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Transcript
MAT 122 Postulates, Theorems, and Corollaries Postulate 1: Through two distinct points, there is exactly one line. Postulate 2: (Ruler Postulate) The measure of any line segment is a unique positive number. Postulate 3: (Segment-Addition Posutlate) If X is a point of AB and A − X − B, then AX + XB = AB. Postulate 4: If two lines intersect, they intersect at a point. Postulate 5: Through three noncollinear points, there is exactly one plane. Postulate 6: If two distinct planes intersect, then their intersection is a line. Postulate 7: Given two distinct points in a plane, the line containing these points also lies in the plane. Theorem 1.3.1: The midpoint of a line segment is unique. Postulate 8: (Protractor Postulate) The measure of an angle is a unique positive number. Postulate 9: (Angle-Addition Postulate) If a point D lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC. Theorem 1.4.1: There is one and only one angle bisector for a given angle. Theorem 1.6.1: If two lines are perpendicular, then they meet to form right angles. Theorem 1.6.2: If two lines intersect, then the vertical angles formed are congruent. Theorem 1.6.3: In a plane, there is exactly one line perpendicular to a given line at any point on the line. Theorem 1.6.4: The perpendicular bisector of a line segment is unique. Theorem 1.7.1: If two lines meet to form a right angle, then these lines are perpendicular. Theorem 1.7.2: If two angle are complementary to the same angle (or to congruent angles), then these angles are congruent. Theorem 1.7.3: If two angle are supplementary to the same angle (or to congruent angles), then these angles are congruent. Theorem 1.7.4: Any two right angles are congruent. Theorem 1.7.5: If the exterior sides of two adjacent acute angle form perpendicular rays, then these angles are complementary. Theorem 1.7.6: If the exterior sides of two adjacent angles form a straight line, then these angles are supplementary. Theorem 1.7.7: If two line segments are congruent, then their midpoints separate these segments into four congruent segments. Theorem 1.7.8: If two angles are congruent, then their bisectors separate these angles into four congruent angles. Theorem 2.1.1: From a point not on a given line, there is exactly one line perpendicular to the given line. Postulate 10: (Parallel Postulate) Through a point not on a line, exacly one line is parallel to the given line. Postulate 11: (Corresponding Angles) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. Theorem 2.1.2: (Alternate Interior Angles) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Theorem 2.1.3: (Alternate Exterior Angles) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Theorem 2.1.4: (Same-Side Interior Angles) If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Theorem 2.1.5: (Same-Side Exterior Angles) If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary. Theorem 2.3.1: (Corresponding Angles ∼ =) If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel. Theorem 2.3.2: (Alternate Interior Angles ∼ =) If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel. Theorem 2.3.3: (Alternate Exterior Angles ∼ =) If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel. Theorem 2.3.4: (Same-Side Interior Angles Supplementary) If two lines are cut by a transversal so that the interior angles on the same side of the transversal are supplementary, then these lines are parallel. Theorem 2.3.5: (Same-Side Exterior Angles Supplemantary) If two lines are cut by a transversal so that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel. Theorem 2.3.6: If two lines are each parallel to a third line, then these lines are parallel to each other. Theorem 2.3.7: If two coplanar lines are each perpendicular to a third line, then these lines are parallel to each other. Theorem 2.4.1: In a triangle, the sum of the measure of the interior angles is 180◦ . Corollary 2.4.2: Each angle of an equiangular triangle measures 60◦ . Corollary 2.4.3 The acute angles of a right triangle are complementary. Corollary 2.4.4: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Corollary 2.4.5: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles. Theorem 2.5.1: The total number of diagonals D in a polygon of n sides is given by the n(n − 3) formula D = . 2 Theorem 2.5.2: The sum S of the measure of the interior angles of a polygon with n sides is given by S = (n − 2) · 180◦ . Note that n > 2 for any polygon. Corollary 2.5.3: The measure I of each interior angle of a regular polygon or equiangular (n − 2) · 180◦ polygon of n sides is I = . n Corollary 2.5.4: The sum of the four interior angles of a quadrilateral is 360◦ . Corollary 2.4.5: The sum of the measure of the exterior angles of a polygon, one at each vertex, is 360◦ . Corollary 2.5.6: The measure E of each exterior angle of a regular polygon or equiangular 360◦ . polygon of n sides is E = n