Geometry Postulates and Theorems (ch.

1-3)
Unique Line Postulate: Through any two points, there is exactly one line. Line Intersection Theorem: Two different lines intersect in at most one point. Distance Formula: If two points on a line have coordinates x and y, the distance between them is |x - y|. Distance Addition Property: If B is on

AC , then AB+BC=AC

Triangle Inequality Postulate: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Angle Addition Property: If (except for point V) is in the interior of mA V C+ mC V B= mA V B.

VC

AVB, then

LPT Linear Pair Theorem: If two angles form a linear pair, then they are supplementary. VAT Vertical Angles Theorem: If two angles are vertical angles, then they have equal measures. Postulates of Equality and Operations: For any real numbers a, b and c: a. RPE Reflexive property of equality: a=a. b. SPE Symmetric property of equality: If a=b, then b=a. c. TPE Transitive property of equality: If a=b, and b=c, then a=c. d. APE Addition (Subtraction) property of equality: If a=b, then a+c=b+c. e. MPE Multiplication (Division) property of equality: If a=b, then ac=bc. Postulates of Inequality and Operations: For any real numbers a, b and c: a. TPI Transitive property of inequality: If a < b, and b < c, then a < c. b. API Addition property of inequality: If a < b, then a+c < b+c. c. MPI Multiplication property of inequality: If a < b and c > 0, then ac < bc. (c is positive) If a < b and c < 0, then ac > bc. (c is negative) Equation to Inequality Property (Part to Whole): If a and b are positive numbers and a+b=c, then c > a and c > b. Substitution property: If a=b, then a may be substituted for b in any expression. CAP Corresponding Angles Postulate: Corresponding angles have the measure if and only if the lines are parallel. Parallel Lines and Slope Theorem: Two nonvertical lines are parallel if and only if they have the same slope. Transitive Property of Parallel Lines: In a plane, if line l is parallel to line m and line m is parallel to line n, then line l is parallel to line n. Two Perpendiculars Theorem: If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other. Perpendicular to Parallels Theorem: In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. Perpendicular Lines and Slopes Theorem: Two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (they are opposite reciprocals).

Geometry Postulates and Theorems (ch. 4-6)

Figure Reflection Theorem: If certain points determine a figure, then its reflection image is the corresponding figure determined by the reflection images of those points. CPCF Corresponding Parts of Congruent Figures Theorem: If two figures (triangles, quads, etc.) are congruent, then any pair of corresponding parts (segments and angles) is congruent. ABCD Theorem: Every Isometry (reflection, translation, rotation or glide reflection) preserves angle measure, betweenness, collinearity (lines), and distance (lengths of segments). Postulates of Congruence: For any figures F, G and H: a. RPC Reflexive property of congruence: b. SPC Symmetric property of congruence: If c. TPC Transitive property of congruence: If

F @F . F @G , then G @F . F @G and G @H , then F @H .

Segment Congruence Theorem: Two segments are congruent if and only if they have the same length. Angle Congruence Theorem: Two angles are congruent if and only if they have the same measure. AIA Alternate Interior Angles Theorem: Two lines cut by a transversal are parallel if and only if alternate interior angles are congruent. AEA Alternate Exterior Angles Theorem: Two lines cut by a transversal are parallel if and only if alternate exterior angles are congruent. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Uniqueness of Parallels Theorem: Through a point not on a line, there is exactly one line parallel to the given line. Triangle-Sum Theorem: The sum of the measures of the angles of a triangle is 180o. Quadrilateral-Sum Theorem: The sum of the measure of the angles of a convex quadrilateral is 360. Polygon-Sum Theorem: The sum of the measures of the angles of a convex n-gon is (n-2)*180. Flip-Flop Theorem: If F and G are points or figures and rm(F) = G, then rm(G) = F. Segment Symmetry Theorem: Every segment has exactly two symmetry lines: its perpendicular bisector, and the line containing the segment. Side-Switching Theorem: If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle. Angle Symmetry Theorem: The line containing the bisector of an angle is a symmetry line of the angle. Circle Symmetry Theorem: A circle is reflection-symmetric to any line through its center. Symmetric Figures Theorem: If a figure is symmetric, then any pair of corresponding parts under the symmetry is congruent. Isosceles Triangle Symmetry Theorem: The line containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle. Isosceles Triangle Base Angles Theorem: If a triangle has two congruent sides, then the angles opposite them are congruent.

Equilateral Triangle Symmetry Theorem: Every equilateral triangle has three symmetry lines, which are bisectors of its angles or perpendicular bisectors of its sides. Equilateral Triangle Angle Theorem: If a triangle is equilateral then it is equiangular. Kite Symmetry Theorem: The line containing the ends of a kite is a symmetry line for the kite. Kite Diagonal Theorem: The symmetry diagonal of a kite is the perpendicular bisector of the other diagonal and bisects the two angles at the ends of the kite. Rhombus Diagonal Theorem: Each diagonal of a rhombus is the perpendicular bisector of the other diagonal. Trapezoid Angle Theorem: In a trapezoid, consecutive angles between a pair of parallel sides are supplementary. Isosceles Trapezoid Symmetry Theorem: The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and a symmetry line for the trapezoid. Isosceles Trapezoid Theorem: In an isosceles trapezoid, the non-base sides are congruent. Rectangle Symmetry Theorem: The perpendicular bisectors of the sides of a rectangle are symmetry lines for the rectangle. Center of a Regular Polygon Theorem: In any regular polygon there is a point (its center) which is equidistant from all of its vertices. Regular Polygon Symmetry Theorem: Every regular n-gon possesses n symmetry lines, which are the perpendicular bisectors of each of its sides and the bisectors of each of its angles. Every regular n-gon possesses n-fold rotational symmetry.