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 $\angle A \cong \angle C$.