Unit 2: Parallel and Perpendicular Lines

Transcription

1 Unit 2: Parallel and Perpendicular Lines Scale for Unit 2 4 I have mastered level 3 and I can determine the angles of a parallelogram. 3 I have mastered level 2 and I can: Identify skewed, parallel, and perpendicular lines from diagrams or on the coordinate plane. I can write the equation of parallel and perpendicular lines 2 I have mastered level 1 and I can: Use the triangle angle sum theorem to find missing values Write proofs to prove information about angles or that lines are parallel 1 I have mastered the Entry level and I can identify and know the relationship (thm/post) between alternate interior angles, alternate exterior angles, corresponding angles, and same-side interior angles. Entry I can define vertical angle and know what the vertical angles theorem is. I can define linear pair and know what the linear pair postulate is. Ranking: Date Level Notes: (what you didn t understand from the chapter and want to work on) Rank Yourself: 1 Notes

2 Level 4 Given that the figure is a parallelogram find the value of a. 1. Identify a line parallel to line AB 2. Identify a line that is skew to line FE 3. Write the slope intercept form of a line perpendicular to y = 3x 5 and passes through (4, -5) Level 3 4. Are the two lines parallel, perpendicular, or neither? 1. Given: a b, c d Prove: 1 and 4 are supplementary. 2. Given: Prove: Level 2 3. Find the value of x, y, and z: Find the value of each angle and justify why you know that, that is the value: Level 1 Find the value of x and y: Entry 2 Notes

3 3.1: Lines and Angles Objective: Students will be able to identify relationships between figures in space and to identify angles formed by two lines and a transversal. Transversal: A line that two or more lines at distinct points. Example: Which segments are parallel to? Your Turn: Answers Which segments are skew to? What are two pairs of parallel planes? What are two segments parallel to plane RUYV? 3 Notes

4 Claim it! Use on of the four diagrams below to play each round of the game. At the beginning of each round, you and your partner must each claim an angle and label it with your initials. Take turns rolling the number cube to determine an angle relationship. 1 = alternate interior 2 = alternate exterior 3 = corresponding 4 = same side interior 5 = vertical angles 6 = linear pair Then initial one angle that has the given relationship to the angle you claimed. On subsequent turns, you may start from any previously initialed angles in the angle relationship. Angles may only be claimed once, so it may not always be possible to claim an angle on your turn. 4 Notes

5 3.2: Properties of Parallel Lines Geogebra Discovery Objective: Students will be able to prove theorems about parallel lines and use properties of parallel lines to find angle measures. Thinking back to the angle pairs we discussed yesterday, what do you think will happen to the angle pairs when we use parallel lines cut by a transversal? Write your thoughts below: When parallel lines are cut by a transversal Rule Alternate Interior angles are Alternate Exterior angles Same- Side Interior Angles Corresponding Angles 5 Notes

6 Each of the previous has a corresponding theorem or postulate that we will use frequently. 3.2: Properties of Parallel Lines 6 Notes

7 Example: Finding missing angle measures: Find the measure of and list what thm or post justifies your answer Your turn: Find the measure of each and list what thm or post justifies your answer: Example: Using Algebra to find missing angles Find the value of x and y: Your turn: Find the value of p: Example: Proofs using parallel lines cut by a transversal Given: Prove: Statement Reason 7 Notes

8 Your turn: Given: Prove: Statement Reason 3-3: Converse of Parallel Lines Theorems We will use Patty Paper to discovery something about lines: 8 Notes

9 Example: What value of the variable will make the lines parallel? Your turn: 9 Notes

10 Example: Proving lines are parallel Statement Reason Constructing perpendicular line: 3-6 Constructing Parallel and Perpendicular Lines Steps: Patty Paper Steps Hand Construction: 10 Notes

11 Construction of parallel lines: Steps: Patty Paper Steps Hand Construction: 3-5: Parallel Lines and Triangles Using half of an index card cut diagonally, label the three angles of your triangle A, B and C. Cut the triangle into three pieces so that the angles are alone. Arrange the angles below so that all three points touch each other. What do you notice? 11 Notes

12 Flow Proof: Arrows show the connection between statements. We will now use a flow proof to prove a very important theorem, the triangle sum theorem. Draw through B, parallel to Given PBC and 3 are supplementary m PBC + m 3 = 180 m PBC = m 1 + m 2 Angles that form a linear pair are Definition of suppl. Angle addition post Supplementary angles. m 1 = m A and m 3 = m C Congruent angles have equal measure. 1 A and 3 C If lines are then alt. int. angles are congruent m 1 + m 2 + m 3 = 180 Substitution property m A + m 2 + m C = 180 Substitution Example: property Using the triangle sum theorem: Find the value of x, y, and z. Your turn: 12 Notes

13 Example: Using the Triangle Exterior Angle Theorem Examples using triangle theorems: The ratio of the angle measures of the acute angles in a right triangle is 1:2. Find the measures of the other two angles. Your turn: The measure of one angle of a triangle is 40. The measures of the other two angles are in a ratio of 3:4. Find the measure of the other two angles. 3-8: Equations of Parallel and Perpendicular Lines in the Coordinate Plane Using the pictures from the bellwork what do you notice about parallel lines? Perpendicular lines? 13 Notes

14 Types of slope: Example: Find the slope between the points (2, -4) and (-1, -3) Your turn: Find the slope between the points (1, -2) and (5, -7) 14 Notes

15 Example: Equations of Parallel and Perpendicular Lines What is an equation in slope-intercept form for the line perpendicular to y=3x + 2 that contains (6, 2) Your turn: What is an equation in point-slope form of a line parallel to y = 4x -2 that contains (-2, -2) Are the following lines parallel, perpendicular or neither? 15 Notes