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Geometry Chapter 3: Parallel and Perpendicular Lines Lesson 1: Lines and Angles Learning Targets Success Criteria LT-1: Identify parallel lines, perpendicular lines, skew lines, and the angles formed by two lines and a transversal. • • • Identify types of lines in the plane. Classify angle pairs. Identify angle pairs and transversals. Parallel lines: Coplanar lines that do not intersect. Perpendicular lines: Lines that intersect at 90º angles. Skew lines: Lines that are not coplanar and do not intersect. Parallel Planes: Planes that do not intersect. Term Example Transversal - A line that intersects two coplanar lines at two different points. The transversal t and the other two lines r and s form eight angles. Corresponding Angles - Angles that lie on the same side of the transversal t, on the same sides of lines r and s. Alternate Interior Angles - Nonadjacent angles that lie on opposite sides of the transversal t, between lines r and s. Alternate Exterior Angles – Angles that lie on opposite sides of the transversal t, outside lines r and s. Same-Side-Interior Angles – Angles that lie on the same side of the transversal t, between r and s. Same-Side-Exterior Angles – Angles that lie on the same side of the transversal t, outside lines r and s. Page 1 Ex #1: Identify Types Of Lines In The Plane. Use the diagram to identify the following: 1a. a pair of parallel segments 1b. a pair of skew segments 1c. a pair of perpendicular segments 1d. a pair of parallel planes 1e. a pair of perpendicular planes Ex #2: Classify Pairs of Angles. Give an example of each angle pair. 2a. corresponding angles 2b. alternate interior angles 2c. alternate exterior angles 2d. same-side interior angles 2e. same-side exterior angles 2f. vertical angles Ex #3: Identify Angle Pairs and Transversals. Use the diagram to identify the transversal and classify each angle pair. 3a. ∠1 and ∠3 3b. ∠2 and∠ 6 3c. ∠4 and ∠6 Page 2 Lesson 2 – Angles Formed by Parallel Lines and Transversals Learning Targets Success Criteria LT-2: Prove and apply the theorems about the angles formed by parallel lines and a transversal. • • Use the Corresponding Angles Postulate. Find angle measures given parallel lines and a transversal. Corresponding Angle Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Same-Side Interior Angle Theorem If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. Same-Side Exterior Angle Theorem If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary. Page 3 Ex #1: Use The Corresponding Angles Postulate. A. Find m∠QRS. B. If CD // EF , find m∠DEF. Ex #2 Find Angle Measures Given Parallel Lines And A Transversal. A. Find x. m∠ABD = (2x + 10)º m∠BDE = (3x - 15)º B. Find m∠CAY. Ex #3: Find Angle Measures Given Parallel Lines And A Transversal. Find x and y in the diagram below. m∠1 = (2x+27)º m∠2 = (2x + y)º Page 4 Lesson 3 – Proving Lines Parallel Learning Targets Success Criteria LT-3: Use the angles formed by a transversal to prove/disprove that two lines are parallel. • • • Prove lines parallel using the converse of: ◦ Corresponding Angles Postulate. ◦ Alternate Interior/Exterior Angles Thm ◦ Same-Side-Interior/Exterior Angles Thm Write proofs to prove lines are parallel. Construct a line parallel to a line through a point not on it. Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Original conditional: If p, then q. converse: If q, then Ex #1: Prove Lines Are Parallel (Using the Converse) Use the diagram at the right, each set of given information, and the theorems that you have learned to show that l // m A. ∠4 ≅ ∠8 B. ∠1 ≅ ∠8 C. m∠3 = (4x – 80), m∠7 = (3x – 50), x = 30 D. m∠3 = (10x + 8), m∠5 (25x – 3), x = 5 Ex #2: Prove Lines are Parallel (Using the Converse) A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m∠1= (8x + 20)° and m∠2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel. Page 5 p. 3.3 Construct a line parallel to a given line through a point not on it. Ex #3: Write A Two-Column Proof To Prove Lines Parallel. Given: l // m, ∠1 ≅ ∠3 Prove: p // r Statements Reasons 1. 2. 3. 4. Ex #4: Write A Two-Column Proof To Prove Lines Parallel. Given: ∠1≅∠2, ∠3 ≅∠1 Prove: XY // WV Statements Reasons 1. 2. 3. Page 6 *Ex #5: Write A Two-Column Proof To Prove Lines Parallel. Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary Prove: l // m Statements Reasons 1. 2. 3. 4. 5. 6. 7. Lesson 4 – Perpendicular Lines Learning Targets Success Criteria LT-4: Construct, prove, and apply theorems about perpendicular lines. • • • Find the shortest distance from a point to a line. Prove properties of lines. Construct a line perpendicular to a given line through a point not on it. Perpendicular Bisector (of a segment): A line perpendicular to a segment at the segment's midpoint. Distance from a point to a line: The length of the perpendicular segment from the point to the line. 3.3 Construct a line perpendicular to a given line through a point not on it. Page 7 Ex #1: Find The Shortest Distance From A Line To A Point Not On It. A. Name the shortest segment from A to BC . B. Write and solve the inequality for x. C. Name the shortest segment from A to BC . D. Write and solve an equation for x. Theorem Hypothesis 3.4.1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. 3.4.2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. 3.4.3 If two coplanar lines are perpendicular to the same line, then the two linew are parallel to each other. Page 8 Conclusion Ex #2: Prove Lines Perpendicular. Given: r // s, ∠1 ≅ ∠2 Prove: r ⊥ t Statements Reasons 1. 2. 3. 4. Ex #3: Prove Lines Perpendicular. Statements Reasons 1. 2. 3. Lesson 5 – Slopes of Lines Learning Targets Success Criteria LT-5: Prove that two lines are parallel or perpendicular by calculating their slopes. • • • Use the slope formula to calculate the slope of a line. Calculate and interpert rate of change. Determine whether lines are parallel, perpendicular, or neither. The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Page 9 Rise: The difference in the y-values of two points on a line. Run: The difference in the x-values of two points on a line. Slope: The ratio of rise to run. Ex #1: Find The Slope Of A Line. Use the slope formula to determine the slope of each line. a. b b. c. d. Ex #2: Calculate And Interpret Rate Of Change. Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line. Page 10 Parallel Line Theorem: In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Ex: Perpendicular Line Theorem: In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. Ex: Ex #3: Determine Whether Lines Are Parallel, Perpendicular, Or Neither. Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. A. B. C. Lesson 6 – Lines in the Coordinate Plane Learning Targets Success Criteria LT-6: Graph lines and write their equations in slope-intercept and point-slope form. • • • • Page 11 Write equations for lines in slope-intercept form: ◦ Given a point and the slope of a line. ◦ Given two points on a line. ◦ Given a point and the y-intercept Write equations for lines in point-intercept form. ◦ Given a point and the slope of a line. ◦ Given two points on a line. ◦ Given a point and the y-intercept Graph lines in the coordinate plane. Classify pairs of lines. The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line. Name Form Point-Slope Form Slope-Intercept Form The equation of a vertical line is: The equation of a horizontal line is: Ex #1: Write The Equation Of Each Line In The Given Form. Ex #2: Graph Each Line In the Coordinate Plane. A. the line with slope 6 through (3, –4) in point-slope form 2. Graph each line. a. y = ½ x + 1 B. the line through (–1, 0) and (1, 2) in slope-intercept form b. y – 3 = -2(x + 4) c. y = -3 C. the line with the x-intercept 3 and y-intercept –5 in point slope form d. y – 1 = - 2 (x + 2) 3 A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. Page 12 Ex #3: Determine Whether Lines Are Parallel, Intersecting, Or Coinciding. A. y = 3x + 7, y = –3x – 4 C. 2y – 4x = 16, y – 10 = 2(x - 1) B. D. 3x + 5y = 2, 3x + 6 = -5y Ex #4: Write Equations For Lines Given A Point And The y-intercept. Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same? Page 13 Lesson Problems 3.1 p. 149 #14-32, 34-43, 58, 60-62 3.2 p. 158 #6-25, 27, 28, 30-32, 34, 35, 41, 42 3.3 p. 166 #12-14, 16-35, 38, 40, 41-45, 58, 60, 64 3.4 p. 175 #6-14, 15 challenge, 16-22, 24 challenge, 26, 27, 31-34, 40. 44 3.5 p. 186 #10-22, 26-28, 29, 30, 34, 38-40 3.6 p. 194 #13-25, 27-34, 37, 39, 45-47, 51, 58-60 Page 14