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Transcript
Chapter 3 Notes Mrs. Myers – Geometry Name ______________________________ Period ______ 3.1 Lines and Angles Parallel Lines: when two lines are coplanar and do not intersect. Skew Lines: lines that do not intersect and are not coplanar. Parallel Planes: two planes that do not intersect. Ex. 1 Identify relationships in space. A) to AB and point D. B) to AB and point D. C) skew to AB and point D. D) plane that is to plane ABEF and point D. * Postulate 13: Parallel Postulate = If there is a line and a point not on the line, 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, there is exactly one line through the point perpendicular to the given line. Transversal: is a line that intersects two or more coplanar lines at different points (line t). Corresponding Angles: two angles that occupy corresponding positions. Alternate Exterior Angles: two angles that lie outside the two lines on opposite sides of the transversal. Alternate Interior Angles: two angles that lie between the two lines on opposite sides of the transversal. Consecutive Interior Angles: two angles that lie between the two lines on the same side of the transversal. Ex. 2 Identify angle relationships A) Corresponding B) Alternate Exterior C) Alternate Interior D) Consecutive Interior 3.2 Proof and Perpendicular Lines * Theorem 3.1: if two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. * Theorem 3.2: if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. * Theorem 3.3: if two lines are perpendicular, then they intersect to form four right angles. Ex. 1 Proof of theorem 3.1 Given: 1 2 1 and 2 are a linear pair Prove: G H 1. 1 and Statements 2 are a linear pair 1. Given 2. 1 and 2 are sup plementary 2. 3. m 1 m 2 180 4. 1 2 Reasons 3. 4. Given 5. m 1 m 2 5. 6. m 1 m 1 180 6. 7. 2 m 1 180 7. Simplify / combine like terms / distributive property of equality 8. m 1 90 8. 9. 9. 1 is a right angle 10. G H 10. Ex. 2 Given: BA BC Prove: 1. BA BC 2. 1 and 2 are complementary Statements Reasons 1. Given ABC is a right angle 2. 3. 3. Definition of right angles 4. m ABC m 1 m 2 4. 5. 90 m 1 m 2 5. Substitution property of equality 6. 6. 1 and 2 are complementary 3.3 Parallel Lines and Transversals * Postulate 15: Corresponding Angles Postulate = If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. If , then 1 2 * Theorem 3.4: Alternate Interior Angles Theorem = If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. If , then 3 4 * Theorem 3.5: Consecutive Interior Angles Theorem = If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. If , then m 5 m 6 180 * Theorem 3.6: Alternate Exterior Angles Theorem = If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. If , then 7 8 * Theorem 3.7: Perpendicular Transversal = 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 Ex. 1 Given m 5 65 . Find the other measurements of the angles and state why. A) m 6 _____ because of __________________________________________ B) m 7 _____ because of __________________________________________ C) m 8 _____ because of __________________________________________ D) m 9 _____ because of __________________________________________ Ex. 2 Proof of the Alternate Interior Angles Theorem Given: P Q Prove: 1 2 Statements Reasons 1. Given 1. P Q 2. 1 3 2. 3. 2 3 3. 4. 1 2 4. Ex. 3 Given: P Q Prove: m 1 m 2 180 Statements Reasons 1. Given 1. P Q 2. 1 3. 2& 4 are a linear pair 3. Definition of linear pair 4. 2& 4 are supplementary 4. Linear pair postulate 5. 1& 2 are supplementary 5. 2. 4 6. m 1 m 2 180 6. Ex. 4 Given: diagram Prove: x 40 Statements 1. m 4 x 15 180 Reasons 1. Linear pair postulate 2. m 4 125 2. 3. 125 x 15 180 3. Substitution property of equality 4. x 140 180 4. Simplify 5. x 40 5. 3.4 Proving Lines are Parallel Postulate 16: Corresponding Angles Converse = If corresponding angles are congruent, then the lines are parallel. If k Theorem 3.8: Alternate Interior Angles Converse = If alternate interior angles are congruent, then the lines are parallel. If ' s are , then j 1 3, then j k Theorem 3.9: Consecutive Interior Angles Converse = If consecutive interior angles add up to 180 (supplementary), then the lines are parallel. If m 1 m 2 180 , then j k Theorem 3.8: Alternate Exterior Angles Converse = If alternate exterior angles are congruent, then the lines are parallel. If 4 5, then j k Ex. 1 Given: m p and m q Prove: p q p 1 2 q Statements 1. m p and m q 1. Given 2. 1 is a right angle 2. 3. 2 is a right angle 3. Definition of perpendicular lines 4. 1 4. 5. p 2 Reasons 5. q Ex. 2 Find the value of x that makes m n . m n 2x + 1 3x - 5 3.5 Using Properties of Parallel Lines * Theorem 3.11: 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 * Theorem 3.12: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m p and n p, then m n Ex.1 Proving theorem 3.11. Given: m n, n k Prove: m k Statements 2. 1 Reasons 1. Given 1. m n 2 2. 3. n k 3. Given 4. 4. Corresponding angles postulate 5. 1 6. m k 3 5. 6. Ex. 2 Another way to prove theorem 3.11. Given: r s, s t Prove: r t r 1 2 s t 3 Statements 1. r 2. 1 Reasons 1. Given s 2 2. 3. s t 3. Given 4. 4. 5. 1 6. r t 3 5. Transitive property of 6. 3.6 Parallel Lines in the Coordinate Plane Slope = rise run * m y2 y1 x2 x1 Ex. 1 Find the slope of a line that passes through the points 3,0 and 4,7 . * Postulate 17: Slope of Parallel Lines = In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical liens are parallel. Ex. 2 If AB has two coordinates 1,1 and 3,5 and CD has two coordinates 6, 4 and 7,6 , are the two lines parallel? Slope-Intercept Form: o y mx b where m slope b y int ercept Ex. 3 Write an equation of the line through the point 4,9 that has a slope of -2. 2 5 through the point 5, 0 . Write the equation of k2 . Ex. 4 Line k1 has the equation y x 3. Line k2 is parallel to line k1 and passes 3.7 Perpendicular Lines in the Coordinate Planes * Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Note: vertical and horizontal lines are perpendicular to each other. Ex. 1 Is P Q ? Line P passes through points 1,3 and 1, 1 . Line Q passes through points 2,0 and 2, 2 . Ex. 2 Is PQ RS ? Point P: Point Q: Point R: Point S: 3,3 5, 2 1, 4 0,1 Ex. 3 Is J N ? 2 Line J: y x 1 3 3 Line N: y x 1 2 Ex. 4 Is B K ? Line B: 2 x 7 y 4 Line K: 7 x 2 y 10 Ex. 5 Line G has the equation y 3x 2 . Find the equation of line H that passes through 3, 4 and is perpendicular to G.