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If a = b and b = c, then a = c. This property is used in proofs to relate elements that are equal to a common element.

If a = b, then b = a. This property is used in proofs to reverse the order of equality between two elements.

If B is between A and C, then AB + BC = AC. Used to relate lengths of segments that are part of a larger segment.

The angle addition postulate states that if point B lies in the interior of \( \angle AOC \), then \( \angle AOB + \angle BOC = \angle AOC \). It enables the calculation of the degree of a larger angle from its smaller parts.

Given a line and a point not on the line, there is exactly one line through the point parallel to the given line. This is one of Euclid's original axioms for geometry.

If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. This theorem is used to analyze parallel lines and the angles within.

If two parallel lines are cut by a transversal, then each pair of corresponding angles are equal. This postulate is vital in establishing angle relationships in geometry.

The vertical angles theorem states that vertical angles, formed by two intersecting lines, are congruent.

For any quantity a, a = a. This is used in proofs to justify that any geometric figure is congruent to itself or any quantity is equal to itself.