After a large effort to rebuild the city, you will now see much of the structure found today. Pay close attention to the roads. Notice that many of the roads run east to west or north and south. This brings us into the topic of this unit.
In this unit, we will study lines and the relationships with angles that are formed. Let’s start with parallel lines and perpendicular lines, which are evident in the Chicago map above!
In this unit, we will study lines and the relationships with angles that are formed. Let’s start with parallel lines and perpendicular lines, which are evident in the Chicago map above!
Parallel lines
Lines that lie on the same plane and do not intersect.
Lines that lie on the same plane and do not intersect.
Perpendicular lines
Lines that lie on the same plane but intersect at a right angle.
Lines that lie on the same plane but intersect at a right angle.
In a close up of our Chicago map, we can identify several lines that are parallel, like Madison Street and Jackson Boulevard. The marking we use to show lines are parallel are arrows pointing in the same direction (see below):
Source:
https://www.google.com/maps/place/Chicago,+IL/@41.8333925,-88.0121586,10z/data=!3m1!4b1!4m5!3m4!1s0x880e2c3cd0f4cbed:0xafe0a6ad09c0c000!8m2!3d41.8781136!4d-87.6297982
The symbol we use to represent parallel lines is: \(\parallel\). So in the diagram above, we can say that \(m\parallel j\). Since Madison Street and Jackson Boulevard are on the same plane but they do not intersect, they represent parallel lines!
We can also identify lines that are perpendicular in our Chicago map, for instance Madison Street and Canal Street. We denote a right angle by the box you see below:
https://www.google.com/maps/place/Chicago,+IL/@41.8333925,-88.0121586,10z/data=!3m1!4b1!4m5!3m4!1s0x880e2c3cd0f4cbed:0xafe0a6ad09c0c000!8m2!3d41.8781136!4d-87.6297982
The symbol we use to represent parallel lines is: \(\parallel\). So in the diagram above, we can say that \(m\parallel j\). Since Madison Street and Jackson Boulevard are on the same plane but they do not intersect, they represent parallel lines!
We can also identify lines that are perpendicular in our Chicago map, for instance Madison Street and Canal Street. We denote a right angle by the box you see below:
The symbol we use to represent perpendicular lines is: \(\perp\). So in the diagram above, we can say that \(m\perp c\), since intersect to meet at a right angle.
If we look at Madison Street, Jackson Boulevard, and Canal Street together, we can now call Canal street a transversal. A transversal is a line that intersects two or more lines. So since line \(c\) intersects lines \(m\) and \(j\), it is called a transversal!
Similar to parallel lines, parallel planes are planes do not intersect. Imagine the most perfect sandwich that you have made for lunch.
If we look at Madison Street, Jackson Boulevard, and Canal Street together, we can now call Canal street a transversal. A transversal is a line that intersects two or more lines. So since line \(c\) intersects lines \(m\) and \(j\), it is called a transversal!
Similar to parallel lines, parallel planes are planes do not intersect. Imagine the most perfect sandwich that you have made for lunch.
The two slices of bread can represent parallel planes since they do not intersect (so long as your sandwich doesn’t get smashed in your backpack!).
Another term that comes up when talking about lines is skew lines.
Another term that comes up when talking about lines is skew lines.
Let’s use the example below to show parallel lines, parallel planes, and skew lines!
Example 1:
The diagram below represents a rectangular prism (it’s like a three dimensional rectangle): a) Identify a line that is parallel to \(\overleftrightarrow{HE}\). b) Identify two planes that are parallel. c) Identify a line that is skew to \(\overleftrightarrow{HE}\). d) Identify two lines that are perpendicular. |
Solution:
a) Recall that parallel lines must lie in the same plane. There are a few options we have here: If we see that \(\overleftrightarrow{HE}\) lies on plane \(HEF\), then we will say that \(\overleftrightarrow{AF}\parallel\overleftrightarrow{HE}\). If we see that lies \(\overleftrightarrow{HE}\) on plane \(HED\), then we will say that \(\overleftrightarrow{CD}\parallel\overleftrightarrow{HE}\). Also \(\overleftrightarrow{BG}\) lies on plane \(HEG\), even though it is not drawn in the diagram. This makes \(\overleftrightarrow{BG}\parallel\overleftrightarrow{HE}\).
b) Since this is a rectangular prism, we have three pairs of parallel planes. Here is just one of several ways to describe them: Plane \(ABG\parallel\) Plane \(HCD\), Plane \(HEF\parallel\) Plane \(CDG\), and Plane \(FED\parallel\) Plane \(ABC\).
c) Since we are trying to find a line that is skew to \(\overleftrightarrow{HE}\), we must find a line that is not on the same plane as \(\overleftrightarrow{HE}\). So \(\overleftrightarrow{GF}\) and \(\overleftrightarrow{GD}\) are examples of lines that are skew to \(\overleftrightarrow{HE}\).
d) Remember that this is a rectangular prism, so each of the corners represent right angles. Therefore there are many pairs of perpendicular lines! Here is just a few of them: \(\overleftrightarrow{AB}\perp\overleftrightarrow{AH}\), \(\overleftrightarrow{FE}\perp\overleftrightarrow{EH}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{DE}\),or \(\overleftrightarrow{BG}\perp\overleftrightarrow{GD}\).
Before we discuss more about parallel and perpendicular lines, we must establish some very important ground work from which several other theorems are created.
a) Recall that parallel lines must lie in the same plane. There are a few options we have here: If we see that \(\overleftrightarrow{HE}\) lies on plane \(HEF\), then we will say that \(\overleftrightarrow{AF}\parallel\overleftrightarrow{HE}\). If we see that lies \(\overleftrightarrow{HE}\) on plane \(HED\), then we will say that \(\overleftrightarrow{CD}\parallel\overleftrightarrow{HE}\). Also \(\overleftrightarrow{BG}\) lies on plane \(HEG\), even though it is not drawn in the diagram. This makes \(\overleftrightarrow{BG}\parallel\overleftrightarrow{HE}\).
b) Since this is a rectangular prism, we have three pairs of parallel planes. Here is just one of several ways to describe them: Plane \(ABG\parallel\) Plane \(HCD\), Plane \(HEF\parallel\) Plane \(CDG\), and Plane \(FED\parallel\) Plane \(ABC\).
c) Since we are trying to find a line that is skew to \(\overleftrightarrow{HE}\), we must find a line that is not on the same plane as \(\overleftrightarrow{HE}\). So \(\overleftrightarrow{GF}\) and \(\overleftrightarrow{GD}\) are examples of lines that are skew to \(\overleftrightarrow{HE}\).
d) Remember that this is a rectangular prism, so each of the corners represent right angles. Therefore there are many pairs of perpendicular lines! Here is just a few of them: \(\overleftrightarrow{AB}\perp\overleftrightarrow{AH}\), \(\overleftrightarrow{FE}\perp\overleftrightarrow{EH}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{DE}\),or \(\overleftrightarrow{BG}\perp\overleftrightarrow{GD}\).
Before we discuss more about parallel and perpendicular lines, we must establish some very important ground work from which several other theorems are created.
Similarly, the Perpendicular Line Postulate states that given a point \(P\) and line \(\ell\), there is only one line that you can draw through point \(P\) that is perpendicular to line \(\ell\).
We use a notation of a square at the intersection of the lines or segments to show that they are perpendicular.
Quick Check
1) Which of the following lines is the transversal?
1) Which of the following lines is the transversal?
2) Use the hexagonal prism below. Assume right angles and parallel lines.
a) Name a pair of perpendicular lines
b) Name a pair of parallel lines
c) Name a pair of skew lines
d) Name a pair of parallel planes
Quick Check Solutions
b) Name a pair of parallel lines
c) Name a pair of skew lines
d) Name a pair of parallel planes
Quick Check Solutions