Geometry Proof Reasons: Theorems, Postulates, and Definitions

Vocabulary flashcards covering geometry proof reasons including properties of equality, congruence, segments, and angles.

Segment Addition

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

Definition of a Midpoint

If $P$ is the midpoint of $\overline{AB}$, then $\overline{AP} \cong \overline{PB}$.

Definition of a Segment Bisector

If a line, segment, ray, or plane is a segment bisector, then it intersects the segment at its midpoint.

Definition of Congruent Segments

If $\overline{DE} \cong \overline{FG}$, then $DE = FG$.

Midpoint Theorem

If $M$ is the midpoint of $\overline{AB}$, then $AM = \frac{1}{2}AB$ and $MB = \frac{1}{2}AB$.

Angle Addition

If point $B$ lies in the interior of $\angle ADC$, then $m\angle ADB + m\angle BDC = m\angle ADC$.

Angle Addition Part II

If $\angle AOC$ is a straight angle and $B$ is any point not on $\overleftrightarrow{AC}$, then $m\angle AOB + m\angle BOC = 180$.

Definition of an Angle Bisector

If $\vec{OC}$ bisects $\angle AOB$, then $\angle AOC \cong \angle COB$.

Definition of Congruent Angles

If $\angle R \cong \angle S$, then $m\angle R = m\angle S$.

Definition of Right Angles

If $\angle 1$ is a right angle, then $m\angle 1 = 90$.

Angle Bisector Theorem

If $\vec{BX}$ is the bisector of $\angle ABC$, then $m\angle ABX = \frac{1}{2}m\angle ABC$ and $m\angle XBC = \frac{1}{2}m\angle ABC$.

Addition of Equality

If $a = b$ and $c = d$, then $a + c = b + d$.

Subtraction of Equality

If $a = b$ and $c = d$, then $a - c = b - d$.

Multiplication of Equality

If $a = b$, then $ac = bc$.

Division of Equality

If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Distributive Property

The property stating $a(b + c) = ab + ac$.

Reflexive of Equality

The property stating $a = a$.

Reflexive of Congruence

The property stating $\overline{DE} \cong \overline{DE}$ and $\angle A \cong \angle A$.

Symmetric of Equality

If $a = b$, then $b = a$.

Symmetric of Congruence

If $\overline{DE} \cong \overline{FG}$, then $\overline{FG} \cong \overline{DE}$; and if $\angle A \cong \angle B$, then $\angle B \cong \angle A$.

Substitution of Equality

If $a = b$ and $x + a = y$, then $x + b = y$.

Transitive of Equality

If $a = b$ and $b = c$, then $a = c$.

Transitive of Congruence

If $\overline{DE} \cong \overline{FG}$ and $\overline{FG} \cong \overline{JK}$, then $\overline{DE} \cong \overline{JK}$; and if $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Reflexive Property of Equality

States that any quantity is equal to itself: $a = a$.

Vertical angle theorem

If 2 angles vertical then they are congruent

Definition of complementary angles

If 2 angles measures have the sum of 90 degrees then they are complementary angles

Definition of supplementary angles

If 2 angles measures have the sum of 180 degrees then they are supplementary angles

Linear pair theorem

If 2 angles form a linear pair then they are supplementary

Definition of perpendicular lines

Perpendicular lines form right angles

If 2 lines are perpendicular

Then they form congruent adjacent angles

If 2 lines form congruent adjacent angles

Then the lines are perpendicular

Perpendicular pair theorem

If the exterior sides of 2 adjacent angles are perpendicular then the angles are complementary

Definition of isosceles triangle

A triangle with 2 congruent sides is an isosceles triangle

Definition of a right triangle

A triangle with one right angle is a right triangle

If the angles are angles of a triangle

Then their sum is 180 degrees

If 2 angles of one triangle are congruent to 2 angles of another triangle

Then the third angles are congruent

If 2 angles are the acute angles of a triangle

Then the angles are complementary

If the angle is the exterior angles of a triangle

Then its measure equals the sum of the remote interior angles

If 2 lines cur by a transversal and corresponding angles are congruent

Then the lines are parallel

Iif 2 lines are cut by a transversal and alternate interior angles are congurent

Then the lines parallel

If 2 lines are out by a transversal and same side interior angles are supplementary

Then the lines parallel

If 2 lines in a plane are perpendicular to the sane line

Then the lines parallel

If 2 lines are parallel to the same line

Then the lines are parallel

If parallel lines are cut by a transversal

Then corresponding angles are congruent

If 2 parallel lines are cut by a transversal

Then alternate interior angles are congruent

If 2 parallel lines are cut by a transversal

Then same side interior angles are supplementary

If a transversal is perpendicualr to one of 2 parallel lines

Then it is perpendicular to the other one also

Congruent supplements theorem

If 2 angles are supplementary to the same angles(or 2 congruent angles) then the 2 angles are congruent.

Congruent complements theorem

If 2 angles are complementary to the same angles (or 2 congruent angles) then the 2 angles are congruent

Right angles congruence theorem

If 2 angles are right angles then they are congruent