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Lesson Explainer: Parallel Lines and Transversals: Other Relationships Mathematics

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 𝐴𝐵 that is perpendicular to another line 𝐸𝐹, then we know that any line parallel to 𝐴𝐵 must have a corresponding angle with a measure of 90.

This gives us the following result.

Property: Orthogonality of Lines Using Parallelism to a Perpendicular Line

If a line is perpendicular to line 𝐴𝐵, then it is perpendicular to any line parallel to 𝐴𝐵.

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.

Let’s 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 𝐸𝐹 intersects 𝐴𝐵 at point 𝐺 and intersects 𝐶𝐷 at point 𝐹, where 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐶𝐷. Which of the following is correct?

  1. 𝐴𝐵 is perpendicular to 𝐶𝐷.
  2. 𝐸𝐺 bisects 𝐴𝐵.
  3. 𝐴𝐵 is parallel to 𝐸𝐺.
  4. 𝐴𝐵 is perpendicular to 𝐸𝐺.
  5. 𝐸𝐹 bisects 𝐶𝐷.

Answer

Let’s begin by adding the fact that 𝐸𝐹𝐶𝐷 means the lines are at right angles to the diagram and the fact that 𝐴𝐵𝐶𝐷.

We can then use corresponding angles to note that 𝐸𝐹𝐴𝐵.

This shows that 𝐴𝐵 is perpendicular to 𝐸𝐺, which is answer D. However, this is a particular case of the fact that if a line is perpendicular to line 𝐿, then it is perpendicular to any line parallel to 𝐿.

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 𝐿 is perpendicular to the parallel lines 𝐴𝐵 and 𝐶𝐷 and that it is not a perpendicular bisector of either line. Thus, we cannot conclude that 𝐸𝐺 bisects line 𝐴𝐵 or that 𝐸𝐹 bisects line segment 𝐶𝐷, so answers A and E are incorrect.

Hence, 𝐴𝐵 is perpendicular to 𝐸𝐺, 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 𝐹 is a point on 𝐴𝐵, 𝐸 is a point on 𝐶𝐷, 𝐴𝐵𝐸𝐹, and 𝐶𝐷𝐸𝐹, which of the following is correct?

  1. 𝐸𝐹 bisects line 𝐴𝐵.
  2. 𝐴𝐵 is parallel to 𝐶𝐷.
  3. 𝐴𝐵 is parallel to 𝐸𝐹.
  4. 𝐴𝐵 is perpendicular to 𝐶𝐷.
  5. 𝐸𝐹 bisects 𝐶𝐷.

Answer

Let’s start by sketching the information we are given, namely that 𝐴𝐵𝐸𝐹 and 𝐶𝐷𝐸𝐹.

We see that 𝐸𝐹 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 𝐴𝐵 and 𝐶𝐷 are perpendicular to the same line, 𝐸𝐹, 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: 𝐴𝐵 is parallel to 𝐶𝐷.

Before we move on to our next example, there is another useful property we can show. Let’s consider a pair of distinct parallel lines, 𝐴𝐵 and 𝐶𝐷, and another distinct line that is also parallel to one of these lines: 𝐶𝐷𝐸𝐹. 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 𝐸𝐹.

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.

Let’s 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 𝐴𝐵𝐶𝐷 and 𝐴𝐵𝐸𝐹, then 𝐶𝐷𝐸𝐹.

Answer

We first recall that if two lines are parallel to the same line, then they must be parallel. Since both 𝐶𝐷 and 𝐸𝐹 are parallel to the same line, 𝐴𝐵, we must have that the two lines are parallel.

Hence, the answer is that 𝐶𝐷𝐸𝐹.

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 𝐿, 𝐿, and 𝐿 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 𝑚𝑀𝑁𝐿=𝑚𝑂𝑁𝑃, as they are vertically opposite angles.

We then see that 𝑂𝑁𝑃 and 𝑀𝑁𝐿 are congruent by the ASA criterion. In particular, this means that 𝐿𝑁=𝑁𝑃. 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.

Let’s 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 𝐴𝐵𝐶𝐷𝐸𝐹 and 𝐶𝐸=2cm, find 𝐴𝐸.

Answer

We can see in the diagram that we are given three parallel lines and two transversals. We can see that 𝐵𝐷=𝐷𝐹, 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, 𝐴𝐶=𝐶𝐸=2cm. We note that 𝐴𝐸=𝐴𝐶+𝐶𝐸, so 𝐴𝐸=2+2=4.cm

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 𝐵𝐹𝐴𝐵, 𝐵𝐹𝐶𝐷, and 𝐵𝐹𝐸𝐹, find the length of 𝐴𝐶, where 𝐵𝐷=𝐷𝐹=3cm and 𝐴𝐸=12cm.

Answer

We are given three lines that are all perpendicular to 𝐵𝐹 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: 𝐴𝐵𝐶𝐷𝐸𝐹.

We can see that 𝐵𝐷=𝐷𝐹=3cm, 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, 𝐴𝐸, has a length of 12 cm, we can halve this length to find the lengths of its segments. We have 𝐴𝐶=𝐴𝐸2=122=6.cm

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 𝐴𝐶𝐵 and lines 𝐴𝑀 and 𝐸𝐷, which are parallel to 𝐶𝐵.

  1. Find the length of 𝐴𝐵.
  2. Find the measure of 𝐴𝐵𝐶.

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 𝐴𝐸=𝐸𝐶, the segments of the other transversal must be equal in length. So, 𝐴𝐷=𝐷𝐵=5mm.

Thus, 𝐴𝐵=𝐴𝐷+𝐷𝐵=5+5=10.mm

Part 2

It appears in the diagram that triangle 𝐴𝐵𝐶 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 𝐿, then it is perpendicular to any line parallel to 𝐿.

Since 𝐸𝐷 is perpendicular to 𝐴𝐶 and 𝐸𝐷 is parallel to 𝐵𝐶, we must have that 𝐵𝐶 is perpendicular to 𝐴𝐶. This means the angle at 𝐶 has a measure of 90, so it is a right triangle. The sum of the measures of the interior angles in a triangle is 180, so 180=35+90+𝑚𝐴𝐵𝐶𝑚𝐴𝐵𝐶=1803590=55.

Alternatively, we can note that 𝐴𝐷𝐸 and 𝐴𝐵𝐶 are corresponding angles of the transversal 𝐴𝐵 of the parallel lines 𝐸𝐷 and 𝐶𝐵.

So, their measures are equal. We can see that 𝑚𝐴𝐸𝐷=90 since it forms a straight angle with a right angle.

We can then determine the measure of 𝐴𝐷𝐸 using the sum of the measures of the internal angles of 𝐴𝐷𝐸. We have 180=𝑚𝐴𝐷𝐸+90+35𝑚𝐴𝐷𝐸=1809035=55.

We then use the fact that 𝐴𝐷𝐸 is congruent to 𝐴𝐵𝐶 to see that 𝐴𝐵𝐶=55.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • If a line is perpendicular to line 𝐴𝐵, then it is perpendicular to any line parallel to 𝐴𝐵.
  • 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.

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