Ruler Postulate: The points on a line can be matched one to one with the REAL numbers. The REAL number that corresponds to a point is the COORDINATE of the point. The DISTANCE between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. Segment Addition Postulate: If B is BETWEEN A and C, then AB + BC = AC. If AB + BC = AC, then B is BETWEEN A and C. Midpoint Formula: The midpoint of a segment is the point halfway across the segment. The midpoint of a segment in the x-y coordinate plane is found by averaging the two x-values and averaging the two y-values. The midpoint of a segment with endpoints (X1, Y1 ) and ( X2, Y2 ) in the x-y coordinate plane is!!!!!,!!!!!!! Protractor Postulate: The rays of any angle can be matched one to one with the real numbers from 0 to 360. Angle Addition Postulate: If ABC is adjacent to DBC then m ABC + m DBC = m ABD. Distance Formula: The distance or length, AB, of AB is the total length across the segment. The distance (d) or length of a segment with endpoints (X1, Y1 ) and (X2, Y2 ) in the x-y coordinate plane is d = (X1 X2)! + (Y1 Y2)! Point, Point, Line, and Plane Postulates: Postulate 5: Through any two points there exists exactly ONE line. Postulate 6: A line contains at least TWO points. Postulate 7: If TWO lines intersect, then their intersection is exactly one POINT. Postulate 8: Through any three noncollinear points there exists exactly one PLANE. Postulate 9: A plane contains at least three NONCOLLINEAR points. Postulate 10: If TWO points lie in a plane, then the LINE containing them lies in the plane. Postulate 11: If TWO planes intersect, then their intersection is a LINE. Algebraic Properties of Equality Addition Property: If a = b, then a + c = b + c. Subtraction Property: If a = b, then a - c = b - c. Multiplication Property: If a = b, then a x c = b x c. Division Property: If a = b, then a / c = b / c. Substitution Property: If a = b, then a can be substituted for b in any equation or expression. The Distributive Property: a(b + c) = (b + c)a = ab + ac, where a, b, and c are real numbers. Reflexive Property: For any real number a, a = a. Symmetric Property: For any real numbers a and b, if a = b, then b = a. Transitive Property For any real numbers a, b, and c, if a = b and b = c, then a = c.
Theorem 2.1 Congruence of Segments: Segment congruence is reflexive, symmetric, and transitive. Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. Theorem 2.3: Right Angles Congruence Theorem All right angles are CONGRUENT. Theorem 2.5: Congruent Supplements Theorem If 1 and 2 are supplementary and 2 and 3 are supplementary, then 1 3. Postulate 12: Linear Pair Postulate If two angles form a linear pair, then the two angles are SUPPLEMENTARY. Theorem 3.1: Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of ALTERNATE interior angles are CONGRUENT. Theorem 3.2: Alternate Exterior Angles Theorem If TWO parallel lines are CUT by a transversal, then the PAIRS of alternate EXTERIOR angles are congruent. Theorem 3.3: Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of CONSECUTIVE interior angles are SUPPLEMENTARY. Postulate 16: Corresponding Angles Converse If two lines are cut by transversal so the pairs of corresponding angles are CONGRUENT, then the lines are PARALLEL. Theorem 2.6: Vertical Angles Congruence Theorem If 1 and 2 are vertical angles, then 1 2. Postulate 13: Parallel Postulate If there is a LINE and a POINT not on the line, then there is exactly ONE line through the point parallel ( ) to the given line. Postulate 14: Perpendicular Postulate If there is a LINE and a POINT not on the line, then there is exactly ONE line through the point perpendicular ( ) to the given line. Postulate 15: Corresponding Angles Postulate If two PARALLEL lines are cut by a TRANSVERSAL then the pairs of CORRESPONDING angles are congruent. Theorem 3.4: Alternate Interior Angles Converse If two lines are cut by transversal so the pairs of alternate interior angles are CONGRUENT, then the lines are PARALLEL Theorem 3.5: Alternate Exterior Angles Converse If two lines are cut by transversal so the pairs of alternate exterior angles are CONGRUENT, then the lines are PARALLEL. Theorem 3.6: Consecutive Interior Angles Converse If two lines are cut by transversal so the pairs of consecutive interior angles are SUPPLEMENTARY, then the lines are PARALLEL. Theorem 3.7: Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are PARALLEL to each other.
Theorem 3.8: If two lines intersect to form a linear pair of congruent angles, then the lines are PERPENDICULAR. Theorem 3.9: If two lines are perpendicular, then they intersect to form FOUR RIGHT ANGLES. Theorem 3.10: If two of two adjacent acute angles are perpendicular, then the angles are COMPLEMENTARY. Theorem 3.11: Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is PERPENDICULAR to the other line. Theorem 4.3: Third Angles Theorem IF TWO ANGLES OF ONE TRIANGLE ARE CONGRUENT TO THE TWO ANGLES OF A SECOND TRIANGLE, THEN THE THIRD ANGLE OF THE FIRST TRIANGLE IS CONGRUENT TO THE THIRD ANGLE OF THE SECOND TRIANGLE. Theorem 4.4: Properties of Congruent Triangles Reflexive Property of Congruent Triangles: ABC ABC Symmetric Property of Congruent Triangles IF ABC DEF, THEN DEF ABC. Transitive Property of Congruent Triangles IF ABC DEF AND DEF GHI, THEN ABC GHI. Theorem 3.12: Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are PARALLEL to each other. Postulate 19: Side-Side-Side (SSS) Congruence Postulate If THREE sides of one triangle are congruent to THREE sides of a second triangle, then the two triangles are CONGRUENT. Theorem 4.1: Triangle Sum Theorem THE ANGLES OF A TRIANGLE ADD UP TO 180. Corollary to the Triangle Sum Theorem THE NON-RIGHT ANGLES OF A RIGHT TRIANGLE ARE COMPLEMENTARY. Theorem 4.2: Exterior Angle Theorem THE MEASURE OF AN EXTERIOR ANGLE OF A TRIANGLE IS EQUAL TO THE SUM OF THE TWO NON-ADJACENT INTERIOR ANGLES. Postulate 20: Side-Angle-Side (SAS) Congruence Postulate IF TWO SIDES AND THE INCLUDED ANGLE OF ONE TRIANGLE ARE CONGRUENT TO TWO SIDES AND THE INCLUDED ANGLE OF A SECOND TRIANGLE, THEN THE TWO Theorem 4.5: Hypotenuse-Leg (HL) Congruence Theorem IF THE HYPOTENUSE AND A LEG OF ONE RIGHT TRIANGLE ARE CONGRUENT TO THE HYPOTENUSE AND A LEG OF A SECOND TRIANGLE, THEN THE TWO
Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate IF TWO ANGLES AND THE INCLUDED SIDE OF ONE TRIANGLE ARE CONGRUENT TO TWO ANGLES AND THE INCLUDED SIDE OF A SECOND TRIANGLE, THEN THE TWO Theorem 4.6: Angle-Angle-Side (AAS) Congruence Theorem IF TWO ANGLES AND THE NON-INCLUDED SIDE OF ONE TRIANGLE ARE CONGRUENT TO TWO ANGLES AND THE CORRESPONDING NON-INCLUDED SIDE OF A SECOND TRIANGLE, THEN THE TWO Theorem 9.1: Translation (Slide) Theorem A translation is an isometry. Theorem 9.2: Reflection (Flip) Theorem A reflection is an isometry. Coordinate Rules for Reflections If preimage (x, y) is reflected over the x-axis (y = 0) its image is (x, -y). If preimage (x, y) is reflected over the y-axis (x = 0) its image is (-x, y). If preimage (x, y) is reflected over the y = x its image is (y, x). If preimage (x, y) is reflected over the y = -x its image is (-y, -x). Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Theorem 9.3: Rotation (Spin) Theorem A rotation is an isometry. IF THERE ARE TWO OR MORE TRIANGLES THAT ARE CONGRUENT, THEN THEIR CORRESPONDING PARTS (SIDES AND ANGLES) ARE CONGRUENT. Theorem 4.7: Base Angles Theorem If two sides of a triangle are congruent, then THE ANGLES OPPOSITE THEM ARE CONGRUENT. Theorem 4.8: Converse of Base Angles Theorem If two angles of a triangle are congruent, then THE SIDES OPPOSITE THEM ARE CONGRUENT. Corollary to the Base Angles Theorem If three sides of a triangle are congruent (equilateral), then IT IS EQUIANGULAR. Corollary to the Converse of Base Angles Theorem If three angles of a triangle are congruent (equiangular), then IT IS EQUILATERAL. Coordinate Rules for Rotations When a point (a, b) is rotated counterclockwise about the origin, For a rotation of 90, (x, y) ( y, x). For a rotation of 180, (x, y) ( x, y). For a rotation of 270, (x, y) (y, x). Theorem 9.4: Composition Theorem A composition of two (or more) isometries is an isometry. Theorem 9.5: Reflections in Parallel Lines Theorem A reflection of a figure over/in/about two parallel lines is A TRANSLATION OF LENGTH TWICE THE DISTANCE BETWEEN THE PARALLEL LINES. Theorem 9.6: Reflections in Intersecting Lines Theorem A reflection of a figure over/in/about two intersecting lines is A ROTATION OF DEGREE TWICE THE ANGLE FORMED BY THE INTERSECTING LINES.
Theorem 5.1: Midsegment Theorem The segment connecting THE MIDPOINTS OF TWO SIDES OF A TRIANGLE (THE MIDSEGMENT) IS HALF THE LENGTH OF THE THIRD SIDE AND PARALLEL TO THE THIRD SIDE OF THE TRIANGLE. Theorem 5.2: Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then THE POINT IS EQUIDISTANT FROM THE SEGMENT S ENDPOINTS. Theorem 5.3: Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then IT IS ON THE PERPENDICULAR BISECTOR OF THE SEGMENT. Theorem 5.4: Concurrency of Perpendicular Bisectors of a Triangle Theorem The perpendicular bisectors of a triangle intersect at a point (circumcenter) that is EQUIDISTANT FROM EACH OF THE TRIANGLE S VERTICES. Theorem 5.5: Angle Bisector Theorem If a point is on the bisector of an angle, then IT IS EQUIDISTANT FROM THE TWO SIDES OF THE ANGLE. Theorem 5.6: Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is EQUIDISTANT FROM THE TWO SIDES OF THE ANGLE, THEN IT IS ON THE ANGLE BISECTOR OF THE ANGLE. Theorem 5.7: Concurrency of Angle Bisectors of a Triangle Theorem The angle bisectors of a triangle intersect at a point (incenter) that is EQUIDISTANT TO EACH OF THE SIDES OF THE TRIANGLE. Theorem 5.8: Concurrency of the Medians of a Triangle Theorem The medians of a triangle intersect at a point (CENTROID) that divides each median into a vertex-side part that is TWO-THIRDS the median s length, and a midpoint-side part that is ONE-THIRD the median s length. Corollary of Theorem 5.8: Concurrency of the Medians of a Triangle Theorem The centroid of a triangle with vertices (X1, Y1), (X2, Y2) and (X3, Y3) in the x-y coordinate plane is!!!!!!!!,!!!!!!!!!! Theorem 5.9: Concurrency of the Altitudes of a Triangle Theorem The altitudes of a triangle INTERSECT AT A POINT (ORTHOCENTER). Theorem 5.10 If one side of a triangle is longer than another side, then THE ANGLE OPPOSITE THE LONGER SIDE IS LARGER THAN THE ANGLE OPPOSITE THE SHORTER SIDE. Theorem 5.11 If one angle of a triangle is larger than another angle, then THE SIDE OPPOSITE THE LARGER ANGLE IS LONGER THAN THE SIDE OPPOSITE THE SMALLER ANGLE. Theorem 5.12: Triangle Inequality Theorem The sum of any two sides of a triangle is GREATER THAN THE LENGTH OF THE THIRD SIDE. Theorem 5.13: Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then THIRD SIDE OF THE FIRST TRIANGLE IS LONGER THAN THE THIRD SIDE OF THE SECOND TRIANGLE. Theorem 5.14: Converse of the Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second, then INCLUDED ANGLE OF THE FIRST TRIANGLE IS LARGER THAN THE INCLUDED ANGLE OF THE SECOND TRIANGLE.