Section 3 1 Parallel and Skew lines Objective
Section 3. 1 ~ Parallel and Skew lines! Objective: to identify angle pairs (corresponding, alternate interior, same-side interior, alternate exterior, same-side exterior)! Quick Definitions Parallel lines: Parallel planes: Skew lines:
A transversal is a _____ that intersects two other ______! In this picture, line ____ is the transversal. This transversal creates ____ angles! Pairs of these angles have special names, depending on their positions. 1 3 42 5 7 8 t n 6 m
Identify the transversal(s) in each picture: n c a b m s t p r k
Special Angle Pairs in Parallel Lines Cut by a Transversal: Angle Pair: Corresponding Angles Alternate Interior Angles Consecutive Interior Angles Alternate Exterior Angles Consecutive Exterior Angles Picture: Relationship:
Practice Identifying the Special Angle Pairs: Corresponding Angles Alternate Interior Angles Consecutive Interior Angles Alternate Exterior Angles Consecutive Exterior Angles 10 12 11 9 1 2 4 3 5 6 8 7
Use the picture to complete each statement: If m 5 = 130, then m 8 =____ because they are… If m 4 = 125, then m 6 =____ because they are… If m 4 = 125, then m 8 =____ because they are… 9 1 2 4 3 12 10 11 5 6 8 7 If m 2 = 45, then m 7 =____ because they are… If m 3 = 50, then m 6 =____ because they are… If m 7 = 42, then m 1 =____ because they are…
MORE PRACTICE: Use the picture to complete each statement. If m 5 = 130, then m 4 =____ because they are… If m 5 = 25, then m 3 =____ because they are… If m 5 = 125, then m 1 =____ because they are… 9 1 2 4 3 12 10 11 5 6 8 7 If m 1 = 50, then m 8 =____ because they are… If m 3 = 50, then m 2 =____ because they are… If m 2 = 51, then m 8 =____ because they are…
Find the value of each variable. 2 x 2 x + 60 3 x - 20 x + 60
Your Turn! 60° 3 y
Section 3. 2 ~ Angles and Parallel Lines Objective: Be able to determine relationships between specific pairs of angles and use algebra to find specific angle measurements. Quick Review: Find the value of x and the measure of each angle. Justify each step! 3 x 2 x + 50
Postulate: When lines are parallel, corresponding angles are ____! Theorem: When lines are parallel, alternate interior angles are ____! Theorem: When lines are parallel, consecutive interior angles are _____! Theorem: When lines are parallel, alternate exterior angles are _____! Theorem: When lines are parallel, consecutive exterior angles are _____!
Let’s use our theorems to find angle measures If find the following angles and state theorem used. 12 3 4 5 6 7 8
Using the picture at the left, find the measure of each angle and tell which postulate or theorem you used. m 1 = 1 2 4 3 9 10 12 11 5 6 8 7 m 2 = m 3 = m 4 = m 5 = m 6 = m 7 =
Find the value of the variables in each picture. Explain your answer. 75° 3 n -15 ° 13 - x 68 +10 x
Write a 2 column proof to solve for y. Given: m 4 = m 5 = 12 3 4 5 6 7 8
Section 3. 5 ~ Proving Line are Parallel Objective: Be able to prove lines are parallel using the properties of the special angle pairs Quick Review: Find the value of x and justify each step. Find each angle measure. 5 x -25 3 x + 45
Converse of Corresponding Angle Postulate: If two lines and a transversal form CORRESPONDING angles that are CONGRUENT, then the two lines are ________! Where are there parallel lines in the pictures? 90° 45° 89° 90° 37° 45°
Converse of Alternating Interior Angle Theorem: If two lines and a transversal form ALTERNATING INTERIOR angles that are CONGRUENT, then the two lines are ________! Where are there parallel lines in the pictures? 50° 130° 100°
Converse of Consecutive Interior Angle Thm: If two lines and a transversal form CONSECUTIVE INTERIOR angles that are SUPPLEMENTARY, then the two lines are ________! Where are there parallel lines in the pictures? 70° 80° 110° 100° 75° 115°
Converse of ALTERNATE EXTERIOR Angle Thm: If two lines and a transversal form ALTERNATE EXTERIOR angles that are CONGRUENT, then the two lines are ________! Solve for x so the lines m and n are // x +116 m n 6 x - 24
Given the following information, is it possible to prove that any of the lines shown are parallel? If so state the postulate or theorem that justifies your answer. 2 = 8 t v 1 2 4 3 r 9 10 12 11 5 67 8 s 14 13 15 12 + 13 = 180 16 4 = 6 14 = 15 9 = 13
Find the value of the variable that would make the lines parallel. State which postulate or theorem justifies your answer. 2 x + 40 80° 2 x - 30 4 x - 40°
Two More Theorems: Theorem: If two lines are parallel to the same line, then they are parallel to each other! Theorem: If two lines are perpendicular to the same line, then they are parallel to each other!
Let’s try a proof! A B 1 2 C 3 D
Last one, YOUR TURN … Are the lines parallel? Explain. 50°
3. 3 Slope and rate of change Objective: We are going to find the rate of change and determine the slope of a line. What does each of the following look like? Positive Slope Negative Slope Zero Slope Undefined Slope
How to find Slope… Anyone remember? When given 2 points (x 1, y 1) and (x 2, y 2) plug them into our slope formula:
Try these Ex 2: Find the slope of the line that passes through (– 1, 4) and (1, – 2). Ex 3: Find the slope of the line that passes through (9, – 3) and (2, 7).
Find the slope of the line that passes through the following points. Ex. 4: (2, 3) and (2, 6) Ex. 5: (-5, 7) and (4, 7)
From a graph! Find the slope of the line. Blue Line: Red Line:
Rate of Change COLLEGE ADMISSIONS In 2004, 56, 878 students applied to UCLA. In 2006, 60, 291 students applied. Find the rate of change in the number of students applying for admission from 2004 to 2006. X – independent variable Y – Dependent variable
Let’s try One More • Find the rate of change for the data in the table.
Parallel and Perpendicular Lines If 2 lines are parallel, there slopes are _____ If 2 lines are perpendicular, there slopes are _____ If m=-4 what is the slope of a line perpendicular and parallel to the line? // m = ______ What about if m = // m = _____ ? m = ______
More Parallel and Perpendicular lines Determine whether and are parallel, perpendicular, or neither for the given set of points. Ex 1: A(1, -3) B(-2, -1) C(5, 0) and D(6, 3) Ex 2: A(3, 6) B(-9, 2) C(5, 4) and D(2, 3)
Section 3. 4 ~ Equations of Lines! Objectives: 1) Be able to graph lines from equations in slope-intercept form! 2) Be able to write the equation of a line using point-slope form! Recall: Coordinate pairs: (x 1, y 1) and (x 2, y 2) SLOPE of a line: Example: Find the slope of the line that passes through (4, 5) and (-1, 2).
Slope-Intercept Form: y = mx + b
Let’s practice s’more! Write the equation of the graph shown:
Point-Slope Form: y = m (x – x 1) + y 1 Write an equation of a line that passes through (2, -3) and has a slope of – 4. Write an equation of a line that passes through (-3, 4) and has a slope of 2/3. Write your final answer in slope-intercept form.
Write an equation of a line that passes through (-2, -1) and (3, 0). Write an equation of a line that passes through (1, 5) and (4, 2). Write your final answer in slope-intercept form.
Graph the equations
Write an equation of a line that passes through (-1, 4) and has a slope of 3. Write an equation of a line that passes through (4, -9) and -1, 1). Write your final answer in slope-intercept form. (
One Last thing… Do we know how to graph horizontal and vertical lines? ? ? y=# Graph of a _______ line x=# Graph of a _______ line What is the equation Graph y = 2 Graph x = -3 of the line? _______
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