Geometry Theorems, Postulates, and Definitions

Addition Properties

**Geometry**

**Theorems**, **Postulates**, **and** **Definitions**

● If the same segment is added to two congruent segments, then the sums are congruent.

(Common Segments Theorem)

● If two congruent segments are added to two congruent segments, then the sums are congruent.

● If the same ∠ is added to two congruent ∠s, then the sums are congruent.

● If two congruent ∠s are added to two congruent ∠s, then the sums are congruent.

Subtraction Properties

● If the same segment is subtracted from two congruent segments, then the differences are

congruent. (Common Segments Theorem)

● If two congruent segments are subtracted from two congruent segments, then the difference are

congruent.

● If the same ∠ is subtracted from two congruent ∠s, then the differences are congruent.

● If two congruent ∠s are subtracted from two congruent ∠s, then the differences are congruent.

Transitive Properties

● If two segments are congruent to the same segment, then they are congruent to each other.

● If two ∠s are congruent to the same ∠, then they are congruent to each other.

● If two segments are congruent to two congruent segments, then they are congruent to each other.

● If two ∠s are congruent to two congruent ∠s, then they are congruent to each other.

Right Angle Congruence

● If two ∠s are right ∠s, then they are congruent.

Vertical Angle Congruence

● If two ∠s are vertical ∠s, then they are congruent.

Definition of Perpendicular Lines

● If two segments (or lines) are perpendicular, then they form a right angle.

● If two segments (or lines) form a right angle, then they are perpendicular.

Definition of Bisection

**Geometry**

**Theorems**, **Postulates**, **and** **Definitions**

● If a point (or segment, ray, or line) bisects a segment, then it divides the segment into two

congruent segments.

● If a point (or segment, ray, or line) divides a segment into two congruent segments, then it bisects

the segment.

● 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 Trisection

● If two points trisect a segment, then they divide the segment into three congruent segments.

● If two rays trisect an angle, then they divide the angle into three congruent angles.

Definition of a Midpoint

● If a point is the midpoint of a segment, then it divides the segment into two congruent segments.

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

Definition of Complementary **and** Supplementary Angles

● If the sum of two angles is a right angle, then they are complementary.

● If the sum of two angles is a straight angle (linear pair), then they are supplementary.

Congruent Complements Theorem

● If two ∠s are complementary to the same ∠, then they are congruent.

● If two ∠s are complementary to two congruent ∠s, then they are congruent.

Congruent Supplements Theorem

● If two ∠s are supplementary to the same ∠, then they are congruent.

● If two ∠s are supplementary to two congruent ∠s, then they are congruent.

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