2.3: Parallel and Perpendicular Lines
Lines and Angles
Marking the Diagram - Sometimes students confuse the marks for parallel and congruent. When introducing them to the arrows that represent parallel lines, review the ticks that represent congruent segments. Seeing the two at the same time helps avoid confusion. Also, explaining that the arrows show that the lines or segments are going in the same direction or have the same slope may help them understand why arrows are used and thus help them remember.
When given the information that two lines or segments are perpendicular, students don’t always immediately see how to mark the diagram accordingly. They need to use the definition of perpendicular and mark one of the right angles created by the lines with a box.
Students also struggle with marking angle bisectors and midpoints (or segments bisectors). Just like the perpendicular lines, they aren’t marking the object directly. They need to mark the results, which are congruent angles or congruent segments in these cases.
Symbol Update - Students should be keeping a list of symbols and how they will be used in this class in their notebooks. Remind them to update this page with the symbols for parallel, \begin{align*}\|\end{align*}, and perpendicular, \begin{align*}\perp\end{align*}.
Construction - The parallel and perpendicular line postulates are used in construction. Constructing parallel and perpendicular lines with a compass and straightedge is a good way to give students kinesthetic experience with these concepts. Construction can also be done with computer software. To construct a parallel or perpendicular line the student will select the line they want the new line to be parallel or perpendicular to, and the point they want the new line to pass through, and chose construct. The way the programs have the students select the line and then the point reinforces the postulates.
Parallel vs Skew Lines - Students often struggle with the difference between these two scenarios. It is difficult for them to picture the skew lines in three dimensions and even harder for them to draw them. Use concrete examples of skew and perpendicular lines (objects in the classroom) to help them visualize the difference. Also, reinforce the requirement for the lines to be coplanar in the definition of parallel lines. If it is not specifically stated that the lines are coplanar, then the lines may be skew.
Naming the Angle Pairs Formed when a Transversal Intersects Two Lines - Students often struggle with memorizing the names of the angle pairs. It helps to go through the names and explain how the names truly describe the angle pairs. Using colored pencils or highlighters to indicate the space in the interior of the parallel lines and the space in the exterior of the parallel lines helps distinguish between alternate exterior and alternate interior. Discussing what alternate vs same side means will help distinguish between these pair of interior angles. Corresponding angles are in the same location with respect to the transversal and the line. Using words like above or below and to the right or left helps students locate the pairs of corresponding angles, particularly in situations where the lines are not parallel and the angles don’t “match”.
Properties of Parallel Lines
The Parallel Hypothesis - So far seven different pairs of angles that may be supplementary or congruent have been introduced. All seven of these pairs are used in the situation where two lines are being crossed by a transversal forming eight angles. Some of these pairs require the two lines to be parallel and some do not. Students sometimes get confused about when they need parallel lines to apply a postulate or theorem, and if a specific pair is congruent or supplementary. A chart like the one below will help them sort it out.
Type of Angle Pair | Relationship | |
---|---|---|
Do Not Require Parallel Lines |
Linear Pairs Vertical Angles |
Supplementary Congruent |
Parallel Lines Required |
Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Consecutive Exterior Angles |
Congruent Congruent Congruent Supplementary Supplementary |
Encourage students who are really struggling to use common sense when deciding whether a pair of angles is supplementary or congruent. When parallel lines are intersected by a transversal, any pair of two angles will be either congruent or supplementary. For students at this level, it is advisable to make these drawings accurate (i.e. make the lines look parallel if they are parallel) so that students can practice using common sense. If the angles look congruent, they are congruent and if one is clearly obtuse and the other is clearly acute, then they are supplementary. This goes against the idea that it is not advisable to encourage students to rely on the appearance of a diagram but for very low level students, this can really help them. In addition, practicing common sense will help them in the real world and in application problems.
Patty Paper Activity - When two lines are intersected by a transversal eight angles are formed in two sets of four. When the lines are parallel, the two sets of four angles are exactly the same. To help students see this relationship, have them darken a set of parallel lines on their binder paper a few inches apart (or they can use the two sides of a ruler to make the parallel lines) and draw a transversal through the parallel lines. Next, they should trace one set of four angles on some thin paper (tracing paper or patty paper). When they slide the set of four angles along the transversal they will coincide with the other set of four angles. Have them try the same thing with a set of lines that are not parallel. This will help students find missing angle measures quickly and remember when they can transfer numbers down the transversal. It does not help them learn the names of the different pairs of angles which is important for communicating with others about mathematical concepts and for writing proofs.
Proving Lines Parallel
When to Use the Converse - It takes some experience before most students truly understand the difference between a statement and its converse. They will be able to write and recognize the converse of a statement, but then will have a hard time deciding which one applies in a specific situation. Tell them when you know the lines are parallel and are looking for angles, you are using the original statements; when you are trying to decide if the lines are parallel or not, you are using the converse.
Proofs of Converse Theorems - The textbook includes a proof of the Converse of the Alternate Interior Angles Theorem. It would be helpful to prove each of the other theorems to help students see when the converse theorems are used.
Example Proof of the Converse of the Alternate Exterior Angle Theorem:
Given: \begin{align*}\angle 1 \cong \angle 8\end{align*}
Prove: \begin{align*}l \ \| \ m\end{align*}
Statements | Reasons |
---|---|
1. \begin{align*}\angle 1 \cong \angle 8\end{align*} | 1. Given |
2. \begin{align*}\angle 1 \cong \angle 4\end{align*} | 2. Vertical Angles Theorem |
3. \begin{align*}\angle 4 \cong \angle 8\end{align*} | 3. Transitive Property of Angle Congruence |
4. \begin{align*}l \ \| \ m\end{align*} | 4. Converse of the Corresponding Angles Postulate |
The Converse of the Alternate Interior Angle Theorem could also be proved using the Converse of the Alternate Exterior Angle Theorem. This would demonstrate to the students that once a theorem has been proved, it can be used in the proof of other theorems. This demonstrates the building block nature of math. Here is one way to do this using the same diagram from above.
Given: \begin{align*}\angle 1 \cong \angle 8\end{align*}
Prove: \begin{align*}l \ \| \ m\end{align*}
Statements | Reasons |
---|---|
1. \begin{align*}\angle 1 \cong \angle 8\end{align*} | 1. Given |
2. \begin{align*}\angle 1 \cong \angle 4\end{align*} | 2. Vertical Angles Theorem |
3. \begin{align*}\angle 5 \cong \angle 8\end{align*} | 3. Vertical Angles Theorem |
4. \begin{align*}\angle 4 \cong \angle 5\end{align*} | 4. Substitution Property of Angle Congruence |
5. \begin{align*}l \ \| \ m\end{align*} | 5. Converse of the Alternate Interior Angles Theorem |
Proving the theorem in several ways gives students a chance to practice with the concepts and their proof writing skills. Similar proofs can be assigned for the other theorems in this section.
Properties of Perpendicular Lines
Complementary, Supplementary, or Congruent - When finding angle measures, students generally need to decide between three possible relationships: complementary, supplementary, and congruent. A good way for them to practice with these and review their equation solving skills, is to assign variable expressions to angle measures, state the relationship of the angles, and have the students use this information to write an equation that when solved will lead to a numerical measurement for the angle. Encourage students to take the time to write out and solve the equation neatly. This process helps them avoid errors. Many times students will find the value of \begin{align*}x\end{align*}, and then stop without plugging in the value to the expression for the angle measures. Have the students find the actual angles measures by plugging their \begin{align*}x\end{align*} value in to verify that their final answers are angle measures that have the desired relationship.
Interpreting “Word” Questions – Students often have difficulty translating the words in a statement into an algebraic equation. Here are a couple examples that might help students interpret verbal questions that are not accompanied by a diagram.
Example 1: Two vertical angles have measures \begin{align*}(2x-30)^\circ\end{align*} and \begin{align*}(x+60)^\circ\end{align*}. Find the measures of the two angles.
Students may wish to make a diagram of vertical angles, then label them with these measures before setting up the equation: \begin{align*}2x-30=x+60\end{align*}. Once they solve for \begin{align*}x\end{align*}, students need to plug this value into both expressions to get the measures of the two angles.
Remind students that these angles should have equal measures. Model for students the process of checking your answers for reasonableness by asking them if the values make sense and how they know. At first they may have trouble answering these questions but if you consistently model this process, it may become second nature to them as well and help them to identify and correct mistakes as they solve problems throughout the course.
Example 2: The outer rays of two adjacent angle with measures \begin{align*}(4x+10)^\circ\end{align*} and \begin{align*}(5x-10)^\circ\end{align*} are perpendicular. Find the measures of each angle.
This example contains a lot of information that students will need to sort out. In this case a diagram is especially useful. Break down the information with students and help them diagram the angles to discover that the sum of the two angles must be \begin{align*}90^\circ\end{align*}. Here is the equation and value of \begin{align*}x\end{align*}:
\begin{align*}4x+10+5x-10&=90\\ x&=10\end{align*}
Using this value of \begin{align*}x\end{align*}, the two angles are \begin{align*}50^\circ\end{align*} and \begin{align*}40^\circ\end{align*}, respectively. Do these measures make sense? How do you know? Practice asking and answering these questions yourself so you can help students answer them correctly.
Example 3: The angles of a linear pair have measures \begin{align*}(3x+45)^\circ\end{align*} and \begin{align*}(2x+35)^\circ\end{align*}. Find the measure of each angle.
Example 4: Perpendicular lines form an angle with measure \begin{align*}(8x+10)^\circ\end{align*}. What is the value of \begin{align*}x\end{align*}?
Again, help students interpret the given information to make a diagram and set up an equation to solve. Then remind students to plug in their \begin{align*}x\end{align*} value to find the actual angle measures. Finally, prompt students to check their work for reasonableness and accuracy.
Answers:
Example 3: \begin{align*}x=20^\circ\end{align*} and the angles are \begin{align*}105^\circ\end{align*} and \begin{align*}75^\circ\end{align*}.
Example 4: \begin{align*}x=10^\circ\end{align*}
Proof Tip - In a proof, students must first state which lines are perpendicular and why, then they can say that all four angles formed by those perpendicular lines are right angles, then right angles are congruent, etc. Students are apt to just jump to the final conclusion because the lines are perpendicular. In a complete proof, these middle steps are important to show understanding of the thought process. Refer to the proof in question 26 of the review problems for an example.
Parallel and Perpendicular Lines in the Coordinate Plane
Order of Subtraction - When calculating the slope of a line using two points it is important to keep straight which point was made point one and which one was point two. It does not matter how these labels are assigned, but the order of subtraction has to stay the same in the numerator and the denominator of the slope ratio. If students switch the order they will get the opposite of the correct answer. If they have a graph of the line, ask them to compare the sign of the slope to the direction of the line. Is the line increasing or decreasing? Does that match the slope? This will give them another opportunity to practice checking their results for reasonableness.
Another strategy to help students with this is to have them write the points vertically and subtract as shown in the diagram below:
\begin{align*}\frac{-3-5}{4-(-2)} = \frac{-8}{6} = \frac{-4}{3}\end{align*}
Graphing Lines with Integer Slopes - The slope of a line is the ratio of two numbers. When students are asked to graph a line with an integer slope they often fail to realize what and where the second number is. Frequently they will make the “run” of the line zero and graph a vertical line. It is helpful to have them write the slope as a ratio over one before they do any graphing. They may only need to do this a few times on paper before they are able to graph the lines correctly.
Zero or Undefined - Students need to make these associations:
Zero in numerator – slope is zero – line is horizontal
Zero in denominator – slope is undefended – line is vertical
Students frequently switch these around. Try to connect these concepts to their experiences. For example explain that when they are walking on a flat surface, they are not going up or down so the slope is zero. Ask them if they can walk up a vertical wall, hopefully they will say, “No.” Then you can explain that the slope of this wall is undefined. After the relationships are explained in class, remind them frequently, maybe have a poster up in the room that shows lines with a positive slope, negative slope, zero slope (horizontal) and undefined slope (vertical). You could also write the relationship on a corner of the board that does not get erased.
Students also struggle with realizing that a line perpendicular to a horizontal line is vertical and vice versa. When they look at these equations, the slope is not evident and so they don’t know what to do. Practice lots of examples with horizontal and vertical lines.
Use Graph Paper - Making a connection between the numbers that describe a line and the line itself is an important skill. Requiring that the students use graph paper encourages them to make nice, thoughtful graphs, and helps them make this connection.
The \begin{align*}y-\end{align*}axis is Vertical - When using the slope-intercept form to graph a line or write an equation, it is common for students to use the \begin{align*}x-\end{align*}intercept instead of the \begin{align*}y-\end{align*}intercept. Remind them that they want to use the vertical axis, \begin{align*}y-\end{align*}intercept, to begin the graph. Requiring that the \begin{align*}y-\end{align*}intercept be written as a point, say (0, 3) instead of just 3, helps to alleviate this problem.
Where’s the Slope - Students are quickly able to identify the slope as the coefficient of the \begin{align*}x-\end{align*}variable when a line is in slope-intercept form. Unfortunately they sometimes extend this to standard form. Remind the students that if the equation of a line is in standard form, or any other form, they must first algebraically convert it to slope-intercept form before they can easily read off the slope. It is wise to do several examples which require changing the equation from standard form into slope-intercept form.
Why Use Standard Form - The slope-intercept form of the line holds so much valuable information about the graph of a line that students probably won’t understand why any other form would ever be used. Mention to them that standard form is convenient when solving systems and putting equations into matrices, things they will be doing in their second year of algebra, to motivate them to learn and remember the standard form.
Organizing Work in Multi-Step Problems - Part of the struggle with these problems is that students get lost in the process. They get wrapped up in a particular step and forget where they are going. It may be helpful to have students write out the steps they will need to follow in the beginning so that they will have a road map to follow. They can even leave some space between each step to go back and fill in with their work. Here is an example of this:
Find the equation of the line that is perpendicular to the line passing through the points (5, 7) and (12, 3) and passes through the second point.
1) Find the slope between the given two points.
\begin{align*}m=\frac{3-7}{12-5}=\frac{-4}{7}\end{align*}
2) Find the opposite reciprocal (perpendicular) slope.
\begin{align*}\perp m = \frac{7}{4}\end{align*}
3) Use the perpendicular slope and the given point (12, 3) to find the \begin{align*}y-\end{align*}intercept.
\begin{align*}3&=\frac{7}{4}(12)+b\\ 3&=21+b\\ -18&=b\end{align*}
4) Write the new equation.
\begin{align*}y= \frac{7}{4} x-18\end{align*}
Many students forget that at the end they need to write an equation with variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. They don’t quite understand that this is an equation that relates all possible pairs of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates. Explain to them that they need to use the given pair to determine what the \begin{align*}y-\end{align*}intercept is, but not in the final equation.
The Distance Formula
The Perpendicular Distance - In theory, measuring along a perpendicular line makes sense to the students, but in practice, when lining up the ruler or deciding which points to put in the distance formula, there are many distractions. Students can evaluate their decision by taking a second look to see if the path they chose was the shortest one possible. This is another opportunity to practice checking their work for reasonableness.
Where to Measure? - Now that the students know to measure along a line that is perpendicular to both parallel lines, they might wonder where along the lines to measure. When working on a coordinate plane it is best to start with a point that has integer coordinates, just to keep the problem simple and accurate. They will get the same distance no matter where they measure though. You may wish to share the alternate definition of parallel lines: Two lines that are a constant distance apart.
Order of Operations - Students might want to cancel the squares with the square root in the distance formula, even though they cannot. See example below:
\begin{align*}\sqrt{\left (x_1-x_2 \right )^2 + \left (y_1-y_2 \right )^2} \neq \left (x_1-x_2 \right ) + \left (y_1-y_2 \right )\end{align*}
Remind students that the square root is like parenthesis. The operations contained inside the square root must be performed before the square root can be taken. Give students lots of opportunity to practice finding the distance between two points to practice this formula.
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