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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.

In a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):

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.

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

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.

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

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

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 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.