Geometry two-column proofs

Given

Statements provided directly in the problem description or diagram (e.g., AB ≅ CD, or m∠1 = 50°).

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 · c = b · c.

Division Property of Equality

If a = b and c ≠ 0, then a / c = b / c.

Substitution Property of Equality

If a = b, then a can replace b in any expression.

Distributive Property

a(b + c) = ab + ac.

Reflexive Property

Any quantity is equal/congruent to itself (a = a or AB ≅ AB).

Symmetric Property

If a = b, then b = a (or if AB ≅ CD, then CD ≅ AB).

Transitive Property

If a = b and b = c, then a = c (or if ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3).

Definition of Segment Congruence

AB ≅ CD means AB = CD.

Definition of Angle Congruence

∠A ≅ ∠B means m∠A = m∠B.

Definition of Midpoint

A point M on AB such that AM = MB (or AM ≅ MB).

Definition of Segment Bisector

A line/segment that intersects a segment at its midpoint.

Definition of Angle Bisector

A ray that divides an angle into two congruent adjacent angles.

Definition of Perpendicular (⊥)

Lines that intersect to form 90° right angles.

Definition of Perpendicular Bisector

A line that is perpendicular to a segment and passes through its midpoint.

Definition of Supplementary Angles

Two angles whose measures add up to 180°.

Definition of Complementary Angles

Two angles whose measures add up to 90°.

Definition of Right Angle

An angle whose measure is 90°.

Segment Addition Postulate

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

Angle Addition Postulate

If D lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Linear Pair Postulate

If two angles form a linear pair, they are supplementary (m∠1 + m∠2 = 180°).

Vertical Angles Theorem

Vertical angles are congruent (∠1 ≅ ∠2).

Right Angle Congruence Theorem

All right angles are congruent.

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or congruent angles), they are congruent.

Congruent Complements Theorem

If two angles are complementary to the same angle (or congruent angles), they are congrue