Common Reasons for proofs

Definition of perpendicular — If two lines (or segments) are perpendicular, they form right angles.

All right angles are congruent — Any two right angles are equal in measure.

Definition of midpoint / segment bisector — A midpoint divides a segment into two congruent segments.

Definition of angle bisector — An angle bisector divides an angle into two congruent angles.

Vertical Angles Theorem — Vertical angles (opposite angles formed by two intersecting lines) are congruent.

Reflexive Property — Any segment or angle is congruent to itself (used for shared sides or angles).

SSS (Side

Side

SAS (Side

Angle

ASA (Angle

Side

AAS (Angle

Angle

CPCTC — Corresponding Parts of Congruent Triangles are Congruent (use after proving triangles congruent).

Corresponding Angles Postulate / Converse — If two parallel lines are cut by a transversal, corresponding angles are congruent. Converse: If corresponding angles are congruent, the lines are parallel.

Alternate Interior Angles Theorem / Converse — If two parallel lines are cut by a transversal, alternate interior angles are congruent. Converse: If alternate interior angles are congruent, the lines are parallel.

Linear Pair / Supplementary Angles — If two angles form a linear pair, they are supplementary (sum to 180°). If two angles are supplementary and one equals a known angle, use that to find the other.

Transversal definition — A line that intersects two (or more) other lines at distinct points; used to create corresponding/alternate interior/exterior angles.

Exterior Angle Theorem — An exterior angle of a triangle equals the sum of the two remote interior angles.