1.3: Parallel and Perpendicular Lines
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Lines and Angles Investigation 3-1 Investigation 3-2 |
Finish Lines and Angles Investigation 3-3 Start Properties of Parallel Lines |
Finish Properties of Parallel Lines Investigation 3-4 |
Quiz 1 Start Proving Lines Parallel |
Finish Proving Lines Parallel Investigation 3-5 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Properties of Perpendicular Lines |
Quiz 2 Start Parallel and Perpendicular Lines in the Coordinate Plane |
Finish Parallel and Perpendicular Lines in the Coordinate Plane | The Distance Formula |
Quiz 3 Start Review of Chapter 3 |
Day 11 | Day 12 | Day 13 | ||
Review Chapter 3 | Chapter 3 Test |
Finish testing (if needed) Start Chapter 4 |
Lines and Angles
Goal
Students will be introduced to parallel, perpendicular, and skew lines in this lesson. Transversals and the angles formed by such are also introduced.
Teaching Strategies
To introduce skew lines, use two pencils and hold them in the air, like skew lines. This will help students visualize that skew lines are in different planes. Use Example 1 as a jumping off point and find more parallel, skew, and perpendicular lines, other than those listed in the solution.
Investigation 3-1 is a useful tool to help students visualize the Parallel Line Postulate. You can decide whether you want to do this activity individually or teacher-led. If you decide to make it a teacher-led demonstration, consider using the overhead and folding a transparency rather than patty paper.
Explore the similarities and differences between the Parallel Line Postulate and the Perpendicular Line Postulate. You can use a Venn diagram to aid in this discussion.
Investigations 3-2 and 3-3 demonstrate the Perpendicular Line Postulate. Guide students through these constructions using a whiteboard compass. If you do not have access to a whiteboard compass, tie a piece of string around your marker and use your finger as the pointer. Make the string at least 8 inches long. If you have access to an LCD screen or computer in the classroom, the website listed in these investigations (www.mathisfun.com) has a great animation of these constructions.
When introducing the different angle pairs, discuss other ways that students can identify the relationships. For example, corresponding angles are in the “same place” on lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*}. Draw a large diagram, like the ones to the left, and find all the linear pairs, vertical angles, corresponding angles, alternate interior angles, alternate exterior angles, and same side interior angles. Use two different pictures to show the different orientations and that the lines do not have to be parallel to have these angle relationships. Explain that vertical angles and linear pairs only use two lines; however these new angle relationships require three lines to be defined. Use Examples 4 and 5. You can also expand on Example 5 and ask:
d) What is a same side interior angle to \begin{align*}\angle 6\end{align*}? \begin{align*}(\angle 7)\end{align*}
e) What is a corresponding angle to \begin{align*}\angle 8\end{align*}? \begin{align*}(\angle 6)\end{align*}
f) What is an alternate exterior angle to \begin{align*}\angle 8\end{align*}? \begin{align*}(\angle 1)\end{align*}
Students might wonder why there is no same side exterior relationship. You can explain that it does exist (\begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*} in the second picture), but not explicitly defined.
Real Life Connection
Discuss examples of parallel, skew, and perpendicular lines and planes in the real world. Examples could be: a table top and the floor (parallel planes), the legs of the table and the table top or floor (perpendicular planes), or the cables in the Brooklyn Bridge (skew lines).
Properties of Parallel Lines
Goal
In this section we will extend 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.
Teaching Strategies
If you discuss the Know What? at the beginning of the lesson, students will only know how to find angle measures that are vertical or a linear pair with \begin{align*}\angle FTS\end{align*} and \begin{align*}\angle SQV\end{align*}. Revisit this at the end of the lesson and use the new-found postulates and theorems to find corresponding angles, alternate interior angles, alternate exterior angles, and same side interior angles. You could also test that the angles in \begin{align*}\Delta FST\end{align*} add up to \begin{align*}180^\circ\end{align*}.
Discuss Example 1 as a refresher on where the corresponding angles are and now, if the lines are parallel, which angles are congruent. Then, guide students through Investigation 3-4. If you prefer, you can do the investigation before introducing the Corresponding Angles Postulate and Example 1, so that students can discover this postulate on their own.
When introducing the alternate interior angles, alternate exterior angles, and same side angles use the results that the students found in Investigation 3-4. Let them draw their own conclusions about all the angles and angle measures. They already know the names of the relationships, so then ask students if any other relationships that they learned in the previous lesson are equal. This will allow you to explain the Alternate Interior Angles Theorem and Alternate Exterior Angles Theorem.
For the Same Side Interior Theorem, ask students which angles are same side interior and then ask what the relationship is. Students should notice that the two angles add up to \begin{align*}180^\circ\end{align*}.
Students may notice that there are other angles that are supplementary or congruent. Encourage students to make these observations even though there are no explicit theorems.
Reinforce to students that these theorems do not apply to parallel lines. Demonstrate this by drawing two non-parallel lines and a transversal. Measure all angles. Students will see the alternate interior angles, corresponding angles, and alternate interior angles are not congruent, nor are the consecutive interior angles supplementary.
Proving Lines Parallel
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
Let students rewrite theorems, postulates, and properties symbolically and using pictures. The converses and the Parallel Lines Property in this section are written in this way to help students understand them better. Continue to use this strategy throughout the text.
Teaching Strategies
Review the concept of a converse from Chapter 2. Then, introduce the converse of the Corresponding Angles Postulate and ask students if they think it is true. Investigation 3-5 is one way to show students that converse of the Corresponding Angle Postulate must be true. Remind students that Postulates do not need to be proven true. However, it is always nice to show students why.
Decide if you would like Investigation 3-5 to be student driven or teacher-led. As a teacher-led investigation, this activity will show students that the converse of the Corresponding Angles Postulate is true. As a student driven activity, encourage students to work in pairs. Before starting, demonstrate how to copy an angle (Investigation 2-2) and then allow students to work through the investigation. Expect it to take 15 minutes.
Investigation 3-5 can also be redone such that students copy the angle and place it in the location of the alternate interior or alternate exterior angle location.
Additional Example: Put the reasons for the proof in the correct order.
Solution: The correct order is C, F or H, G, A, F or H, B, D, E.
Properties of Perpendicular Lines
Goal
This section further explains the properties of perpendicular lines and how they affect transversals.
Perpendicular Lines Investigation
On the whiteboard, draw a linear pair such that the shared side is perpendicular to the non-adjacent sides (see picture). Ask students what the angle measure of each angle is and what they add up to. Once students arrive at the correct conclusion, reiterate that a congruent linear pairs is the same as a linear pair formed by perpendicular lines and the angles will always be \begin{align*}90^\circ\end{align*}.
Second, extend \begin{align*}\overrightarrow{BD}\end{align*} to make a line and add a parallel line to \begin{align*}\overleftrightarrow{AC}\end{align*} (see picture). Now, discuss the effects of a perpendicular transversal. Steer this discussion towards Theorems 3-1 and 3-2 and see if the converses of either of these theorems are true. Again, reiterate that all eight angles in this picture will be \begin{align*}90^\circ\end{align*}.
Prove Move
In a proof involving perpendicular lines, the following three steps must be included to say that the angles are \begin{align*}90^\circ\end{align*}.
- Two lines are perpendicular (usually the given)
- The angles formed are right angles (definition of perpendicular lines)
- The angles formed are \begin{align*}90^\circ\end{align*} (definition of right angles)
To say that two right angles are congruent, the following steps must be included:
- Two lines are perpendicular (usually the given)
- The angles formed are right angles (definition of perpendicular lines)
- The right angles are congruent/equal (all right angles are congruent or congruent linear pairs)
This can seem repetitive to students because many of them will feel that it should be inferred that if two lines are perpendicular, then all the angles will be equal/right/\begin{align*}90^\circ\end{align*}. This is not the case. Explain that they are writing a proof to someone who knows nothing about math or the definitions of perpendicular lines or right angles. They cannot assume that it is the math teacher that is reading each proof. See Example 3 in this section as an example of these steps.
Additional Example: Algebra Connection Solve for \begin{align*}x\end{align*}.
Solution: The three angles add up to \begin{align*}180^\circ\end{align*}, or \begin{align*}(5x - 6)^\circ\end{align*} and \begin{align*}(4x + 15)^\circ\end{align*} add up to \begin{align*}90^\circ\end{align*}.
\begin{align*}(5x - 6)^\circ + (4x + 15)^\circ & = 90^\circ\\ 9x + 9^\circ & = 90^\circ\\ 9x & = 81^\circ\\ x & = 9^\circ\end{align*}
Parallel and Perpendicular Lines in the Coordinate Plane
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. Then, we will apply the concepts learned in this chapter to the coordinate plane.
Real Life 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. Discuss synonyms for slope: grade, slant, incline. Then, have students brainstorm further. Relate this back to the Know What? for this section. Explain how the slope and the grade are related. For example, in the Know What? the slope of the California Incline is \begin{align*}\frac{3}{25}\end{align*} (see FlexBook). The grade of this incline is a percentage, so \begin{align*}\frac{3}{25} \cdot 100 \% = 12 \%\end{align*}.
Relevant Review
Before discussing standard form for a linear equation, make sure students can clear fractions.
Additional Example: Solve the following equations for \begin{align*}x\end{align*}.
a) \begin{align*}\frac{5}{6} x = 30\end{align*}
b) \begin{align*}\frac{2}{3} x+3 = 9\end{align*}
c) \begin{align*}\frac{7}{6} x + \frac{1}{4} = \frac{1}{2}\end{align*}
Solution: Multiply each number by what the lowest common denominator would be.
a) \begin{align*}6 \cdot \left( \frac{5}{6} x = 30 \right )\!\\ 5x = 180\!\\ x = 36\end{align*}
b) \begin{align*}3 \cdot \left( \frac{2}{3} x + 3 = 9 \right )\!\\ 2x + 9 = 27\!\\ 2x = 18\!\\ x = 9\end{align*}
c) \begin{align*}12 \cdot \left( \frac{1}{3} x + \frac{1}{4} = \frac{3}{2} \right )\!\\ 4x + 3 = 18\!\\ 4x = 15\!\\ x = 3.75\end{align*}
Of course, there are other ways to approach these problems, but this method of clearing fractions will help students change slope-intercept form into standard form. Show students these alternate ways of solving these problems and let them decide which is easier. Then, apply both to changing a slope-intercept equation into standard form.
Additional Example: Change \begin{align*}y =\frac{3}{4} x - \frac{1}{2}\end{align*} into standard form using two different methods.
Solution: Method #1: Clear fractions
\begin{align*}& 4 \cdot \left( y = \frac{3}{4} x- \frac{1}{2} \right )\\ & 4y = 3x - 2\\ & -3x + 4y = -2\\ & \text{or} \quad 3x - 4y = 2\end{align*}
Method #2: Find a common denominator
\begin{align*}\frac{4}{4} y & = \frac{3}{4} x - \frac{2}{4}\\ - \frac{3}{4} x + \frac{4}{4} y & = - \frac{2}{4}\\ -3x + 4y & = -2\\ 3x - 4y & = 2\end{align*}
Students generally want to avoid fractions, so Method #1 should seem for desirable to them.
Teaching Strategies
When discussing the rise over run triangles, begin making the right triangle connection to students, demonstrating that every rise/run triangle will form a \begin{align*}90^\circ\end{align*} angle. When students are asked to find the distance between two points, they can use the Pythagorean Theorem.
Have students trace the top and bottom edges of a ruler onto a coordinate plane (use graph paper). 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. Recall that this is an easy way to draw parallel lines (Investigation 3-4).
The Distance Formula
Goal
Students are introduced to the Distance Formula and its applications.
Teaching Strategies
Students should be familiar with the Pythagorean Theorem and possibly even the Distance Formula from previous classes. A quick review of the Pythagorean Theorem might be helpful. The reason neither are derived at this time is because we have not yet introduced triangles or the properties of right triangles, which is in Chapter 9. At this point, students can accept both formulas as true and they will be proven later.
Even though the Distance Formula is written \begin{align*}d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\end{align*}, where \begin{align*}(x_1, y_1)\end{align*} is the first point and \begin{align*}(x_2, y_2)\end{align*} is the second point the order does not matter as long as the same point’s coordinate is first. So, if students prefer, they can use \begin{align*}d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*}. Using Example 1, show students that they can use \begin{align*}d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*} and the answer will be the same.
\begin{align*}d = \sqrt{(4-(-10))^2 + (-2 - 3)^2} = \sqrt{14^2 + (-5)^2} = \sqrt{196 + 25} = \sqrt{221}\end{align*}
Finding the distance between two parallel lines can be quite complicated for some students because there are so many steps to remember. For this text, we have simplified this subsection to only use lines with a slope of 1 or -1. Reinforce the steps used to find the distance between two parallel lines from Example 5.
Additional Example: Find the shortest distance between \begin{align*}y = x + 4\end{align*} and \begin{align*}y = x - 4\end{align*}.
Solution: First, graph the two lines and find the \begin{align*}y-\end{align*}intercept of the top line, which is (0, 4).
The perpendicular slope is -1. From (0, 4) draw a straight line with a slope of -1 towards \begin{align*}y = x - 4\end{align*}. This perpendicular line intersects \begin{align*}y = x - 4\end{align*} at (4, 0). Use these two points to determine how far apart the lines are.
\begin{align*}d & = \sqrt{(0 - 4)^2 + (4 - 0)^2}\\ & = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66 \ units\end{align*}