Geometry Definitions Used in Proofs

Reflexive Property of Equality

Any angle or segment is congruent to itself. "true for real numbers, segments, length, and angle measure." for a, a=a; for AB, AB=AB; for angle A, m∠A=m∠A

Transitive Property of Equality

If A = B and B = C, then A = C. If two segments (or angles) are congruent to the same segment (or angle), then they are congruent. If two segments (or angles) are congruent to congruent segments (or angles), then they are congruent.

Postulate 2: Segment Additon Postulate

If B is between A and C, then AB + BC = AC.

Postulate 4: Angle Addition Postulate

If P is in the interior of angle RST, then m<RSP + m<PST = m<RST.

Definition of Segment Bisector

If a point (or line, ray) divides a segment into two congruent segments, then it is a segment bisector. If a point (or line, ray) is a segment bisector, then it divides a segment into two congruent segments.

Definition of Midpoint

If a point divides a segment into two congruent segments, then it is a segment bisector. If a point is a midpoint, then it divides a segment into two congruent segments.

Definition of an Angle Bisector

If a ray bisects an angle, then it divides the angle into two congruent angles. If a ray divides an angle into two congruent angles, then it bisects the angle.

Definition of an Angle Trisector

If a ray trisects an angle, then it divides the angle into three congruent angles. If a ray divides an angle into three congruent angles, then it trisects the angle.

Definition of Equiangular polygons

If all interior angles of a polygon are congruent, then the polygon is equiangular. If a polygon is equiangular, then all interior angles of the polygon are congruent.

Definition of Equilateral polygons

If all sides of a polygon are congruent, then the polygon is equilateral. If a polygon is equilateral, then all the sides of the polygon are congruent.

Division Theorem

If segments (or angles) are congruent, then their like divisions are congruent.

Multiplication Theorem

If segments (or angles) are congruent, then their like multiples are congruent.

Definition of a Straight Angle

If the measure of an angle is 180, then the angle is a straight angle. If an angle is a straight angle, then the measure of the angle is 180.

Definition of a Right Angle

If the measure of an angle is 90, then it is a right angle. If an angle is a right angle, then the measure of the angle is 90.

Definition of an Acute Angle

If the measure of an angle is between 0 and 90, then the angle is acute. If an angle is acute, then the measure of the angle is between 0 and 90.

Definition of an Obtuse Angle

If the measure of an angle is between 90 and 180, then the angle is obtuse. If an angle is obtuse, then the measure of the angle is between 90 and 180.

Definition of Supplementary Angles

If the measure of two angles add to 180, then they are supplementary angles. If two angles are supplementary, then their angle measures add to 180.

Definition of Complementary Angles

If the measure of two angles add to 90, then they are complementary angles. If two angles are complementary, then their angle measures add to 90.

Addition Theorem

If the same segment (or angle) is added to congruent segments (or angles), then the sums are congruent. If congruent segments (or angles) are added to congruent segments (or angles), then the sums are congruent.

Subtraction Theorem

If the same segment (or angle) is subtracted from congruent segments (or angles), then the differences are congruent. If congruent segments (or angles) are subtracted from congruent segments (or angles), then the differences are congruent.

Postulate 13: Parallel Postulate

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

Postulate 13: Perpendicular Postulate

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

Definition of Congruent Angles

If two (or more) angles have the same measure, then the angles are congruent. If two (or more) angles are congruent, then the angles have the same measure.

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.

Right Angle Congruence Theorem

If two angles are right angles, they are congruent. All right angles are congruent.

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary. If two angles are supplementary, then they form a linear pair.

Definition of a Linear Pair

If two angles form a straight angle, then they are a linear pair. If two angles form a linear pair, then they for a straight angle. If two angles form a linear pair, then their measures add to 180.

Definition of Perpendicular Lines

If two lines form right angles, then they are perpendicular. If two lines are perpendicular, then the form right angles. If two lines form a 90 angle, then they are perpendicular. If two lines are perpendicular, then then form a 90 angle.

Definition of Congruent segments

If two segments are congruent, then their measures are equal. If two segements have equal measures, then they are congruent.