Transversals and Parallel Lines
In this section, we’ll be discussing the properties of parallel lines and transversals.
Parallel Lines
Parallel lines are lines on the same plane that never intersect. They run side by side at the same distance, infinitely, forever, without meeting. Parallel lines are indicated by drawing an arrow mark on each parallel line. In linear algebra, they have the same slope.
In the image above, you can see there’s a little arrow indicating the lines are parallel.
To show that the lines are parallel using a statement, you write two vertical bars like \(l||m\) which means “line l is parallel to line m.”
Parallel Planes
Parallel planes are planes that never intersect with one another. You can picture it as one plane floating above another, or the top and bottom planes of a cube.
Transversals
Transversals are lines that pass through two or more other lines at different points on the same plane. Look at the following examples of transversals.
Angles Formed By Transversals
The angles formed by transversals have specific names.
Interior Angles are the angles formed between the two lines the transversal intersects.
Exterior Angles are the angles formed outside the two intersecting lines.
Alternate Interior Angles are the interior angles on opposite sides of the transversal.
Alternate Exterior Angles are the exterior angles on opposite sides of the transversal.
Same Side Interior Angles are the interior angles on the same side of the transversal.
Corresponding Angles are the angles in the same relative location around each intersection of the transversal.
Note that lines do not need to be parallel for these angle definitions to apply.
Special Angle Pairs
If we are given two parallel lines and a transversal, we can immediately determine certain properties about the angles they create.
Corresponding angles are congruent if a transversal intersects two parallel lines.
Alternate interior angles are congruent if a transversal intersects two parallel lines.
Alternate exterior angles are congruent if a transversal intersects two parallel lines.
Same side interior angles are supplementary if a transversal intersects two parallel lines.
Proving Parallel Lines With Converse Rules
Using the converse of the above postulates and theorems, we can prove that two lines are parallel.
Converse of Corresponding Angles Postulate
If the corresponding angles formed by two lines and a transversal are congruent, then the two lines are parallel.
Converse of Alternate Interior Angles Theorem
If the alternate interior angles formed by two lines and a transversal are congruent, then the two lines are parallel.
Converse of Same-Side Interior Angles Theorem
If the same-side interior angles formed by two lines and a transversal are supplementary, then the lines are parallel.
Converse of Alternate Exterior Angles Theorem
If the alternate exterior angles formed by two lines and a transversal are congruent, then the lines are parallel.
Given the information above, we can now figure out all sorts of useful information about the angles in transversals. Here are some worked out examples.
1. Calculate: Determine the measures of all angles one through four.
Solution:
Angle 2 corresponds with the 115° angle, and must also be 115° because of the Corresponding Angles Postulate. Therefore, \(m\angle 2 = 115°\)
Angle 2 and angle 3 are vertical angles, so they are also congruent. \(m\angle 3 = 115°\)
Angles 1 and 2 form a linear pair, so we know \(m\angle 1 = 180°-115°\) which equals 65°. So \(m\angle 1 = 65°\)
Angles 1 and 4 are vertical angles, so we know they’re congruent. \(m\angle 4 = 65°\) You could have also figured this out by solving with angles 2 and 4 being a linear pair.
2. Definition: What theorem or postulate would you use to show angles 1 and 8 are congruent. What about angles 2 and 6?
Solution:
Angles 1 and 8 are congruent by the alternate exterior angles theorem. Angles 2 and 6 are congruent by the corresponding angles postulate.
3. Calculate: Determine 𝑚∠𝐸𝐷𝐹 and the value of x.
Solution:
There are several angle pairs we could use to solve this. I have used corresponding angles to determine that \(m\angle CDF\) is congruent to the 75° angle, so \(m\angle CDF=75°\)
\(\angle CDF\) and \(\angle EDF\) form a linear pair, so \(m\angle CDF\ + m\angle EDF = 180°\)
\(m\angle EDF = 180°-75°=105°\)
Solve for x
\((3x+15)°=105°\)
\(3x=90\)
\(x=30\)
4. Determine: Assume this diagram was not drawn to scale. Are lines l and m parallel? Why?
Solution:
They are not parallel. The converse of same-side interior angles theorem says that the two same-side interior angles must be supplementary (add up to 180°) for the lines to be parallel.
115° and 75° add up to 190° so lines l and m cannot be parallel.
5. Identify: What are the transversals of \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{BD}\)
Solution:
The transversals of \(\overleftrightarrow{AB}\) are \(\overleftrightarrow{AC}\) and \(\overleftrightarrow{BD}\)
The transversals of \(\overleftrightarrow{BD}\) are \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)
6. Calculate: Given \(m\angle 1=70°\), determine the measures of all other fifteen angles in this diagram using whatever theorems or postulates you would like.
Solution:
There are multiple ways to solve this. Here is one example:
\(\angle 1 \cong \angle 4\) because they’re vertical angles. So \(m\angle 4=70°\)
\(\angle 1\) and \(\angle 2\) form a linear pair, so \(m\angle 2 = 180°-70° = 110°\)
Angles 2 and 3 are vertical angles, so \(m\angle 3=110°\)
Now that we know the measures of angles 1 through 4, we can use the corresponding angles postulate to determine the measures of angles 5 through 8 and 9 through 12.
We can use the corresponding angles postulate again to determine the measures of the remaining angles 13 through 16.
So angles 1, 4, 5, 8, 9, 12, 13, and 16 are all 70°
The remaining angles are all 110°
