Angle Relationships, Properties of Equality, and Angle Proofs

Vocabulary flashcards covering geometric angle relationships, algebraic properties of equality, definitions, postulates, and theorems used in geometric proofs.

Adjacent Angles

Two angles that are next to each other and share a common side.

Vertical Angles

Two angles across from each other on intersecting lines; they are always congruent.

Linear Pair

Two angles that are adjacent and supplementary, forming a straight line where $m\angle 1 + m\angle 2 = 180^\circ$.

Complementary Angles

Any two angles whose measures have a sum of $90^\circ$.

Supplementary Angles

Any two angles whose measures have a sum of $180^\circ$.

Addition Property of Equality

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

Subtraction Property of Equality

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

Multiplication Property of Equality

If $a = b$, then $a \cdot c = b \cdot c$.

Division Property of Equality

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

Distributive Property

If $a(b + c)$, then $a(b + c) = a \cdot b + a \cdot c$.

Substitution Property of Equality

If $a = b$, then $a$ may be replaced (substituted) by $b$ in any expression or equation.

Reflexive Property of Equality

For any real number $a$, $a = a$ (a value will always equal itself).

Symmetric Property of Equality

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

Transitive Property of Equality

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

Definition of Congruence

The measures of two angles are equal if and only if the angles are congruent ($m\angle A = m\angle B \leftrightarrow \angle A \cong \angle B$).

Definition of a Right Angle

An angle measures $90^\circ$ if and only if it is a right angle ($m\angle A = 90^\circ \leftrightarrow \angle A\text{ is a right angle}$).

Definition of an Angle Bisector

A line or ray that divides an angle into two equal parts.

Definition of Perpendicular

Perpendicular lines form right angles.

Angle Addition Postulate

If point $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.

Vertical Angles Theorem

If two angles are vertical, then they are congruent.

Complement Theorem

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

Linear Pair Theorem (Supplement Theorem)

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

Congruent Complements Theorem

If two angles are complementary to the same angle, then they are congruent (if $\angle A$ is complementary to $\angle B$ and $\angle C$ is complementary to $\angle B$, then $\angle A \cong \angle C$).

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent (if $\angle A$ is supplementary to $\angle B$ and $\angle C$ is supplementary to $\angle B$, then $\angle A \cong \angle C$).