3.1: Lines and Angles
Learning Objectives
- Define parallel lines, skew lines, and perpendicular planes.
- Understand the Parallel Line Postulate and the Perpendicular Line Postulate.
- Identify angles made by two lines and a transversal.
Review Queue
- What is the equation of a line with slope -2 and \begin{align*}y-\end{align*}intercept 3?
- What is the slope of the line that passes through (3, 2) and (5, -6)?
- Find the \begin{align*}y-\end{align*}intercept of the line from #2. Write the equation too.
- Define parallel in your own words.
Know What? To the right is a partial map of Washington DC. The streets are designed on a grid system, where lettered streets, A through \begin{align*}Z\end{align*} run east to west and numbered streets, \begin{align*}1^{st}\end{align*} to \begin{align*}30^{th}\end{align*} run north to south. Every state also has its own street that runs diagonally through the city.
Which streets are parallel? Which streets are perpendicular? How do you know?
If you are having trouble viewing this map, look at the interactive map: http://www.travelguide.tv/washington/map.html
Defining Parallel and Skew
Parallel: Two or more lines that lie in the same plane and never intersect.
To show that lines are parallel, arrows are used.
Label It | Say It |
---|---|
\begin{align*}\overleftrightarrow{AB} || \overleftrightarrow{MN}\end{align*} | Line \begin{align*}AB\end{align*} is parallel to line \begin{align*}MN\end{align*} |
\begin{align*}l || m\end{align*} | Line \begin{align*}l\end{align*} is parallel to line \begin{align*}m\end{align*}. |
Lines must be marked parallel with the arrows in order to say they are parallel. Just because two lines LOOK parallel, does not mean that they are.
Recall the definition of perpendicular from Chapter 1. Two lines are perpendicular when they intersect to form a \begin{align*}90^\circ\end{align*} angle. Below \begin{align*}l \perp \overline{AB}\end{align*}.
In the definitions of parallel and perpendicular, the word “line,” is used. Line segments, rays and planes can also be parallel or perpendicular.
The image to the left shows two parallel planes, with a third blue plane that is perpendicular to both of them.
An example of parallel planes could be the top of a table and the floor. The legs would be in perpendicular planes to the table top and the floor.
Skew lines: Lines that are in different planes and never intersect.
In the cube:
\begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{FH}\end{align*} are skew
\begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{EF}\end{align*} are skew
Example 1: Using the cube above, list:
(a) A pair of parallel planes
(b) A pair of perpendicular planes
(c) A pair of skew lines.
Solution: Remember, you only need to use three points to label a plane. Below are answers, but there are other possibilities too.
(a) Planes \begin{align*}ABC\end{align*} and \begin{align*}EFG\end{align*}
(b) Planes \begin{align*}ABC\end{align*} and \begin{align*}CDH\end{align*}
(c) \begin{align*}\overline{BD}\end{align*} and \begin{align*}\overline{CG}\end{align*}
Parallel Line Postulate
Parallel Postulate: For any line and a point not on the line, there is one line parallel to this line through the point.
There are infinitely many lines that go through \begin{align*}A\end{align*}, but only one that is parallel to \begin{align*}l\end{align*}.
Investigation 3-1: Patty Paper and Parallel Lines
Tools Needed: Patty paper, pencil, ruler
1. Get a piece of patty paper (a translucent square piece of paper). Draw a line and a point above the line.
2. Fold up the paper so that the line is over the point. Crease the paper and unfold.
3. Are the lines parallel?
Yes. This investigation duplicates the line we drew in #1 over the point. This means that there is only one parallel line through this point.
Perpendicular Line Postulate
Perpendicular Line Postulate: For any line and a point not on the line, there is one line perpendicular to this line passing through the point.
There are infinitely many lines that pass through \begin{align*}A\end{align*}, but only one that is perpendicular to \begin{align*}l\end{align*}.
Investigation 3-2: Perpendicular Line Construction; through a Point NOT on the Line
Tools Needed: Pencil, paper, ruler, compass
1. Draw a horizontal line and a point above that line. Label the line \begin{align*}l\end{align*} and the point \begin{align*}A\end{align*}.
2. Take the compass and put the pointer on \begin{align*}A\end{align*}. Open the compass so that it reaches past line \begin{align*}l\end{align*}. Draw an arc that intersects the line twice.
3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc below the line. Repeat this on the other side so that the two arc marks intersect.
4. Take your straightedge and draw a line from point \begin{align*}A\end{align*} to the arc intersections below the line. This line is perpendicular to \begin{align*}l\end{align*} and passes through \begin{align*}A\end{align*}.
To see a demonstration of this construction, go to:
http://www.mathsisfun.com/geometry/construct-perpnotline.html
Investigation 3-3: Perpendicular Line Construction; through a Point on the Line
Tools Needed: Pencil, paper, ruler, compass
1. Draw a horizontal line and a point on that line. Label the line \begin{align*}l\end{align*} and the point \begin{align*}A\end{align*}.
2. Take the compass and put the pointer on \begin{align*}A\end{align*}. Open the compass so that it reaches out horizontally along the line. Draw two arcs that intersect the line on either side of the point.
3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc above or below the line. Repeat this on the other side so that the two arc marks intersect.
4. Take your straightedge and draw a line from point \begin{align*}A\end{align*} to the arc intersections above the line. This line is perpendicular to \begin{align*}l\end{align*} and passes through \begin{align*}A\end{align*}.
To see a demonstration of this construction, go to:
http://www.mathsisfun.com/geometry/construct-perponline.html
Example 2: Construct a perpendicular line through the point below.
Solution: Even though the point is below the line, the construction is the same as Investigation 3-2. However, draw the arc marks in step 3 above the line.
Angles and Transversals
Transversal: A line that intersects two other lines.
The area between \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is the interior.
The area outside \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is the exterior.
Looking at \begin{align*}t, l\end{align*} and \begin{align*}m\end{align*}, there are 8 angles formed. They are labeled below.
There are 8 linear pairs and 4 vertical angle pairs.
An example of a linear pair would be \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*}.
An example of vertical angles would be \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 8\end{align*}.
Example 3: List all the other linear pairs and vertical angle pairs in the picture above.
Solution:
Linear Pairs: \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 4, \ \angle 3\end{align*} and \begin{align*}\angle 4, \ \angle 1\end{align*} and \begin{align*}\angle 3, \ \angle 5\end{align*} and \begin{align*}\angle 6, \ \angle 6\end{align*} and \begin{align*}\angle 8, \ \angle 7\end{align*} and \begin{align*}\angle 8, \ \angle 5\end{align*} and \begin{align*}\angle 7\end{align*}
Vertical Angles: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4, \ \angle 2\end{align*} and \begin{align*}\angle 3, \ \angle 6\end{align*} and \begin{align*}\angle 7\end{align*}
There are also 4 new angle relationships.
Corresponding Angles: Two angles that are on the same side of the transversal and the two different lines. Imagine sliding the four angles formed with line \begin{align*}l\end{align*} down to line \begin{align*}m\end{align*}. The angles which match up are corresponding.
Above, \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} are corresponding angles.
Alternate Interior Angles: Two angles that are on the interior of \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, but on opposite sides of the transversal.
Above, \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are alternate exterior angles.
Alternate Exterior Angles: Two angles that are on the exterior of \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, but on opposite sides of the transversal.
Above, \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 7\end{align*} are alternate exterior angles.
Same Side Interior Angles: Two angles that are on the same side of the transversal and on the interior of the two lines.
Above, \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are same side interior angles.
Example 4: Using the picture above, list all the other pairs of each of the newly defined angle relationships.
Solution:
Corresponding Angles: \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 7, \ \angle 1\end{align*} and \begin{align*}\angle 5, \ \angle 4\end{align*} and \begin{align*}\angle 8\end{align*}
Alternate Interior Angles: \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 5\end{align*}
Alternate Exterior Angles: \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 7\end{align*}
Same Side Interior Angles: \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 6\end{align*}
Example 5: For the picture below, determine:
(a) A corresponding angle to \begin{align*}\angle 3\end{align*}?
(b) An alternate interior angle to \begin{align*}\angle 7\end{align*}?
(c) An alternate exterior angle to \begin{align*}\angle 4\end{align*}?
Solution:
(a) \begin{align*}\angle 1\end{align*}
(b) \begin{align*}\angle 2\end{align*}
(c) \begin{align*}\angle 5\end{align*}
Know What? Revisited For Washington DC, all of the lettered and numbered streets are parallel. The lettered streets are perpendicular to the numbered streets. We do not have enough information about the state-named streets to say if they are parallel or perpendicular.
Review Questions
- Questions 1-3 use the definitions of parallel, perpendicular, and skew lines.
- Question 4 asks about the Parallel Line Postulate and the Perpendicular Line Postulate.
- Questions 5-9 use the definitions learned in this section and are similar to Example 1.
- Questions 10-20 are similar to Examples 4 and 5.
- Question 21 is similar to Example 2 and Investigation 3-2.
- Questions 22-30 are Algebra I review.
- Which of the following is the best example of parallel lines?
- Railroad Tracks
- Lamp Post and a Sidewalk
- Longitude on a Globe
- Stonehenge (the stone structure in Scotland)
- Which of the following is the best example of perpendicular lines?
- Latitude on a Globe
- Opposite Sides of a Picture Frame
- Fence Posts
- Adjacent Sides of a Picture Frame
- Which of the following is the best example of skew lines?
- Roof of a Home
- Northbound Freeway and an Eastbound Overpass
- Longitude on a Globe
- The Golden Gate Bridge
- Writing What is the difference between the Parallel Line Postulate and the Perpendicular Line Postulate? How are they similar?
Use the figure below to answer questions 5-9. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.
- Find two pairs of skew lines.
- List a pair of parallel lines.
- List a pair of perpendicular lines.
- For \begin{align*}\overline{AB}\end{align*}, how many perpendicular lines would pass through point \begin{align*}V\end{align*}? Name this line.
- For \begin{align*}\overline{XY}\end{align*}, how many parallel lines would pass through point \begin{align*}D\end{align*}? Name this line.
For questions 10-16, use the picture below.
- What is the corresponding angle to \begin{align*}\angle 4\end{align*}?
- What is the alternate interior angle with \begin{align*}\angle 5\end{align*}?
- What is the corresponding angle to \begin{align*}\angle 8\end{align*}?
- What is the alternate exterior angle with \begin{align*}\angle 7\end{align*}?
- What is the alternate interior angle with \begin{align*}\angle 4\end{align*}?
- What is the same side interior angle with \begin{align*}\angle 3\end{align*}?
- What is the corresponding angle to \begin{align*}\angle 1\end{align*}?
Use the picture below for questions 17-20.
- If \begin{align*}m\angle 2 = 55^\circ\end{align*}, what other angles do you know?
- If \begin{align*}m\angle 5 = 123^\circ\end{align*}, what other angles do you know?
- If \begin{align*}t \perp l\end{align*}, is \begin{align*}t \perp m\end{align*}? Why or why not?
- Is \begin{align*}l || m\end{align*}? Why or why not?
- Construction Draw a line and a point not on the line. Construct a perpendicular line to the one your drew.
Algebra Review Find the slope of the line between the two points, \begin{align*}\frac{y_2-y_1}{x_2-x_1}\end{align*}.
- (-3, 2) and (-2, 1)
- (5, -9) and (0, 1)
- (2, -7) and (5, 2)
- (8, 2) and (-1, 5)
- Find the equation of the line from #22. Recall that the equation of a line is \begin{align*}y = mx + b\end{align*}, where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept.
- Find the equation of the line from #23.
- Find the equation of the line from #24.
- Is the line \begin{align*}y=-x+3\end{align*} parallel to the line in #26? How do you know?
- Is the line \begin{align*}y=-x+3\end{align*} perpendicular to the line in #26? How do you know?
Review Queue Answers
- \begin{align*}y = -2x + 3\end{align*}
- \begin{align*}m = \frac{-6-2}{5-3} = \frac{-8}{2} = -4\end{align*}
- \begin{align*}{\;} \ \ \ 2 = -4(3) + b\!\\ {\;} \ \ \ 2 = -12 + b\!\\ -14 = b\!\\ {\;} \ \ \ y = -4x + 14\end{align*}
- Something like: Two lines that never touch or intersect and in the same plane. If we do not say “in the same plane,” this definition could include skew lines.