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1.3: Parallel and Perpendicular Lines

Created by: CK-12

Lines and Angles

Pacing: This lesson should take one to one and one-half class periods

Goal: Students will be introduced to parallel, perpendicular, and skew lines in this lesson. Transversals and the angles formed by such are also introduced.

While example 1 shows students that it is possible for streets to be perpendicular or parallel, challenge students to find roads that begin as parallel then intersect (or begin perpendicular and then become parallel).

Visualization! Show a map that has zoned roads in the fashion of example 1, perhaps in rural Ohio or Kansas. Encourage students to compare this map with one of Atlanta, New York City, or Chicago.

Physical Model! Give each student a cube; it could be a die, box, etc. When discussing the definition of skew lines, have students point to the lines you are referencing. This provides students a physical model in addition to allowing you to do a quick assessment.

In Class Activity! Have students trace the top and bottom of a ruler to create pair of parallel lines. Then construct an oblique line crossing through both parallel lines. Instruct students to number all 8 angles and color code each of the terms found on page 131. Ask students to summarize each definition and how it relates to another angle in the diagram.

Vocabulary! There are five ways to determine parallel lines: showing congruent corresponding angles or congruent alternate interior angles, proving same side interior angles are supplementary, showing both lines are parallel to a third line or by showing both lines are perpendicular to the same line.

Have students prove the following lines are parallel using one of the above methods.

Challenge! There are 13 red and white alternating stripes on the United States Flag. Explain why the top red stripe must be parallel to the third white stripe. Answer: Using the syllogism property and the idea of parallel lines, since each lines is parallel to the one before it, then the first red stripe must be parallel to the third white stripe. Also, the flag may look weird if the stripes were not parallel!


Parallel Lines and Transversals

Pacing: This lesson should take one class period

Goal: The textbook further extends the notion of transversals and parallel lines to illustrate the corresponding angles postulate and the alternate interior angles postulate. Additional theorems and postulates are proven in this lesson.

Use the in-class activity from the previous lesson as a refresher and guide to the lesson opener. This lesson provides several key theorems: corresponding angles postulate and alternate interior angles postulate. Students have experienced the definitions of these angles in previously lessons and have also been given brief introductory proofs. The goal of this lesson is to use these notions to prove alternate exterior angles are congruent and consecutive interior angles of parallel lines are supplementary.

In-Class Activity! Divide your class into six sections of pairs. Provide enough copies of the Corresponding Angles Postulate, its converse, the Alternate Interior Angles Postulate, its converse, and the Alternate Exterior Angles Postulate and its converse. Instruct each pair to prove their theorem, and then group homogenous sections in order to discuss the results. Taking a pair of each theorem and its respective converse to form a team of four, have the students discuss the proofs. As an assignment, have the groups create a visual poster of the proof of the theorem, the converse and its respective proof.

In-Class Activity! Demonstrate to students that these theorems do not apply to non-parallel lines. Each student should create two non-parallel lines and a transversal. Label the 8 angles formed, having students measure all angles. Students will see the alternate interior angles, corresponding angles, and vertical angles are not congruent, nor are the consecutive interior angles supplementary.

Proving Lines Parallel

Pacing: This lesson should take one class period

Goal: The converse of the previous lesson’s theorems and postulates are provided in this lesson. Students are encouraged to read through this lesson and follow along with the proofs.

Vocabulary! The Parallel Lines Property can be stated, “If line l is parallel to line m, and line m is parallel to line n, then lines l and n are also parallel.” Ask students to write the converse of this property and determine if it is true. If students determine the converse false, have them provide a counterexample.

Slopes of Lines

Pacing: This lesson should take one to two class periods

Goal: Students should feel comfortable with slopes and lines. Use this lesson as a review of key concepts needed to determine parallel and perpendicular lines in the coordinate plane.

Fun Fact! The word slope comes from the Middle English word sloop, meaning at an angle.

Most students have experienced slope in Algebra. However, students rarely have seen the delta symbol when determining slope. Use the following to stress the connection to pre-calculus:

\text{Slope}	= \text{rate of change}\ \frac{\text{Rise}}{\text{run}} = \frac{\triangle y}{\triangle x} = \frac{y_2 - y_1}{x_2 - x_1}

Vocabulary Connection! Ask students to brainstorm the many different interpretations of the word slope. Apply these to real world situations such as the slope of a mountain, or the part of a continent draining into a particular ocean (Alaska’s North Slope), the slope of a wheelchair ramp, etc.

Include the synonym for slope – grade. Students should come up with more examples using this word.

Real Life Connection! Eldred Street in Los Angeles, California has a grade of 33 \%, Baldwin Street In Dunedin, New Zealand boasts a 35 \% incline, and Banton Avenue in Pittsburg, Pennsylvania officially measurers 37 \%! Have students reconstruct the incline of these streets using the rise over run notion of slope.

When discussing the rise over run triangles, begin making the right triangle connection to students, demonstrating that every rise/run triangle will form a 90\;\mathrm{degree} angle. When students are asked to find the distance between two oblique points, the distance formula is a derivation of Pythagorean’s Theorem.

Fun Tip! To illustrate why vertical lines have an undefined slope, ask for volunteers for the following demonstration.

To illustrate a horizontal line, run a length of masking tape on your floor. Ask a student to walk over the line. Onlookers should see the student is walking at a zero incline (or slope).

To illustrate an oblique line, lay a 2” by 4” piece of wood on top of a chair, or something sturdy, creating a 2 \% - 3 \% incline. Ask a student to walk up the hill. Relate the percentage to a fraction, relating rise over run.

To illustrate a vertical, ask students to place their feet on a wall, lying parallel to the floor. Instruct the students to walk up the wall in this position, similar to what Spiderman can do. Students will tell you this is impossible! Dividing by zero is also impossible, thus illustrating why vertical lines have undefined (impossible) slopes.

To further demonstrate perpendicular slopes, use the formula to your advantage \mathrm{slope} = \frac{(y_2 - y_1)} {(x_2 - x_1)} so the slope of the line perpendicular must be -\frac{(x_2- x_1)} {(y_2- y_1 )}

Students may find that making an xy T-chart is an easy way to construct a line. Whichever your preference, make sure students can see a variety of ways to begin to solve a problem.

Equations of Lines

Pacing: This lesson should take one to two class periods

Goal: This lesson reinforces key concepts learned during Algebra to prepare students for geometric connections. Students will review slope intercept form, standard form for a linear equation, and introduce equations for parallel and perpendicular lines.

Alternative Ways to Think! An alternative way to express slope-intercept form is y = b + mx. In some situations, this form will make much more sense to students that the “original” way. You could also try substituting m with a. Linear regressions found on graphing calculators often use this formula: y = ax + b. Students tend to feel frustrated with the constant replacement of variables. Determine which variable appear in later textbooks and feel free to use that variable from the beginning.

Inquiry based learning! Have students trace the top and bottom edges of a ruler onto a coordinate plane. Ask students to determine the equations for each line and compare the results. Students should notice that, if done correctly, the slopes will be equal. Follow this activity with the equations for parallel lines section.

Use the graph provided in example 3 for this activity. Once students have found the slope of the graphed equation, incorporate the previous lesson’s concept to find the slope of the line perpendicular. Ask students to place a dot anywhere on the y-axis and use the newly found slope to construct a line. Using a projection device, ask several students to graph their equations. Students should come to the conclusion there are infinitely many lines perpendicular in a coordinate plane.

Algebra Review! Before discussing standard form for a linear equation, make sure students can clear fractions, something that is widely forgotten. During the warm up or opening set, ask students to clear the following fractions:

\frac{5}{6} x = 30  && \frac{2}{3} x + 3 = 9 && \frac{7}{6} x + \frac{1}{4} = \frac{1}{2}

This will allow you to determine the level of which you may have to re-teach before moving on to standard form.

Why do I need this? Many students ask why they need to know standard form. One reason is because many real life problems take form in a linear combination (standard form) approach. For example, one cheeseburger is \$1.69 and a small French fry is \$1.39. How many of each can you buy with \$15.25, excluding tax? The equation begins in standard form and many students will rewrite this into slope-intercept form.

Perpendicular Lines

Pacing: This lesson should take one class period

Goal: Students will extend their learning to include angle pairs formed with perpendicular lines. The properties presented in this lesson hold only for perpendicular lines pairs.

Extension: Connect the introduction of this lesson with circles. Draw a circle around the origin of the Cartesian plane found on page 180. Students should already know the sum of the degrees of a circle (360^\circ). Demonstrate to students what angles are formed when the axes split the circle into four congruent segments. This will aid students when discussing circles in Chapter 9.

Example 3 can also be solved using the notion of vertical angles. To find m\angle WHO = 90^\circ, instruct students to visualize these two angles as being vertical angles.

Extension: Extend example 4 to review vertical angles. Turn ray L into a line and have students apply the Vertical Angle Theorem to the angles found in quadrant four. This will help keep Vertical Angles fresh in students’ minds.

Take time to review how perpendicular angles are formed – the product of slopes of perpendicular lines must equal -1. Continuous review will help students prepare for the test.

Why Is This So? Students may question why the lesson is entitled “Perpendicular Lines” when most of the material presented is regarding angles. Explain to students that these properties only hold for lines intersecting at 90\;\mathrm{degree} angles. To further illustrate, have students construct non-perpendicular lines and attempt to draw in complementary adjacent angles.

Perpendicular Transversals

Pacing: This lesson should take one class period

Goal: The goal of this lesson is to introduce students to the concept that parallel lines are equidistant from each other and to prove lines parallel using the converse of the perpendicular to parallels theorem.

As students work through example 4, ask them to look at the slopes of the lines. Students should realize the slopes are the same, thus they will never intersect. Have students create their own property describing this concept.

Additional Example: Have students place a ruler in any direction on a coordinate plane. Then, by tracing the top and bottom of the ruler, the students will create parallel lines. Ask each student to find their equations for their personal lines. Check with a partner to see if the equations are correct.

Non-Euclidean Geometry

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to extend students’ understanding of geometry beyond parallel and perpendicular lines, angle pairs, and abstract drawings. Most students will enjoy this lesson due to the real life application. However, even if students are unfamiliar with taxicabs, extend this lesson to rural areas with roads that intersect.

History Connection! Take time to discuss Euclid during the lesson. Show the following picture of his book, Elements. Go through the first five postulates. Use the following website to gather additional information. Or, have students write mini-reports of the impact Euclid had on present day Geometry. Offer “Euclid Day,” a day of celebration on behalf of Euclid. The possibilities are endless!

Create a class discussion regarding Euclid’s 5^{\mathrm{th}} Postulate. “If two lines are cut by a transversal, and consecutive interior angles have a total measure of less than 180\;\mathrm{degrees}, then the lines will intersect on that side of the transversal.” Mathematicians tried to prove this true, thus making it a theorem as opposed to a postulate for 2000 years. Since many mathematicians did not regard this as truth, non-Euclidean geometries were founded.

Other types of non-Euclidean geometry are: spherical geometry, hyperbolic geometry and elliptic geometry. In spherical geometry, straight lines are great spheres, so any two lines meet in two points. There are also no parallel lines (think longitude lines meeting at the poles). Hyperbolic geometry satisfied all Euclid’s postulates except the parallel postulate, replacing it with “For any infinite straight line L and any point P not on it, there are many other infinitely extending straight lines that pass through P and which do not intersect L.” Elliptic geometry replaces Euclid’s parallel postulate with “through any point in the plane, there exist no lines parallel to a give line.”  

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