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Transcript
Geometry Chapter 3 Key Vocabulary Parallel Lines – coplanar lines that do not intersect Skew Lines – non-coplanar lines that do not intersect Parallel Planes – planes that do not intersect Transversal – line that intersects two or more coplanar lines at different points Angles Formed by Transversals Corresponding Angles (CA) 1 2 3 4 Alternate Exterior Angles (AEA) Alternate Interior Angles (AIA) 6 5 7 8 Consecutive Interior Angles (CIA) x intercept – where graph intersects the x axis; x -intercept (a,0) y intercept – where graph intersects the y axis; y -intercept (0, b) Postulates/ Theorems Postulate 13 – Parallel Postulate If there is a line and a point not on the line, then there exists only 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 exists only one line through the point perpendicular to the given line. 1 Geometry Chapter 3 Postulate 15 – Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the corresponding angles are congruent. If p q , then 1 5 Theorem 3.1 – Alternate Interior Angles Theorem 1 2 4 3 If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. If p q , then p 4 6 5 6 Theorem 3.2 – Alternate Exterior Angles Theorem 8 q 7 If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. If p q , then 2 8 Theorem 3.3 – Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary. If p q , then m 3 m 6 180 Postulate 16 – Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. If 1 5 , then a b Theorem 3.4 – Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. If 3 6 , then a b a b 1 2 3 4 5 6 8 7 Theorem 3.5 – Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. If 2 8 , then a b 2 Geometry Chapter 3 Theorem 3.6 – Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. If m 3 m 5 180 , then a b Theorem 3.7 – Transitive Property of Parallel Lines p q r If two lines are parallel to the same line, then they are parallel to each other. If p q and q r , then p r Postulate 17 – Slopes of Parallel Lines Parallel lines have the same slope Vertical lines are parallel Postulate 18 – Slopes of Perpendicular Lines Slopes of perpendicular lines are opposite reciprocals Vertical and horizontal lines are perpendicular Product of slopes is -1 Theorem 3.8 e If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If 1 2 , then e f 1 2 Theorem 3.9 f a If two lines are perpendicular, then they intersect to form four right angles. If a b , then it creates for right angles b Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. If j k , then 1, 2 are complementary j 1 2 k 3 Geometry Chapter 3 Theorem 3.11 – Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. If h k and j h , then j k Theorem 3.12 – Lines Perpendicular to a Transversal Theorem m n In a plane, it two lines are perpendicular to the same line, then they are parallel to each other. p If m p and n p , then m n Concepts Slopes of Lines Positive Slope – rises from left to right Negative Slope – falls from left to right Slope of Zero – horizontal line Undefined Slope – vertical line Slope-Intercept Form of a Line y mx b Standard Form of a Line Ax By C Formulas Slope Formula – used to find slope of line given two points Slope rise run or m y2 y1 x2 x1 4