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Transcript
nat If Bs Ways to Prove Angles Congruent • Vertical angles are congruent. • If two angles are supplements of congruent angles (or the same angle) the two angles are congruent. • If two angles are complements of congruent angles (or of the same angle) the two angles are congruent. • If two lines are perpendicular they form congruent adjacent angles. • If parallel lines are cut by a transversal, the corresponding angles are congruent. • If parallel lines are cut by a transversal, the alternate interior angles are congruent. • If an angle is bisected, it divides it into two congruent angles. • If two angles are equal in measure, then they are congruent. (Definition of Congruent Angles) • If two angles in one triangle are congruent to two angles in another triangle, the third angles are congruent. • The exterior angle of a triangle equals the sum of the two remote interior angles. Ways to Prove Angles Complementary • If the exterior sides of 2 adjacent angles are perpendicular the angles are complementary. • If the sum of the measure of two angles is 90 degrees, the angles are complementary (Definition of complementary angles) • The acute angles of a right triangle are complementary. Ways to Prove Angles Supplemenatry • If parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. • If two angles add up to 180 degrees, then they are supplementary (Defintion of Supplementary Angles) Angle AngleBisector AdditionTheorem Axiom Ways to Prove Lines Perpendicular Segment Addition Axiom Midpoint Theorem B is isinthe thebisector interior of angle ABC: CDA then the measure of angle • If BX The measure of angle ABX = CDB the measure of angle BDA the measure of angle 1/2 the+ measure of angle ABC; The=Measure of angle SBCCDA = 1/2 the measure of angle ABC • If a line is perpendicular to one of two parallel lines, it is perpendicular to the other one as well. • If two lines form right angles, then they are perpindicular. (Definition of perpendicular lines) • If B is between A and C then AB + BC = AC (whole = sum of its parts) • If M the midpoint of AB: AM = 1/2 AB; BM = 1/2 AB Ways to Prove Lines Parallel Lines, Points, and Planes • If two lines are perpendicular to the same line, they are parallel to each other. • If two lines are parallel to a third line, then the two lines are parallel to each other. • If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parrallel. • If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. • If two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel. • In a plane two lines perpendicular to the same line are parallel. • If 2 lines intersect, they intersect in exactly one point. • Though a line and a point not in the line there is exactly one plane. • If two lines intersect exactly one point contains the lines. • A line contains at least two points. • A plane contains at least three points not all in one line. • Space contains at least four points not all in one plane. • Through any two point there is exactly on line. • Through any three point there is at least on plane, and through any three non-collinear points there is exactly on plane. • If two points are in a plane, then the line that contains the points in in that plane. • If two planes intersect, then their intersection is a line. • Through a point outside a line, there is exactly one line parallel to the given line. Definitions Line Segment: A part of a line with two end points. Midpoint of a Segment: Divides the segment into two equal parts. Bisector of a Segment: Intersects the segment at its midpoint; could be a line or a segment. Congruent Segments: Two segments that have the same measure. Perpendicular Lines: Two lines that intersect to form right angles. Parallel Lines: Two lines on the same plane that never intersect. Ray: A line that only goes in one direction. Opposite Rays: Two rays that have the same endpoint that form a straight line. Angle: The union of two rays with a common endpoint (vertex). Adjacent Angles: Two angles that share a ray, share a vertex, and do not overlap. Obtuse Angle: An angle was a measure greater than 90 degrees but less than 180 degrees. Acute Angle: An angle with a measure between 0 and 90 degrees. Right Angle: An angle with a measure of 90 degrees. Bisector of an Angle: A ray, line, or sector that cuts an angle into two equal halves. Vertical Angles: Two non adjacent angles formed by two intersecting lines. Supplementary Angles: Angles that add up to 180 degrees when combined. Complimentary Angles: Angles that add up to 90 degrees when combined. Congruent Angles: Two angles that have the same measure. Collinear: Two points on the same line. Coplanar: Two points on the same plane. Isosceles triangle – a triangle with 2 sides congruent Scalene triangle – a triangle with no sides the same length Equilateral triangle: a triangle with all sides congruent Right triangle- a triangle with one right angle Obtuse triangle- a triangle with one obtuse angle Acute triangle – a triangle with all acute angles. Properties of Equality Subtraction Addition Multiplication Division Substituion Reflexive Symmetric: Transitive • If a = b and c = d then a - c = b - d • If a = b and c = d then a + c = b + d • If a = b then ac = bc • If a = b and c ≠ d then a/c = b/c • If a = b then you can substitute a for b or b for a in an equation • a=a • If a = b then b = a