geo proofs and properties.

Addition Property of Equality

If two things are equal, you can add the same value to both sides and they’ll still be equal.

Subtraction Property of Equality

If two things are equal, you can subtract the same value from both sides and they’ll stay equal.

Multiplication Property of Equality

If two things are equal, you can multiply both sides by the same number and they’ll still be equal.

Division Property of Equality

If two things are equal and you divide both sides by the same number (not zero), they’ll still be equal.

Distributive Property

You can multiply a number across a group inside parentheses, like a(b + c) = ab + ac.

Substitution Property

If two things are equal, you can replace one with the other in an equation or expression.

Reflexive Property of Equality

Any number or segment is equal to itself. (a = a)

Symmetric Property of Equality

If one thing equals another, you can flip them around. (If a = b, then b = a)

Transitive Property of Equality

If a = b and b = c, then a = c — like a chain of equal values.

Reflexive Property of Congruence

Any segment or angle is congruent to itself. It’s the same on both sides.

Symmetric Property of Congruence

If one segment is congruent to another, you can switch them. (If AB ≅ CD, then CD ≅ AB)

Transitive Property of Congruence

If AB ≅ CD and CD ≅ EF, then AB ≅ EF — congruence carries through like a chain.

Definition of Congruence

Segments or angles are congruent if they have the same length or measure — congruent means “equal in size.”

Definition of Midpoint

A midpoint splits a segment into two equal parts — both halves are congruent.

Segment Addition Postulate

If a point is between two other points, you can add the smaller segments to get the whole (AB + BC = AC).

Definition of Angle Congruence

Angles are congruent if their measures are the same — they open the same amount.

Definition of Angle Bisector

A ray or line that cuts an angle exactly in half, making two equal angles.

Definition of Complementary Angles

Two angles that add up to 90° — like a corner.

Definition of Supplementary Angles

Two angles that add up to 180° — like a straight line.

Definition of Perpendicular

Perpendicular lines meet to form right (90°) angles.

Definition of a Right Angle

A right angle measures exactly 90° — like the corner of a square.

Angle Addition Postulate

If a point splits an angle, the two smaller angles add up to the big angle (m∠1 + m∠2 = m∠whole).

Vertical Angles Theorem

Vertical angles (the opposite ones when lines cross) are always congruent.

Complement Theorem

If two angles form a right angle together, they are complementary (add to 90°).

Supplement Theorem

If two angles form a straight line (linear pair), they are supplementary (add to 180°).

Congruent Complements Theorem

If two angles are complementary to the same angle, then they are congruent to each other.

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent to each other.