Geometry Basic Postulates, theorems, and definitions

Ruler Postulate

The points on any line or line segment can be put into one-to-one correspondence with real numbers.

Segment Addition Postulate

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

Reflexive Property of Segment Congruence

AB ≅ AB (a segment is congruent to itself).

Symmetric Property of Segment Congruence

If AB ≅ CD, then CD ≅ AB.

Transitive Property of Segment Congruence

If AB ≅ CD and CD ≅ EF, then AB ≅ EF.

Midpoint Theorem / Definition of a Midpoint

If M is the midpoint of AB, then AM ≅ MB.

Definition of Congruent Segments

If AB ≅ CD, then AB = CD (and vice versa).

Protractor Postulate

Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.

Angle Addition Postulate

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

Supplement Theorem

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

Complement Theorem

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

Reflexive Property of Angle Congruence

∠1 ≅ ∠1

Symmetric Property of Angle Congruence

If ∠1 ≅ ∠2, then ∠2 ≅ ∠1.

Transitive Property of Angle Congruence

If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3.

Congruent Supplements Theorem

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

Congruent Complements Theorem

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

Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.

Right Angles Theorem (1)

Perpendicular lines intersect to form four right angles.

Right Angles Theorem (2)

All right angles are congruent.

Right Angles Theorem (3)

Perpendicular lines form congruent adjacent angles.

Right Angles Theorem (4)

If two angles are congruent and supplementary, then each angle is a right angle.

Right Angles Theorem (5)

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

Definition of Congruent Angles

If ∠ABC ≅ ∠DEF, then m∠ABC = m∠DEF (and vice versa).

Definition of Angle Bisector

A ray that divides an angle into two congruent angles.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary (their measures add to 180°).

Transitive Property (of equality)

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

Reflexive Property (of equality)

a = a

Symmetric Property (of equality)

If a = b, then b = a.

Substitution Property

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

Definition of Congruence

Figures are congruent if and only if they have the same measure (∠A ≅ ∠B ↔ m∠A = m∠B; AB ≅ CD ↔ AB = CD).

Definition of Complementary

Two angles are complementary if and only if their measures add up to 90°.

Definition of Supplementary

Two angles are supplementary if and only if their measures add up to 180°.

Definition of Perpendicular

Two lines are perpendicular if and only if they intersect to form a right angle (90°).

Definition of Bisector

A segment, ray, or line that divides a segment or angle into two congruent parts.

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.

Distributive Property

a(b + c) = ab + ac

Angle Addition Postulate (reason to use it)

Use when a point or ray lies inside an angle and splits it into two smaller angles.

Segment Addition Postulate (reason to use it)

Use when a point lies between the endpoints of a segment and splits it into two smaller segments.

Linear Pair

Two adjacent angles whose noncommon sides are opposite rays. They are supplementary.

Vertical Angles

Two nonadjacent angles formed by two intersecting lines. They are congruent.

Adjacent Angles

Two angles that share a common vertex and a common side but have no interior points in common.

Midpoint

The point that divides a segment into two congruent segments.

Right Angle

An angle that measures exactly 90°.

Given

Information stated in the problem. Use it as a reason for any statement that was handed to you.

Statement: "∠1 and ∠2 are a linear pair." What is the reason?

Definition of linear pair (then use Linear Pair Postulate to get supplementary).

Statement: "m∠1 + m∠2 = 180°" from a linear pair. What is the reason?

Linear Pair Postulate / Supplement Theorem

Statement: "AM ≅ MB" from "M is the midpoint of AB." What is the reason?

Definition of midpoint (Midpoint Theorem)

Statement: "m∠1 = m∠2" from "∠1 ≅ ∠2." What is the reason?

Definition of congruent angles

Statement: "AB = CD" from "AB ≅ CD." What is the reason?

Definition of congruent segments

Statement: "∠1 ≅ ∠3" from "∠1 and ∠3 are vertical angles." What is the reason?

Vertical Angles Theorem

Statement: "∠1 ≅ ∠2" from "∠1 and ∠2 are supplements of ∠3." What is the reason?

Congruent Supplements Theorem

Statement: "∠1 ≅ ∠2" from "∠1 and ∠2 are complements of ∠3." What is the reason?

Congruent Complements Theorem

Statement: "∠1 is a right angle" from "m∠1 = 90°." What is the reason?

Definition of a right angle

Statement: "∠ABD ≅ ∠DBC" from "BD bisects ∠ABC." What is the reason?

Definition of angle bisector

Statement: "m∠1 + m∠2 = 90°" from "∠1 and ∠2 are complementary." What is the reason?

Definition of complementary angles

Statement: "AB ⊥ CD" gives you what?

Four right angles (Right Angles Theorem), and the definition of perpendicular gives a 90° angle.

Statement: "x + 5 = 12, so x = 7." What is the reason?

Subtraction Property of Equality

Statement: "2x = 10, so x = 5." What is the reason?

Division Property of Equality

Statement: "AB = AB" What is the reason?

Reflexive Property

Statement: "m∠1 = 50° and m∠1 + m∠2 = 90°, so 50° + m∠2 = 90°." What is the reason?

Substitution Property