Reasons for Proofs

Reflexive Property of Equality

A mathematical principle stating that any quantity is equal to itself, expressed as $a = a$ for any number or mathematical expression.

Transitive Property of Equality

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

Substitution Property of Equality

A principle stating that if two quantities are equal, one can be substituted for the other in any expression or equation, expressed as if $a = b$, then any occurrence of $a$ can be replaced with $b$.

Symmetric Property of Congruence

If one geometric figure is congruent to another, then the second figure is congruent to the first, expressed as if $A \text{ is congruent to } B$, then $B \text{ is congruent to } A$.

Definition of Congruent

Two figures are congruent if they have the same shape and size, allowing for rotation, reflection, or translation.

Midpoint Theorem

The Midpoint Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is half of that side.

Congruent Supplements Theorem

The Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.

Congruent Compliments Theorem

The Congruent Complements Theorem states that if two angles are each complemented by the same angle, then the two angles are congruent.

Perpendicular Lines Theorem

If two lines are perpendicular, they form congruent, adjacent angles.

Congruent Adjacent Angles Theorem

If two lines for congruent, adjacent angles, then the lines are perpendicular