In this explainer, we will learn how to use parallelism of lines to find a missing length of a line segment in a transversal line cut by parallel lines.
Definition: Transversal
A transversal is a line that intersects two or more lines in the same plane at distinct points.
The lines that a transversal intersects do not have to be parallel, but in each of the problems we will look at, they are.
Two examples of transversals are and in the following figure, because they both intersect each of the three parallel lines at distinct points. We can see that the transversal intersects the lines at points , , and , while the transversal intersects them at points , , and .
Notice that the intersections of the three parallel lines and the two transversals create four different line segments. A segment with endpoints at and and another segment with endpoints at and both lie on the transversal . Likewise, a segment with endpoints at and and another segment with endpoints at and both lie on the transversal .
When a transversal is cut by parallel lines, the corresponding angles are congruent. Thus, we know that quadrilaterals and in the figure are similar. Each of our four segments is a side of one of these quadrilaterals, with corresponding to and corresponding to . The fact that corresponding sides of similar figures are proportional leads us to a theorem of parallel lines and transversals.
Theorem: The Basic Proportionality Theorem (Thalesβs Theorem)
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
Based on this theorem, we know that, in our figure, the ratio of the length of to that of is equal to the ratio of the length of to that of . We can write this as the proportion
Notice that, in this proportion, the ratio on each side of the equation contains the lengths of segments on the same transversal. It is worth noting that we could also write a proportion in which the ratio on each side of the equation contains the lengths of corresponding segments on the two different transversals. That is,
Solving each of these proportions for an unknown segment length will give the same result, because cross-multiplying will lead to either the equation or an equivalent equation, regardless of which of the two proportions is used.
Suppose now that our figure had appeared as follows, indicating that and are congruent.
This would tell us that , which would allow us to substitute into the proportion for , giving us
We would then be able to simplify the right side of the equation to 1, so we would know that . This leads us to another theorem of parallel lines and transversals.
Theorem: Thalesβs Special Theorem
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Letβs now use these theorems in the problems that follow to find missing segment lengths on transversals when two transversals are intersected by three or more lines.
Example 1: Using the Properties of Parallel Lines and Transversals to Find a Missing Side Length
Using the information in the figure, determine the length of .
Answer
In the figure, we can see that we have three parallel lines:
We can also see that these parallel lines are cut by the two transversals and . Remember that a transversal is a line that intersects two or more lines in the same plane at distinct points. The basic proportionality theorem (Thalesβs theorem) tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. For this reason, we know that the ratio of the length of to that of must be equivalent to the ratio of the length of to that of . Thus, we can write the proportion
The figure shows us that , , and , so we can substitute these values into the proportion to get
Cross-multiplying then gives us the equation which we can simplify to
Finally, dividing both sides of the equation by 47, we get
Thus, we know that the length of in the figure is 144 cm.
Note
Another proportion we could write to solve the problem is
We can see that, in this proportion, the ratio on each side of the equation contains the lengths of corresponding segments on the two different transversals rather than the lengths of segments on the same transversal. Substituting into this proportion gives us and by multiplying both sides of the equation by 141, we arrive at
Thus, we get the same length for regardless of which of the two proportions we use. We again find that the length of in the figure is 144 cm.
In the next example, we will also use the basic proportionality theorem (Thalesβs theorem) to find the length of a line segment. This time, however, we will also need to use the fact that the length of a line segment is equal to the sum of the lengths of the shorter disjoint segments that form it to solve the problem.
Example 2: Using the Properties of Parallel Lines and Multiple Transversals to Find a Missing Side Length
In the figure, lines , , , and are all parallel. Given that , , , and , what is the length of ?
Answer
In this problem, we are told that
The lengths of , , , and are also given to us, so we can label the figure as shown:
We can see that each of the lines , , , and is cut by the two transversals and . Recall that the basic proportionality theorem (Thalesβs theorem) states that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. This tells us that the ratio of the length of to that of must be equal to the ratio of the length of to that of . This allows us to write the proportion
Notice, however, that we are not given the length of either or . We are given the lengths of both and , though, so we can use the fact that the length of a line segment is equal to the sum of the lengths of the shorter disjoint segments that form it to rewrite the proportion in a way that will allow us to solve the problem. Since , we get
After substituting the given values of , , , and into this proportion, we then arrive at which we can rewrite as
Next, cross-multiplying gives us the equation which we can simplify to
Finally, to solve for , we can divide both sides of the equation by 12 to get
We can therefore conclude that the length of in the figure is 10.
Next, letβs look at a problem in which we can use Thalesβs special theorem to determine a line segmentβs length.
Example 3: Finding the Lengths of Proportional Line Segments using the Properties of Parallel Lines and Multiple Traversals
Given that , find the length of .
Answer
We can see that, in this problem, we have four parallel lines:
We can also see that these parallel lines cut off , , and on the transversal , with , as well as , , and on the transversal , with the length of being 9 cm. Based on this information, we are asked to find the length of .
Remember that Thalesβs special theorem tells us that if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Thus, since we know that , we also know that .
To begin, we can use the fact that the length of a line segment is equal to the sum of the lengths of the shorter disjoint segments that form it to write the equation
Since , we know that . This allows us to substitute into the equation for both and to get which we can then rewrite as
Recall that we were told that the length of is 9 cm, so we can now substitute 9 into the equation for to get and divide both sides by 3, giving us
Remember that and have the same length, so we also know that
Next, because the length of a line segment is equal to the sum of the lengths of the segments of which it is comprised, we can write the equation and after substituting 3 for both and , we arrive at
Thus, the length of in the figure is 6 cm.
Note
An alternate solution method is to use the basic proportionality theorem (Thalesβs theorem). According to this theorem, if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. In order to find the length of using this theorem, we must set up and solve a proportion using the information we have been given.
Letβs begin by assuming that the length of is . That is,
Because the length of a line segment is equal to the sum of the lengths of the shorter disjoint segments that form it, we can write the equation
Substituting into the equation for then gives us
Since , we know that . Therefore, we can now substitute into the equation for both and to get which we can rewrite as
Next, dividing both sides by 3, we arrive at
Since we were told that is congruent to , we also know that . Thus, again using the fact that the length of a line segment is equal to the sum of the lengths of the segments of which it is comprised, we can determine the length of in terms of :
Now that we know that the length of is and that the length of is , we can set up a proportion that we can then solve to find the length of . In the figure, we can see that corresponds to and that corresponds to , so we can write
Next, substituting for , for , and 9 for , we get
This equation can be simplified to and after multiplying both sides by 9, we arrive at
Thus, we have again determined that the length of in the figure is 6 cm.
The lengths of line segments are not always integers. They can also be expressed in terms of variables. In the next problem, we will look at an example of this, and we will solve for the variable, using the basic proportionality theorem (Thalesβs theorem) as we do so.
Example 4: Using the Properties of Parallel Lines and Solving Linear Equations to Determine the Value of an Unknown and the Length of a Line Segment
In the diagram below, , , , , and . Find the value of and the length of .
Answer
Letβs begin by using the lengths of , , , , and that have been given to us to label the diagram as shown:
We can see that, in the diagram, we have four parallel lines:
We can also see that each of these lines is cut by the two transversals and . The basic proportionality theorem (Thalesβs theorem) tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. Thus, we know that that the ratio of the length of to that of must be equal to the ratio of the length of to that of . This allows us to write the proportion
Substituting the given lengths for , , , and into this proportion then gives us and after simplifying the right side, we get
Now, to eliminate the denominator on the left side, we can multiply both sides by to get and to solve the resulting equation for , we can subtract 1 from both sides to get
Next, to find the length of , we can again use the basic proportionality theorem (Thalesβs theorem) to write the proportion
Substituting the given values of , , and into this proportion then gives us
Notice that the numerators of the fractions on both sides of the proportion are the same. We, therefore, know that the denominators of the fractions must also be the same, so
In summary, in the diagram, we can deduce that and .
As a final example, we will solve for two variables instead of one, this time using Thalesβs special theorem to help us find the answer.
Example 5: Finding the Lengths of Sides Using the Properties of Parallel Lines to Form Linear Equations
In the given figure, find the values of and .
Answer
By examining the figure, we can see that we have three parallel lines:
We can also see that these parallel lines cut off and on the transversal , as well as and on the transversal , with .
Remember that the Thalesβs special theorem tells us that if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Thus, since we know that , we also know that .
Congruent segments have lengths that are equal, so it follows that
The figure shows us that , , , and , so we can substitute the appropriate expression from the figure for each segment length, giving us the equations
Letβs solve each of these equations in turn. First, to solve the equation , we can subtract from both sides to get and then add 20 to both sides so that our equation becomes
Finally, we can divide both sides by 2 to arrive at
Next, to solve the equation , we can subtract from both sides to get and then add 25 to both sides so that our equation becomes
Finally, we can divide both sides by 2 to arrive at
Thus, in the figure we can deduce that and .
Note
An alternate solution method is to use the basic proportionality theorem (Thalesβs theorem). Recall that this theorem tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. For this reason, we know that the ratio of the length of to that of must be equivalent to the ratio of the length of to that of . Therefore, we can write the proportion
We can then substitute the appropriate expression into the proportion for each segmentβs length to get
Because we know that , we also know that . It follows that which we can rewrite as the separate equations
To solve the equation , we can begin by multiplying both sides by to eliminate the denominator on the left side. This gives us which is the same equation we solved with our previous method to find the value of .
To solve the equation , we can again begin by eliminating the denominator, this time by multiplying both sides of the equation by . After doing so, we get which is the same equation we solved with our previous method to find the value of . Therefore, we will get the same - and -values by using this method. Again, we will find that and .
Now, letβs finish by recapping some key points.
Key Points
- A transversal is a line that intersects two or more lines in the same plane at distinct points.
- The basic proportionality theorem (Thalesβs theorem) states that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
- Thalesβs special theorem states that if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
- The length of a line segment is equal to the sum of the lengths of the shorter disjoint segments that form it.
