Geometry Chapter 2 Learning Targets! By the unit of the chapter, you should be able to:  Identify the relationships between two lines or two planes  Name angle pairs formed by parallel lines and transversals  Use theorems to determine the relationships between specific pairs of angles  Use algebra to find angle measurements  Find slopes of lines  Use slope to identify parallel and perpendicular lines  Write an equation of a line given information about the graph  Solve problems by writing equations  Recognize angle pairs that occur with parallel lines, and prove that two lines are parallel using angle relationships

Section 1 ~ Parallel and Skew lines! L.T. #1: Be able 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 ______! t m n In this picture, line ____ is the transversal. This transversal creates ____ angles! Pairs of these angles have special names, depending on their positions

Identify the transversal(s) in each picture: a b c k m n r s t p

Special Angle Pairs in Parallel Lines Cut by a Transversal: Corresponding Angles Alternate Interior Angles Consecutive Interior Angles Alternate Exterior Angles Consecutive Exterior Angles Angle Pair:Picture:Relationship:

Practice Identifying the Special Angle Pairs: Corresponding Angles Alternate Interior Angles Consecutive Interior Angles Alternate Exterior Angles Consecutive Exterior Angles

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… 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… 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. 2x2x 2x + 60 x x - 20

Your Turn! 60° 3y3y

Section 3.2 ~ Angles and Parallel Lines L.T.: 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! 3x3x 2x + 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 the theorem used

Using the picture at the left, find the measure of each angle and tell which postulate or theorem you used m  1 = m  2 = m  3 = m  4 = m  5 = m  6 = m  7 =

75° 3n -15 ° x 13 - x Find the value of the variables in each picture. Explain your answer.

Write a 2 column proof to solve for y. Given: m  4 = m  5 =

L.T.: Be able to prove lines are parallel using the properties of the special angle pairs Section 3.5 ~ Proving Line are Parallel Quick Review: Find the value of x and justify each step. Find each angle measure. 3x x -25

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? 45° 37° 90° 89°

Converse of Alternating Interior Angle Theorem: If two lines and a transversal form ALTERNATING INTERIOR angles that are CONGRUENT, then the two lines are ________________! 100° Where are there parallel lines in the pictures? 50° 130°

Where are there parallel lines in the pictures? 70° 110°100° 80° 75° 115° Converse of Consecutive Interior Angle Thm: If two lines and a transversal form CONSECUTIVE INTERIOR angles that are SUPPLEMENTARY, then the two lines are ________________!

Converse of ALTERNATE EXTERIOR Angle Thm: If two lines and a transversal form ALTERNATE EXTERIOR angles that are CONGRUENT, then the two lines are ________________! 6x - 24 x +116 Solve for x so the lines m and n are // m n

r s v t 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  12 +  13 = 180  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. 2x ° 2x 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!

A D C B 1 23

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 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 Try these Ex 2: Find the slope of the line that passes through (–1, 4) and (1, –2). 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 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 X – independent variable Y – Dependent variable

Let’s try One More Find the rate of change for the data in the table.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 = _____ 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)

L.T.#1: Be able to graph lines from equations in slope- intercept form! L.T.#2: Be able to write the equation of a line using point-slope form! Section 3.4 ~ Equations of Lines! 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 = m x + b

Let’s practice s’more! Write the equation of the graph shown:

Write an equation of a line that passes through (2, -3) and has a slope of –4. Point-Slope Form: y = m (x – x 1 ) + y 1 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?_______