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Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines 1

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Geometry 1 Unit 3 3.1 Lines and Angles

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Lines and Angles Parallel Lines

Parallel lines are lines that are coplanar and do not intersect.

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Some examples of parallel lines

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Lines and Angles Skew Lines

Lines that are not coplanar and do not intersect

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Lines and Angles Parallel Planes Planes that do not intersect

Parallel capacitors

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Lines and Angles Example 1 A D B C G E F Name two parallel lines

Name two skew lines Name two perpendicular lines A B E C D G F A

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Lines and Angles Example 2

Think of each segment in the diagram as part of a line. Which of the lines appear fit the description? a. Parallel to TW and contains V b. Perpendicular to TW and contains V c. Skew to TW and contains V d. Name the plane(s) that contain V and appear to be parallel to the plane TPQ T P U W Q V S R Line UV Line VW Line RV, line SV Plane RVW

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Lines and Angles Parallel Postulate l

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. P l There is exactly one line through P parallel to l.

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Lines and Angles 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. There is exactly one line through P perpendicular to l. P l

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Constructing Perpendicular Lines

Step 1: Draw a line, and a point not on the line

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Constructing Perpendicular Lines

Step 2:

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Constructing Perpendicular Lines

Step 3:

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Constructing Perpendicular Lines

Step 4:

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Constructing Perpendicular Lines

Step 4 – completed this is what your paper should look like

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Constructing Perpendicular Lines

Draw a line through The intersection and The point not on the line

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Constructing Perpendicular Lines

You now have two perpendicular lines – they intersect at 90o – the line you constructed passes through the point that you drew at the beginning Way to Go!

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Lines and Angles Transversal

A line that intersects two or more coplanar lines at different points

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Lines and Angles Corresponding Angles

Two angles that occupy corresponding positions 1 2 5 6

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Lines and Angles Alternate Exterior Angles

Angles that lie outside two lines on the opposite sides of the transversal 1 2 5 6

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Lines and Angles Alternate Interior Angles

Angles that lie between the two lines on opposite sides of the transversal 1 2 5 6

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Lines and Angles Consecutive Interior Angles

Also called same side interior Angles that lie between two lines on the same side of a transversal 1 2 5 6

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Lines and Angles Transversal

A line that intersects two or more coplanar lines at different points

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Lines and Angles Corresponding Angles

Two angles that are formed by two lines and a transversal and occupy corresponding positions. L O V E A H T M

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Lines and Angles Alternate Exterior Angles

Two angles that are formed by two lines and a transversal and that lie between the two lines on opposite sides of the transversal L O V E A H T M

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Lines and Angles Alternate Interior Angles

Two angles that are formed by two lines and a transversal and that lie outside the two lines on opposite sides of the transversal 1 L O V E A H T M

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Lines and Angles Consecutive Interior Angles

Two angles that are formed by two lines and a transversal and that lie between the two lines on the same side of the transversal also called “same side interior angles” L O V E A H T M

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Lines and Angles Example 3 Label your diagram.

List all pairs of angles that fit the description. a. Transversal b. Corresponding c. Alternate exterior angles d. Alternate interior angles d. Consecutive interior angles 2 3 4 1 6 7 5 8

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3.2 Proof and Perpendicular Lines

Geometry 1 Unit 3 3.2 Proof and Perpendicular Lines 29

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Proof and Perpendicular Lines

Review of 2.5 A two-column proof has numbered ____________________ on one side, and _______________ that show the logical order of an argument on the other. In the two-column proof, the reasons must use one of the following: __________________________ ______________________; a ______________________; or a ______________________ ______________________ _________________ Statements Reasons Given information A definition A property A postulate A previously proven theorem

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Proof and Perpendicular Lines

3 types of Proofs Two-Column Proof The most formal type of proof. It lists numbered statements in the left-hand column and a reason for each in the right hand column Paragraph Proof Flow Proof

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Proof and Perpendicular Lines

3 types of Proofs Two-Column Proof The most formal type of proof. It lists numbered statements in the left-hand column and a reason for each in the right hand column Paragraph Proof Describes the logical argument with sentences. It is more conversational than a two-column proof. Flow Proof

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Proof and Perpendicular Lines

3 types of Proofs Two-Column Proof The most formal type of proof. It lists numbered statements in the left-hand column and a reason for each in the right hand column Paragraph Proof Describes the logical argument with sentences. It is more conversational than a two-column proof. Flow Proof Uses the same statements as a two column proof, but the logical flow connecting the statements are connected by arrows

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Proof and Perpendicular Lines

Theorem Explanation Sketch Congruent Linear Pair Theorem If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular Adjacent Complementary Angle Theorem Perpendicular Lines Intersection Theorem

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Proof and Perpendicular Lines

Theorem Explanation Sketch Congruent Linear Pair Theorem If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular Adjacent Complementary Angle Theorem If two sides of two adjacent acute angles are perpendicular then the angles are complementary Perpendicular Lines Intersection Theorem

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Proof and Perpendicular Lines

Theorem Explanation Sketch Congruent Linear Pair Theorem If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular Adjacent Complementary Angle Theorem If two sides of two adjacent acute angles are perpendicular then the angles are complementary Perpendicular Lines Intersection Theorem If two angles are perpendicular then they intersect to form four right angles

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Proof and Perpendicular Lines

Example 1-Method 1 Given: AB = CD Prove: AC = BD A C B D Statements Reasons 1. 2. 3. 4. Teacher edition Page 137 Example 1

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Proof and Perpendicular Lines

Example 1- Method 2 Given: AB = CD Prove: AC = BD A C B D Teacher edition Page 137 Example 1

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Proof and Perpendicular Lines

2 1 Example 2- Method 1 Given: BA perpendicular to BC Prove: 1 and 2 are complementary Statements Reasons 1. 2. 3. 4. 5. 6. Teacher edition page 137 example 2

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Proof and Perpendicular Lines

2 1 Example 2- Method 3 Given: BA perpendicular to BC Prove: 1 and 2 are complementary Teacher edition page 137 example 2

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Proof and Perpendicular Lines

Example 3- Method 1 Two Column Proof Given: 5 and 6 are a linear pair 6 and 7 are a linear pair Prove: 5 z 7 Statements Reasons

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Proof and Perpendicular Lines

Example 3- Method 2 Paragraph Proof

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Proof and Perpendicular Lines

Example 3- Method 3 Flow Chart Proof

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3.3 Parallel Lines and Transversals

Geometry 1 Unit 3 3.3 Parallel Lines and Transversals

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Parallel Lines and Transversals

Activity: Measuring angles of parallel lines and their transversals Objective: Discover the relationships between the angles of parallel lines and their transversals Question: What is the relationship between the angles and the lines? Step 1: Construct a segment Step 2: Construct 2 parallel lines crossing that segment Step 3: Number the angles 1 – 8 Step 4: Measure each angle with a protractor, write that measure on the figure Step 5: Write, in paragraph form, the relationships you see

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Parallel Lines and Transversals

Step 1: Construct a segment

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Parallel Lines and Transversals

Construct 2 parallel lines crossing that Segment

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Parallel Lines and Transversals

Step 3: Number the angles 1 – 8

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Parallel Lines and Transversals

Step 4: Measure each angle with a protractor, write that measure on the figure _____o _____o _____o _____o _____o _____o _____o _____o

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Parallel Lines and Transversals

Step 5: Write, in paragraph form, the relationships you see

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Parallel Lines and Transversals

Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 1 z 2

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Parallel Lines and Transversals

Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4 3 z 4

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Parallel Lines and Transversals

Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6 m5 + m2 = 180°

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Parallel Lines and Transversals

Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8 7 z 8

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Parallel Lines and Transversals

Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. h k j is perpendicular to k

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Parallel Lines and Transversals

3 1 6 8 7 5 2 4 q p Example 1 Given: p || q Prove: m1 + m2 = 180° Statements Reasons 1. 2. 3. 4. Teacher edition page 144 example 1

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Parallel Lines and Transversals

Solve for x = x x = 150 3. 12x = x + 1 = 151

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Parallel Lines and Transversals

5. (2x + 1) = (7x + 15) = 81

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Parallel Lines and Transversals

X = 15 4. X = 75 X = 75 5. X = 75 X = 4.5 6. X = 8 Answers

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Parallel Lines and Transversals

Example 2 Given that m5 = 65°, find each measure. Tell which postulate or theorem you used to find each one. a. b. c d. 6 7 5 9 p q 8

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Parallel Lines and Transversals

Example 3 How many other angles have a measure of 100°? AB || CD AC || BD A B C 100° D 7

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Parallel Lines and Transversals

Example 4 Use properties of parallel lines to find the value of x. (x – 8)° 72° 116°

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Parallel Lines and Transversals

Example 5 Find the value of x. (x – 20)° x° 70° 65°

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3.4 Proving Lines are Parallel

Geometry 1 Unit 3 3.4 Proving Lines are Parallel

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Proving Lines are Parallel

Corresponding Angle Converse Postulate If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel j k j || k

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Proving Lines are Parallel

Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent then the lines are parallel. 3 1 If 1 z 3, then j || k j k

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Proving Lines are Parallel

Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the two lines are parallel j k 1 2 If m1 + m2 = 180°, then j || k.

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Proving Lines are Parallel

Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. j k 4 5 If 1 z 3, then j || k.

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Proving Lines are Parallel

m 1 2 p q Example 1 Given: m p, m q Prove: p || q Statements Reasons 1. 2. 3. 4. Teacher edition page 151 example 1

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Proving Lines are Parallel

4 B D 6 5 C Example 2 Given: 5 z 6, 6 z 4 Prove: AD || BC

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Proving Lines are Parallel

Example 3 Find the value of x that makes m || n. (2x + 1)° (3x – 5)° m n 6

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Proving Lines are Parallel

Example 4 Is AB || DC? Is BC || AD? 155° 65° 40° 115° D C A B Teacher edition page 152 example 5

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Proving Lines are Parallel

Example 5 When the lines r and s are cut by a transversal, 1 and 2 are same side interior angles. If m1 is three times m2, can r be parallel to line s? Explain Yes, if m2 = 45.

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Proving Lines are Parallel

The sum of the interior degrees of a triangle is ___180°___. The sum of the degrees of a pair of complementary angles is ___90°___. The sum of the degrees of a pair of supplementary angles is ___180°___. The sum of the degrees of consecutive interior angles if transversal crosses parallel lines is ___180°___. Parallel lines have slopes that are congruent. The students need to fill theirs in, then they can have these notes 74

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3.5 Using Properties of Parallel Lines

Geometry 1 Unit 3 3.5 Using Properties of Parallel Lines

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Using Properties of Parallel Lines

Lines Parallel to a Third Line Theorem If two lines are parallel to the same line, then they are parallel to each other. If p || q and q || r, then p ||r p q r

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Using Properties of Parallel Lines

Lines Perpendicular to a Third Line Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n p If m p and n p, then m ||n

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Using Properties of Parallel Lines

1 2 4 3 Example 1 Given: r || s and s || t Prove: r || t Statements Reasons 1. 2. 3. 4. 5. 6.

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Using Properties of Parallel Lines

Example 2 The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe. S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 Describe your thinking as you prove that S1 and s13 are parallel

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Using Properties of Parallel Lines

Example 3 You are building a CD rack. You cut the sides, bottom, and top so that each corner is composed of two 45o angles. Prove that the top and bottom front edges of the CD rack are parallel. Given: Prove: Angle Addition Postulate Given Angle Addition Postulate Given Substitution Property Substitution Property Definition of a right angle Substitution Property Definition of perpendicular lines Definition of perpendicular lines In a plane, 2 lines ⊥ to the same line are ║

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3.6 Parallel Lines in the Coordinate Plane

Geometry 1 Unit 3 3.6 Parallel Lines in the Coordinate Plane

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Parallel Lines in the Coordinate Plane

The slope of a line is usually represented by the variable m. Slope is the change in the rise, or vertical change, over the change in the run, or horizontal change.

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Parallel Lines in the Coordinate Plane

Example 1 Cog railway A cog railway goes up the side of a Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section. rise =________ run =________ slope = =

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Parallel Lines in the Coordinate Plane

Example 2 The cog railway covers about 3.1 miles and gains about 3600 feet of altitude. What is the average slope of the track?

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Parallel Lines in the Coordinate Plane

Example 3 Find the slope of a line that passes through the points (0,6) and (5,2). x1= y1 = x2= y2 = slope = =

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Parallel Lines in the Coordinate Plane

Slopes of Parallel Lines Postulate In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

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Parallel Lines in the Coordinate Plane

Example 4 Find the slope of each line.

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Parallel Lines in the Coordinate Plane

Example 5 Find the slope of each line. Which lines are parallel?

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Parallel Lines in the Coordinate Plane

In algebra, you learned that you can use the slope m of a non-vertical line to write the equation of the line in slope intercept form. slope y-intercept y = mx + b

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Parallel Lines in the Coordinate Plane

Example 6 y = 2x y = -½x – 3 What is the slope? What is the y-intercept? Do you have enough information to graph the line?

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Parallel Lines in the Coordinate Plane

Example 7 Write the equation of a line through the point (2,3) with a slope of 5. Step 2: Substitute the values above into the equation y = mx + b. SOLVE FOR b. Step 1: x = y = m = ___________ = (_______) (_________) + b y m x Step 3 Rewrite the equation of the line in slope-intercept form, using m and b from your solution to the equation above y = _______ x + _________ m b

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Parallel Lines in the Coordinate Plane

Example 8 Line k1 has the equation y = 2/5 x + 3. Line k2 is parallel to k1 and passes through the point (-5, 0). Write the equation of k2. Y = 2/5 x + 2

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Parallel Lines in the Coordinate Plane

Example 9 Write an equation parallel to the line What do you have to keep the same as the original equation? What did you change? Page 170 #45

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Parallel Lines in the Coordinate Plane

Example 10 A zip line is a taut rope or a cable that you can ride down on a pulley. The zip line below goes from a 9 foot tall tower to a 6 foot tower 20 feet away. What is the slope of the zip line?

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3.7 Perpendicular Lines in the Coordinate Plane

Geometry 1 Unit 3 3.7 Perpendicular Lines in the Coordinate Plane

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Perpendicular Lines in the Coordinate Plane

Activity: Investigating Slope of Parallel Lines You will need: an index card, a pencil and the graph below. Place the index card at any angle – except straight up and down – on the coordinate plane below, with a corner of the card placed on an intersection. Use the edge of the card like a ruler, draw to lines, that will intersect at the corner of the card that lines up with the intersection on the coordinate plane. Name the lines ‘o’ and ‘p’. Move the index card and select, then label, two points on line. These should be points where the line goes directly through an intersection on the coordinate plane. Using the equation for slope, find the slope of each line. Activity can be found on page 172 of the textbook I need to have a large piece of notepad backing to use as an index card. If it is white I can put the lines on it. Have the student supplies ready: Straight edge, index card, pencil 96

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Perpendicular Lines in the Coordinate Plane

Example 1 Label the point of intersection And the x-intercept of each line. Find the slope of each line. Multiply the slopes. Question: What do you notice? Look at the activity from the start of class. Multiply the slopes of those lines. What is true about the product of the slopes of perpendicular lines?

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Perpendicular Lines in the Coordinate Plane

Example 2 Decide whether and are perpendicular. A D C B What is the product of the slopes of perpendicular lines? __________________________ Are these lines perpendicular? ____________

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Perpendicular Lines in the Coordinate Plane

Example 3 A B C D Decide whether and are perpendicular. What is the product of the slopes of perpendicular lines? __________________________ Are these lines perpendicular? ____________

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Perpendicular Lines in the Coordinate Plane

Example 4 Decide whether these lines are perpendicular. line h: line j: What is the product of the slopes of perpendicular lines? __________________________ Are these lines perpendicular? ____________

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Perpendicular Lines in the Coordinate Plane

Example 5 Decide whether these lines are perpendicular. line r: line s: What is the product of the slopes of perpendicular lines? __________________________ Are these lines perpendicular? ____________

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Perpendicular Lines in the Coordinate Plane

Slope of a line Slope of the perpendicular line Product of the slopes 7 4 -1

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Perpendicular Lines in the Coordinate Plane

Example 6 Line l1 has equation y = -2x +1. Find an equation for the line, l2 that passes through point (4, 0) and is perpendicular to l1. What is the slope of l1? ______________ What form is l1 written in? _______________________________ What does the slope of l2 need to be if they are perpendicular? __________ With the point known (4, 0) , (it is in the original question), and the slope known for l2 , Can you find the y-intercept, b, of the perpendicular line? x = ________________ y = ________________ What is the equation of the perpendicular line? m = _______________ b = ______________

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Perpendicular Lines in the Coordinate Plane

Example 7 Line g has equation y = 3x – 2. Find an equation for the line h that passes through point (3, 4) and is perpendicular to g. What is the slope of g? ______________ What form is g written in? _______________________________ What does the slope of h need to be if they are perpendicular? __________ With the point known (3, 4), (it is in the original question), and the slope known for h , Can you find the y-intercept, b, of the perpendicular line h? x = ________________ y = ________________ m = ________________ What is the equation of line h? b = ______________

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Perpendicular Lines in the Coordinate Plane

Example 8 What is the equation of a line a, which passes through point (-2, 0) that is perpendicular to line z, What is the slope of z? ______________ What form is z written in? _______________________________ What does the slope of a need to be if they are perpendicular? __________ With the point known (-2, 0) , (it is in the original question), and the slope known for z , Can you find the y-intercept, b, of the perpendicular line? x = ________________ y = ________________ m = ________________ b = ________________ What is the equation of the perpendicular line? ______________________

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Perpendicular Lines in the Coordinate Plane

Example 9 . Find an equation for the line s that passes through point (3, 1) and is perpendicular to g. Line g has equation What is the slope of g? ______________ What form is g written in? _______________________________ What does the slope of s need to be if they are perpendicular? With the point known (3, 1) , what is the equation of the perpendicular line s? x = ________________ y = ________________ m = ________________ b = ________________

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