In mathematics, we have postulates which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct logical arguments for new facts of mathematics. A new fact is called a theorem and its argument is called its proof.

We can write proofs in a few different ways.

An example of each type of proof can be found in the worked examples.

This topic also introduces the following theorems:

Congruent complements theorem

If two angles are complementary to the same angle then they are congruent.

In the given example, \angle 1 \cong \angle 3.

Congruent supplements theorem

If two angles are supplementary to the same angle then they are congruent.

In the given example, \angle 1 \cong \angle 3.

Right angle congruence theorem

All right angles are congruent.

Double right angle theorem

If two angles are congruent and supplementary, then each is a right angle.

To prove these theorems, we may need to use some other properties of angles and angles.

Properties of segment congruence tell us that segment congruence is reflexive, symmetric, and transitive

• Reflexive property of congruent segments: Any segment is congruent to itself, so \overline{AB} \cong \overline{AB}
• Symmetric property of congruent segments: If \overline{AB} \cong \overline{CD}, then \overline{CD} \cong \overline{AB}
• Transitive property of congruent segments: If \overline{AB} \cong \overline{CD} and \overline{CD} \cong \overline{EF}, then \overline{CD} \cong \overline{EF}

Properties of angle congruence tell us that angle congruence is reflexive, symmetric, and transitive

• Reflexive property of congruent angles: Any angle is congruent to itself, so \angle A \cong \angle A
• Symmetric property of congruent angles: If \angle A \cong \angle B, then \angle B \cong \angle A
• Transitive property of congruent angles: If \angle A \cong \angle B and \angle B \cong \angle C, then \angle A \cong \angle C

Worked examples