Unit 6: Parallel and Perpendicular Lines Lesson 6.1: Identify Pairs of Lines and Angles Lesson 3.1 from textbook Objectives Identify relationships between lines such as parallel and skew. Understand and Apply the Parallel and Perpendicular Postulates. Identify the different angles formed by transversals such as corresponding, alternate interior, alternate exterior, and consecutive interior. PARALLEL AND SKEW LINES WITH PARALLEL PLANES Lines m and n are. Lines m and k are. Planes T and U are. Lines k and n are, and there is a plane (not shown) containing them. Example 1 Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description? Line(s) parallel to CD and containing point A. Line(s) skew to CD and containing point A. Lines(s) perpendicular to CD and containing point A. Plane(s) parallel to plane EFG and containing point A. Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.
Example 2 Use the diagram to answer the following statements. Name a pair of parallel lines. Name a pair of perpendicular lines. Is PN KM? Explain. Is PQ NP? Explain. ANGLES FORMED BY TRANSVERSALS Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Example 3 Identify all pairs of angles of the given type. Corresponding Alternate Interior Alternate Exterior Consecutive Interior
Unit 6: Parallel and Perpendicular Lines Lesson 6.2: Use Parallel Lines and Transversals Lesson 3.2 from textbook Objectives Find the measures of angles formed by parallel lines and transversals Prove that these angles are congruent or supplementary using properties, definitions, theorems and postulates. ACTIVITY: 1. Draw a pair of parallel lines and name them AB and CD 2. Draw a third line through points A and D. 3. What is the relationship between <BAD and <CDA? 4. Find the measure of the following: m<bad = m<cda = Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are. Alternate Exterior Angles Postulate If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are.
Example 1 If m<7 = 123, find the measures of the other seven angles in the figure and tell the postulate or theorem you used to find the measure. m<1 = m<2 = m<3 = m<4 = m<5 = m<6 = m<5 = Example 2 Find the value of x. x = x = Example 3 Find the value of x and y. x = y =
Unit 6: Parallel and Perpendicular Lines Lesson 6.3: Prove Lines are Parallel Lesson 3.3 from textbook Objectives Use angle relationships along with definitions, properties, theorems, and postulates to prove that two lines are parallel. Corresponding Angles Converse Postulate If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Theorem If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel. Consecutive Interior Angles Converse Theorem If two lines are cut by a transversal so the consecutive interior angles are congruent, then the lines are parallel. Example 1 a) Find the value of x that makes m n. x = b) Is there enough information to conclude m n? Explain.
Example 2 Given: r s and <1 <3. Prove: p q. Statements Reasons 1. r s 1. 2. <1 <2 2. 3. <1 <3 3. 4. <2 <3 4. 5. p q 5. Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel. Example 3 The flag of the United States of America has 13 stripes. Each stripe is parallel to the one below it. Explain why the top stripe is parallel to the bottom stripe.
Unit 6: Parallel and Perpendicular Lines Lesson 6.4 Find and Use Slopes of Lines Lesson 3.4 from textbook Objectives Find the slope of a line in the coordinate plane. Identify parallel and perpendicular lines based on their slopes. Graph parallel and perpendicular lines in the coordinate plane. Slope m = rise run = y x 2 2 y1 x 1 Different Slopes in the Coordinate Plane Negative: from left to right. Positive: from left to right. Zero: Undefined: Example 1: Find the slopes of line a and d. Slope of a = Slope of d = Slopes of Parallel and Perpendicular Lines a) Two lines are parallel if and only if they. b) Two lines are perpendicular if and only if they. *Example: If lines a and b are parallel and line a has a slope of 6, then the slope of line b is. If lines a and c are perpendicular and line a has a slope of 4, then the slope of line c is.
Example 2 Find the slope of each line. Determine whether the lines are perpendicular. k 1 = k 2 = k 3 = Parallel lines = Example 3 Example 4 Tell whether the lines through the given points are parallel, perpendicular, or neither. Tell which line through the given points is steeper. Line 1: (-1, 2), (2, 3) Line 1: (0, 3), (4, 2) Line 2: (0, 0), (3, 1) Line 2: (3, 1), (-3, 4) Example 5 The line h goes through (3, 0) and (7, 6). Graph the line perpendicular to h that passes through (2, 5).
Unit 6: Parallel and Perpendicular Lines Lesson 6.5 Write and Graph Equations of Lines Lesson 3.5 from textbook Objectives Write equations of lines in slope-intercept and standard form Graph lines in the coordinate plane given their equation. Write equations of parallel lines in a coordinate plane. Write equations of perpendicular lines in the coordinate plane. WRITING EQUATIONS OF LINES Slope-intercept form: y = mx + b m = b = Example 1 Write an equation of the line with the given slope and y-intercept. m = -3 b = 6 Equation: Example 2 Write an equation of the line in slope intercept form. Example 3 Write an equation of the line that passes through the given point P and has the slope m. m = 2 P(3, 0) Equation: Example 4 Write an equation of a line that passes through P(2, -1) and is parallel to the line with the equation y = 3x 5. Parallel means their slopes are. Equation:
Example 5 Write an equation of a line that passes through P(3, 5) and is perpendicular to the line with the equation y = 2 1 x 5. Perpendicular means their slopes are. Equation: Example 6 Identify the x- and y-intercepts of the line. Use the intercepts to write an equation of a line. x-intercept y-intercept Equation STANDARD FORM equation of a line: Ax +By = C EX) 4x 5y = 6 To convert STANDARD FORM into SLOPE-INTERCEPT. Example 7 Example 8 Graph a line with the equation written Graph the line with the equation in standard form. y 3 = -3x + 2 3x + 4y = 12