In this explainer, we will learn how to apply the angle relationships between a pair of parallel lines and a transversal to establish and use other relationships between parallel lines and transversals.
There are many uses for the angle relationships between parallel lines and transversals. The most common use is finding the measures of other angles using results such as the alternate interior angles being congruent. However, there are other uses, such as proving relationships between transversals of lines.
To see this in action, we first recall that a transversal of a pair of parallel lines will have congruent corresponding angles. This is a useful result since we can use this to determine the angles between two lines using the angle between one line and a parallel line.
In particular, if we have a line that is perpendicular to another line , then we know that any line parallel to must have a corresponding angle with a measure of .
This gives us the following result.
Property: Orthogonality of Lines Using Parallelism to a Perpendicular Line
If a line is perpendicular to line , then it is perpendicular to any line parallel to .
This is not the only property we can show using properties of transversals. We recall that a transversal of a pair of parallel lines will have congruent corresponding angles. We can also recall that if the corresponding angles in a transversal of two lines are congruent, then the lines are parallel.
Therefore, if a transversal of two lines is perpendicular to the two lines, then the lines must be parallel since they have congruent corresponding angles.
This gives us the following property.
Property: Orthogonality of Two Lines to Another Line Implies They Are Parallel
If two lines are perpendicular to the same line, then they must be parallel.
Letβs now see some examples of applying these properties to determine relationships between lines.
Example 1: Identifying That Two Lines Are Perpendicular Because One of Them Is Perpendicular to a Line Parallel to the Other
If intersects at point and intersects at point , where and . Which of the following is correct?
- is perpendicular to .
- bisects .
- is parallel to .
- is perpendicular to .
- bisects .
Answer
Letβs begin by adding the fact that means the lines are at right angles to the diagram and the fact that .
We can then use corresponding angles to note that .
This shows that is perpendicular to , which is answer D. However, this is a particular case of the fact that if a line is perpendicular to line , then it is perpendicular to any line parallel to .
It is worth noting that although the diagram looks like the transversal bisects the parallel lines, this does not need to be the case since we can translate the transversal and still have the image being perpendicular to the parallel lines.
We see that line is perpendicular to the parallel lines and and that it is not a perpendicular bisector of either line. Thus, we cannot conclude that bisects line or that bisects line segment , so answers A and E are incorrect.
Hence, is perpendicular to , which is answer D.
In our next example, we will determine the relationship between two lines that are both perpendicular to a given line.
Example 2: Identifying That Two Lines Are Parallel Because Both Are Perpendicular to a Third Line
If is a point on , is a point on , , and , which of the following is correct?
- bisects line .
- is parallel to .
- is parallel to .
- is perpendicular to .
- bisects .
Answer
Letβs start by sketching the information we are given, namely that and .
We see that is a transversal and the corresponding angles of this transversal have equal measures. This means that the lines must be parallel.
In particular, we can recall that if two lines are perpendicular to the same line, then they must be parallel. Since both and are perpendicular to the same line, , we must have that the two lines are parallel.
This is an application of the more general result that if a transversal cuts two lines with corresponding angles of equal measures, then the lines must be parallel.
Hence, the answer is B: is parallel to .
Before we move on to our next example, there is another useful property we can show. Letβs consider a pair of distinct parallel lines, and , and another distinct line that is also parallel to one of these lines: . We can sketch this as shown.
It appears as though all three of the lines are parallel. We can prove this is the case by sketching a transversal perpendicular to .
Using corresponding angles, we can show that all three lines are perpendicular to the transversal. Then, we recall that if two lines are perpendicular to the same line, then they must be parallel to each other. Hence, all three lines are parallel. We have proven the following result.
Property: Parallelism of Lines Is a Transitive Relation
If two distinct lines are parallel to a third distinct line, then all three lines are parallel.
Letβs now see an example of using this property to determine the relationship between given lines.
Example 3: Identifying That Two Lines Are Parallel Because Both Are Parallel to a Third Line
Fill in the blank: If and , then .
Answer
We first recall that if two lines are parallel to the same line, then they must be parallel. Since both and are parallel to the same line, , we must have that the two lines are parallel.
Hence, the answer is that .
There is one final property we want to show about the transversals of parallel lines, and we will introduce this property with an example. Consider the following diagram where , , and are all parallel and the lengths of the line segments between the parallel lines are as shown.
In the diagram, it appears that if the lengths of the line segments of a transversal between three parallel lines are equal, then the lengths of the line segments of any transversal between three parallel lines will be equal.
We can prove that this is the case using congruent triangles. We can start with a perpendicular transversal that is split into two sections of equal length by the parallel lines as shown.
We can note that , as they are vertically opposite angles.
We then see that and are congruent by the ASA criterion. In particular, this means that . Hence, the other transversal is also split into two sections of equal length. We can always add in a transversal perpendicular to these lines, so the proof of this result in general is very similar. We have shown the following property.
Property: Comparing Segment Lengths of Transversals of Parallel Lines
If a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length.
Letβs now see some examples of using this property to find the length of a transversal between parallel lines.
Example 4: Using the Property of Parallel Lines Dividing a Transversal into Segments of Equal Length to Solve a Problem
If and , find .
Answer
We can see in the diagram that we are given three parallel lines and two transversals. We can see that , so in this case the parallel lines divide one of the transversals into segments of equal length. We can recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Therefore, it will divide the other transversal into sections of equal length.
Thus, . We note that , so
In our next example, we will use this property to determine the length of one of the segments of a transversal between three parallel lines.
Example 5: Identifying Whether Lines Are Parallel to Solve a Problem
If , , and , find the length of , where and .
Answer
We are given three lines that are all perpendicular to and two transversals of these lines. Since these three lines are all perpendicular to the same line, we can note that they must be parallel to each other: .
We can see that , so in this case the parallel lines divide this transversal into segments of equal length. We can then recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Therefore, they will divide the other transversal into sections of equal length.
Since the whole line segment between the parallel lines, , has a length of 12 cm, we can halve this length to find the lengths of its segments. We have
In our final example, we will apply multiple properties of the transversals of parallel lines to a triangle with a side bisected by parallel lines.
Example 6: Using Properties of Parallel Lines to Solve a Problem
Consider triangle and lines and , which are parallel to .
- Find the length of .
- Find the measure of .
Answer
Part 1
We are given three parallel lines and two transversals of these lines. We can then recall that if a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. Since , the segments of the other transversal must be equal in length. So, .
Thus,
Part 2
It appears in the diagram that triangle is a right triangle. However, we need to justify why this is the case. We can do this by recalling that if a line is perpendicular to line , then it is perpendicular to any line parallel to .
Since is perpendicular to and is parallel to , we must have that is perpendicular to . This means the angle at has a measure of , so it is a right triangle. The sum of the measures of the interior angles in a triangle is , so
Alternatively, we can note that and are corresponding angles of the transversal of the parallel lines and .
So, their measures are equal. We can see that since it forms a straight angle with a right angle.
We can then determine the measure of using the sum of the measures of the internal angles of . We have
We then use the fact that is congruent to to see that
Letβs finish by recapping some of the important points from this explainer.
Key Points
- If a line is perpendicular to line , then it is perpendicular to any line parallel to .
- If two lines are perpendicular to the same line, then they must be parallel.
- If two distinct lines are parallel to a third distinct line, then all three lines are parallel.
- If a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length.
