Geometric Proofs and Theorems: Properties, Congruence, and Angle Relationships

Reflexive POE

AB = AB

Symmetric POE

If AB=CD, then CD=AB.

Transitive POE

If AB=CD and CD=EF, then AB=EF.

Reflexive POC

𝐴𝐴𝐴𝐴≅𝐴𝐴𝐴𝐴

Symmetric POC

If 𝐴𝐴𝐴𝐴≅𝐶𝐶𝐶𝐶, then 𝐶𝐶𝐶𝐶≅𝐴𝐴𝐴𝐴.

Transitive POC

If 𝐴𝐴𝐴𝐴≅𝐶𝐶𝐶𝐶 and 𝐶𝐶𝐶𝐶≅𝐸𝐸𝐸𝐸, then 𝐴𝐴𝐴𝐴≅𝐸𝐸𝐸𝐸.

Definition of Congruence

2 or more segments with the same measure

Definition of Bisector

A segment, ray, or line that intersects a segment at its midpoint

Definition of Midpoint

A point on a segment exactly halfway between the endpoints of a segment

Definition of Vertical Angles

2 nonadjacent angles formed by 2 intersecting lines

Definition of Complementary

2 angles whose sum is 90 degrees

Definition of Supplementary

2 angles whose sum is 180 degrees

Definition of Linear Pair

2 adjacent angles whose nonadjacent sides form opposite rays

Definition of Right Angle

An angle that is exactly 90 degrees

Definition of Perpendicular

2 lines that intersect to form a right angle

Segment Addition Postulate

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

Midpoint Theorem

If M is the midpoint of 𝐴𝐴𝐴𝐴, then 𝐴𝐴𝐴𝐴≅𝑀𝑀𝑀𝑀.

Angle Addition Postulate

If D is in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Supplement Theorem

If 2 angles form a linear pair, then they are supplementary.

Complement Theorem

If 2 adjacent angles form a right angle, then they are complementary.

Congruent Supplements Theorem

If 2 angles are both supplementary to a 3rd angle, then those 2 angles are congruent.

Congruent Complements Theorem

If 2 angles are both complementary to a 3rd angle, then those 2 angles are congruent.

Vertical Angles Theorem

If 2 angles are vertical angles, then they are congruent.

Right Angle Theorem

If 2 lines are perpendicular, then they form 4 right angles.

Summary

All right angles are congruent.