Geometric Proofs

Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles

Ex: If angle ABC is a right angle then we know that angle ABD and angle DBC are complementary (assuming that ray BD intersects angle ABC)

Congruent Supplements Theorem

If angles are supplementary to the same angle or congruent angles, then they are congruent to each other

Ex: If angle D and angle E are supplementary angles and angle F and angle E are supplementary angles, then angle D is congruent to angle F

Congruent Complements Theorem

If angles are complementary to the same angle or congruent angles, then they are congruent to each other.

Ex: If angle D and angle E are complementary angles and angle F and angle E are complementary angles, then angle D is congruent to angle F

Relfexive Property of Angle Congruence

All angles are congruent to themselves

_{Ex: Angle 1 is congruent to Angle 1}

Symmetric Propertry of Angle Congruence

If angle 1 is congruent to angle 2, then angle 2 is congruent to angle 1

Transitive Propertry of Angle Congruence

If angle 1 is congruent to angle 2 and angle 2 is congruent to angle 3, then angle 1 is congruent to angle 3

Midpoint Therorem

If M is the midpoint of line segment AB, then line segment AM and line segment MB = ½(line segment AB)

Reflective Property of Segment Congruence

Line segment 1 is congruent to line segment 1

Symmetrical Property of Segment Congruence

If line segment 1 is congruent to line segment 2, then line segment 2 is congruent to line segment 1

Transitive Property of Segment Congruence

If line segment 1 is congruent to line segment 2 and line segment two is congruent to line segment 3, then line segment 1 is congruent to line segment 3

Plane-Line Postulate

If two points lie in the a plane, then the line containing them lies in a plane

Plane Intersection Postulate

If two planes intersect, their intersection is a line

Plane-Point Postulate

A plane contains at least three noncolinear points

Three Point Postulate

Through exactly three noncolinear points, there exists a plane

Line Intersection Postulate

If two lines intersect, then their intersection is exactly one point

Line-Point Postulate

A line contains two points

Two Point Postulate

Through any two points there exists a line

Distributive Property

a(b+c) = ab + ac

Corollary

A statement easily proved from a theorem

Theorem

A statement that is proved by using deductive reasoning from hypothesis to conclusion

Proof

A logical argument showing that a given statement is true. Each statement used in a proof must be supported by valid reasoning.

Def. of Linear Pair

A pair of adjacent angles with noncommon sides that are opposite rays

Linear Pair Postulate

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

Def. of an Angle Bisector

A ray that divides an angle into two congruent angles

Angle Addition Postulate

If point K lies in the interior of angle ABC, then angle ABK + angle KBC = the measure of angle ABC

Def. of Vertical Angles

Two nonadjacent angles that share a common vertrex and are formed by two intersecting lines

Def of. Adjacent Angles

Two angles that lie in the same plane, have a common vertex, have a common side, but do not have any common interior points

Def. of an Angle

A figure formed by two rays that have common endpoint. Measured in degrees or radians.

Def. of the Midpoint

Midway between two points. Divides a line segment into two congruent parts.

Def. of Congruent Segments

Line segments that are equal in length; have the same measure. 

Def. of Segment Bisector

A line segment, line, ray, or plane that divides a line segment into two congruent line segments

Segment Addition Postulate

If point B is between points A and C, the line segment AB + line segment BC = line segment AC