Essential Geometry Postulates and Theorems Reference

Fundamental Geometry Postulates

• 1.1 Ruler Postulate: Points on a line can be matched one-to-one with real numbers. The real number corresponding to a point is its coordinate.
• 1.2 Segment Addition Postulate: If B is between A and C, then AB + BC = AC.
• 1.3 Protractor Postulate: Consider ray OB and a point A on one side of OB. The rays of the form OA can be matched one-to-one with real numbers from 0 to 180.
• 1.4 Angle Addition Postulate: If P is in the interior of ∠RST, then the measure of ∠RSP + the measure of ∠PST = the measure of ∠RST.
• 2.1 Two Point Postulate: Through any two points, there exists exactly one line.
• 2.2 Line-Point Postulate: A line contains at least two points.
• 2.3 Line Intersection Postulate: If two lines intersect, their intersection is exactly one point.
• 2.4 Three Point Postulate: Through any three noncollinear points, there exists exactly one plane.
• 2.5 Plane-Point Postulate: A plane contains at least three noncollinear points.
• 2.6 Plane-Line Postulate: If two points lie in a plane, the line containing them lies in the plane.
• 2.7 Plane Intersection Postulate: If two planes intersect, their intersection is a line.
• 2.8 Linear Pair Postulate: If two angles form a linear pair, they are supplementary.
• 3.1 Parallel Postulate: Given a line and a point not on it, there is exactly one line through the point parallel to the given line.
• 3.2 Perpendicular Postulate: Given a line and a point not on it, there is exactly one line through the point perpendicular to the given line.
• 4.2 Reflection Postulate: A reflection is a rigid motion.
• 4.3 Rotation Postulate: A rotation is a rigid motion.
• 10.1 Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Logic and Algebraic Properties

• Law of Detachment: If the hypothesis of a true conditional statement is true, the conclusion is also true.
• Law of Syllogism: If p → q and q → r are true, then p → r is true.
• 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 (c ≠ 0).
• Division Property of Equality: If a = b, then a/c = b/c (c ≠ 0).
• Substitution Property: If a = b, then a can be substituted for b in any equation or expression.
• Distributive Property: a(b+c) = ab + ac or a(b-c) = ab - ac.
• Associative Property: (a+b) + c = a + (b+c) or (a•b) • c = a • (b•c).
• Commutative Property: a + b = b + a or a • b = b • a.
• Zero Property: If a • b = 0, then a = 0 or b = 0.
• Symmetric Property: If a = b, then b = a.
• Reflexive Property: a = a, AB = AB, and m∠A = m∠A.
• Transitive Property: If a = b and b = c, then a = c.

Theorems on Angles and Lines

• 2.6 Vertical Angles Congruence Theorem: Vertical angles are congruent.
• 3.1 Corresponding Angles Theorem: If two parallel lines are cut by a transversal, corresponding angles are congruent.
• 3.2 Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
• 3.3 Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, alternate exterior angles are congruent.
• 3.4 Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, consecutive interior angles are supplementary.
• 3.5–3.8 Converses: If angles are congruent/supplementary as described above, the lines are parallel.
• 3.9 Transitive Property of Parallel Lines: If two lines are parallel to the same line, they are parallel to each other.
• 3.10 Linear Pair Perpendicular Theorem: If two lines intersect to form a linear pair of congruent angles, the lines are perpendicular.
• 3.11 Perpendicular Transversal Theorem: In a plane, if a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other.
• 3.12 Lines Perpendicular to a Transversal: In a plane, if two lines are perpendicular to the same line, they are parallel.
• 3.13 Slopes of Parallel Lines: Nonvertical lines are parallel if and only if they have the same slope (m1 = m2).
• 3.14 Slopes of Perpendicular Lines: Nonvertical lines are perpendicular if and only if the product of their slopes is -1 (m1 • m2 = -1).

Triangle Theorems and Properties

• 5.1 Classifying Triangles: Scalene, isosceles, equilateral, acute, right, obtuse, and equiangular.
• 5.1 Triangle Sum Theorem: Interior angles sum to 180°.
• 5.2 Exterior Angles Theorem: Exterior angle equals the sum of the two nonadjacent interior angles.
• 5.3 Properties of Congruence: Reflexive, symmetric, and transitive.
• 5.4 Third Angles Theorem: If two angles of one triangle are congruent to two angles of another, the third angles are congruent.
• 5.5–5.11 Congruence Theorems: SAS, SSS, HL, ASA, AAS.
• 5.6–5.7 Base Angles Theorem: If two sides are congruent, opposite angles are congruent (and vice versa).
• 5.7 CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
• 6.1–6.4 Bisector Theorems: Perpendicular and angle bisector properties.
• 6.5–6.7 Triangle Centers: Circumcenter, incenter, and centroid properties.
• 6.8 Triangle Midsegment Theorem: Parallel to the third side and half its length.
• 6.11 Triangle Inequality Theorem: Sum of any two sides > third side.
• 6.12–6.13 Hinge Theorem: Relates side lengths to included angles.

Polygons and Quadrilaterals

• 7.1 Polygon Interior Angles: Sum = (n - 2) • 180°.
• 7.2 Polygon Exterior Angles: Sum = 360°.
• 7.3–7.6 Parallelogram Theorems: Opposite sides/angles congruent, consecutive angles supplementary, diagonals bisect each other.
• 7.7–7.10 Parallelogram Converses: Methods to prove a quadrilateral is a parallelogram.
• 7.11–7.13 Special Quadrilaterals: Properties of rhombi, rectangles, and squares regarding diagonals and angles.