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Transcript
Postulates and Theorems Chapter 1 *Through any two points there is exactly one line. *If two distinct lines intersect, then they intersect in exactly one point. *If two distinct planes intersect, then they intersect in exactly one line. *Through any three noncollinear points there is exactly one plane. *Ruler Postulate: Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real numbers. *Segment Addition Postulate: If three points A, B and C are collinear and B is between A and C, the AB + BC = AC. *Protractor Postulate: Consider OB and a point A on one side of OB . Every ray of the form OA can be paired one to one with a real number from 0 to 180. *Angle Addition Postulate: If point B is in the interior of , then mAOB mBOC mAOC . *Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. *Midpoint Formula for a Number Line: The coordinate ab . 2 of the midpoint M of AB is *Midpoint Formula for a Coordinate Plane: Given AB where A x1 , y1 and B x2 , y2 , the coordinates of x1 x2 y1 y2 , . 2 2 the midpoint of AB are M *The Distance Formula: The distance between two points A x1 , y1 and B x2 , y2 is d x2 x1 y2 y1 2 2 . *Area Addition Postulate: The area of a region is the sum of the areas of its nonoverlapping parts. Chapter 2 *Law of Detachment: If the hypothesis of a true conditional is true, then the conclusion is true. In symbolic form: If p q and p is true, then q is true. *Law of Syllogism: If p q is true and q r is true, then p r is true. *Vertical Angles Theorem: Vertical angles are congruent. *Congruent Supplements Theorem: If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. *Congruent Complements Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. *All right angles are congruent. *If two angles are congruent and supplementary, then each is a right angle. Postulates and Theorems Chapter 3 *Corresponding Angles Postulate: If a transversal intersects two parallel lines, then corresponding angles are congruent. *Alternate Interior Angles Theorem: If a transversal intersects two parallel lines, then alternate interior angles are congruent. *Same-Side Interior Angles Theorem: If a transversal intersects two parallel lines, then same-side interior angles are supplementary. *Alternate Exterior Angles Theorem: If a transversal intersects two parallel lines, then alternate exterior angles are congruent. *Converse of the Corresponding Angles Postulate: If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. *Converse of the Alternate Interior Angles Theorem: If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. *Converse of the Same-Side Interior Angles Theorem: If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. *Converse of the Alternate Exterior Angles Theorem: If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. *If two lines are parallel to the same line, then they are parallel to each other. *In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. *Perpendicular Transversal Theorem: In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. *Parallel Postulate: Through a point not on a line, there is one and only one line parallel to the given line. *Triangle Angle-Sum Theorem: The sum of the measures of the angles of a triangle is 180. *Triangle Exterior Angle Theorem: The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. *Corollary to the Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. *Spherical Geometry Parallel Postulate: Through a point not on a line, there is no line parallel to the given line. *Perpendicular Postulate: Through a point not on a line, there is one and only one line perpendicular to the given line. *Slopes of Parallel Lines: If two nonvertical lines are parallel, then their slopes are equal. If the slopes of two distinct nonvertical lines are equal, then the lines are parallel. Any two vertical lines or horizontal lines are parallel. *Slopes of Perpendicular Lines: If two nonvertical lines are perpendicular, then the product of their slopes is -1. If the slopes of two lines have a product of -1, then the lines are perpendicular. Any horizontal line and vertical line are perpendicular.