Properties, Definitions, Theorems for Proofs

Reflexive Property of Equality

any value is equal to itself

a = a

AB = AB

m∠A = m∠A

Symmetric Property of Equality

If one value is equal to another, then the second value is equal to the first

If a = b, then b = a.

If AB = 4, then 4 = AB.

If m∠A = 3x, then 3x = m∠A.

Transitive Property of Equality

If one value is equal to a second value, and the second value is equal to a third value, then the first value is equal to the third value.
If a = b and b = c, then a = c.

If AC = AB + BC and AB + BC = 14, then AC = 14.

If m∠A = m∠B and m∠B = 50°, then m∠A = 50°.

Reflexive Property of Congruence

Every figure is congruent to itself.

segment BC ≅ segment BC

∠A ≅ ∠A

Symmetric Property of Congruence

If one figure is congruent to another, then the second figure is congruent to the first.

If segment BC ≅ segment AB, then segment AB ≅

If ∠A ≅ ∠B, then ∠B ≅ ∠A.

Transitive Property of Congruence

If one figure is congruent to a second value, and the second figure is congruent to a third figure, then the first figure is congruent to the third value.

If AC ≅ AB and AB ≅ BC, then AC ≅ BC.

If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

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 ac = bc

Division Property of Equality

If a = b, then a/c = b/c, as long as c ≠ 0.

Substitution (property of equality)

If a = b, then a can be substituted for b in any expression.

Use this for combining like terms instead of Simplify

Segment Addition Postulate

If points A, B, and C are collinear and point B is between points A and C, then AB + BC = AC.

Angle Addition Postulate

If ray BD is inside angle ABC, then m∠ABD + m∠DBC = m∠ABC.

Parallel Postulate

Given a line and a point not on the line, there exists exactly 1 line through the given point parallel to the given line.

Definition of congruence

If AB = CD then segment AB ≅ segment CD.

works both ways to move from congruence to equality or back the other way

Definition of midpoint

A point that divides a segment into two equal parts, such that if M is the midpoint of segment AB, then AM = MB.

Definition of segment bisector

A line, ray, or segment that divides a segment into two equal parts, which means if line l is a segment bisector of segment AB, then AM = MB.

Definition of angle bisector

A line, ray, or segment that divides an angle into two equal parts, meaning if ray AC is an angle bisector of ∠BAD, then m∠BAC = m∠CAD.

Definition of complementary angles

Two angles whose measures have a sum of 90 degrees

m∠A + m∠B = 90^o

Definition of supplementary angles

Two angles whose measures have a sum of 180^o

m∠A + m∠B = 180^o

Definition of perpendicular

If figures are perpendicular, then they intersect to form right angles.

If figures intersect to form right angles, then they are perpendicular.

Linear Pair Theorem

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

Definition of linear pair

a pair of adjacent angles whose non-shared side are opposite rays

Definition of vertical angles

Two nonadjacent angles that are opposite each other when two lines intersect

Vertical Angles Theorem

Vertical angles are congruent.

Complement Theorem

If the non-common sides of two adjacent angles form a right angle, then the angles are complementary.

Congruent Supplements Theorem

Angles supplementary to the same angle (or congruent angles) are congruent.

m∠A + m∠B = 180^o and m∠A + m∠C = 180^o, then ∠B ≅ ∠C

Congruent Complements Theorem

Angles complementary to the same angle (or congruent angles) are congruent.

m∠A + m∠B = 90^o and m∠A + m∠C = 90^o, then ∠B ≅ ∠C