectangle.