| A | B |
|---|
| Figure Reflection Theorem | If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points. |
| Corresponding Parts of Congruent Figures (CPCF) Theorem | If two figures are congruent, then any pair of corresponding parts is congruent. |
| A-B-C-D Theorem | Every isometry preserves Angle measure, Betweenness, Collinearity (lines), and Distance (lengths of segments). |
| Reflexive Property of Congruence | F is congruent to F |
| Symmetric Property of Congruence | If F is congruent to G, then G is congruent to F. |
| Transitive Property of Congruence | If F is congruent to G and G is congruent to H, then F is congruent to 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. |
| Alternate Interior Angles Theorem | If two parallel lines are cut by a transversal, then the alternate interior 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. |
| Alternate Exterior Angles Theorem | If two lines are cut by a transversal and form congruent alternate exterior angles, then the lines are parallel. |
| Uniqueness of Parallels Theorem (Playfair's Parallel Postulate) | 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 measures of the angles of a convex quadrilateral is 360º. |
| Polygon-Sum Theorem | The sum of the measures of the angles of a convex n-gon is (n - 2)•180. |
| Linear Pair Theorem | If two angles form a linear pair, then they are supplementary. |
| Vertical Angles Theorem | If two angles are vertical angles, then they have equal measures. |
| Parallel Lines and Slopes Theorem | Two nonvertical lines are parallel if and only if they have the same slope. |
| Transitivity of Parallelism Theorem | In a plane, if m // n and n // r, then m // r. |
| Two Perpendiculars Theorem | If two coplanar lines m and n 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. |
| Reflexive Property of Equality | a = a |
| Symmetric Property of Equality | If a = b, then b = a. |
| Transitive Property of Equality | If a = b and b = c, then a = c. |
| Angle Addition Property | If ray VC is in the interior of angle AVB, then the measure of angle AVC + CVB = AVB. |
| Unique Line Assumption (Point-Line-Plane Postulate) | Through any two points, there is exactly one line. |
| Substitution Property | If a = b, then a may be substituted for b in any expression. |
| Corresponding Angles Postulate | If two coplanar lines are cut by a transversal and corresponding angles are equal, then the lines are parallel. |
| Angle Bisector | The ray with points in the interior of an angle that forms two angles of equal measure with the sides of the angle. |
| Auxiliary Figure | A figure that is added to a given figure, often to aid in completing proofs. |
| Circle | The set of points in a plane at a certain distance (its radius) from a certain point (its center). |
| Congruent Figures | A figure F is congruent to a figure G if and only if G is the image of F under a reflection or a composite of reflections. |
| Midpoint | The point on a segment equidistant from the segment's endpoints. |
| Reflection | A transformation in which each point is mapped onto its reflection image over a line or plane. |
| Perpendicular Bisector | The line containing the midpoint of the segment and perpendicular to the segment. |
| Flip-Flop Theorem | If F and G are points/figures and r(F) = G, then r(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 lie containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle. |
| Isosceles Triangle Coincidence Theorem | In an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, and the median to the base determine the same line. |
| 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 (and 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 Thorem | 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. |
| Definition of Parallelogram | A quadrilateral with both pairs of opposite sides parallel. |
| Definition of Rhombus | A quadrilateral with four sides of equal length. |
| Definition of Rectangle | A quadrilateral with four right angles. |
| Definition of Square | A quadrilateral with four equal sides and four right angles. |
| Definition of Kite | A quadrilateral with two distinct pairs of consecutive sides of the same length. |
| Definition of Trapezoid | A quadrilateral with AT LEAST one pair of parallel sides. |
| Definition of Isosceles Trapezoid | A trapezoid with a pair of base angles equal in measure. |
| Reflection-Symmetric Figure | A figure F for which there is a reflection r such that r(F) = F. |
| Definition of Isosceles Triangle | A triangle with at least two sides equal in length. |
| 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 and n-fold rotation symmetry. |
| Definition of Regular Polygon | a convex polygon whose angles are all congruent and whose sides are all congruent. |
| SSS Congruence Theorem | If, in two triangles, three sides of one are congruent to three sides of the other, then the triangles are congruent. |
| SAS Congruence Theorem | If, in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, then the triangles are congruent. |
| ASA Congruence Theorem | If, in two triangles, two angles and the included side of one are congruent to two angles and the included side of the other, then the two triangles are congruent. |
| AAS Congruence Theorem | If, in two triangles, two angles and a non-included side of one are congruent respectively to two angles and the CORRESPONDING non-included side of the other, then the triangles are congruent. |
| Isosceles Triangle Base Angles Converse Theorem | If two angles of a triangle are congruent, then the sides opposite them are congruent. |
| HL Congruence Theorem | If, in two right triangles, the hypotenuse and a leg of one are congruent to the hypotenuse and a leg of the other, then the triangles are congruent. |
| SsA Congruence Theorem | If two sides and the angle opposite the longer of the two sides in one triangle are congruent, respectively, to two sides and the corresponding angle in another triangle, then the triangles are congruent. |
| Properties of a Parallelogram Theorem | In any parallelogram, (a) opposite sides are congruent; (b) opposite angles are congruent; (c) the diagonals intersect at their midpoints |
| Parallelogram Symmetry Theorem | Ever parallelogram has 2-fold rotation symmetry about the intersection of its diagonals |
| Sufficient Conditions for a Parallelogram Theorem | If, in a quadrilateral, (a) one pair of sides is both parallel and congruent, or (b) both pairs of opposite sides are congruent, or (c) the diagonals bisect each other, or (d) both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram. |
| Exterior Angle Theorem | In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. |
| Exterior Angle Inequality | In a triangle, the measure of an exterior angle is greater than the measure of the interior angle at each of the other two vertices. |
| Unequal Sides Theorem | If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. |
| Unequal Angles Theorem | If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. |
