In this explainer, we will learn how to find missing lengths in a triangle containing two or three parallel lines using proportionality.
Recall that when two parallel lines are cut by a transversal, the resulting corresponding angles are equal.
By adding a second transversal as pictured below, we can form two triangles.
Giving each vertex a label, we can define the larger triangle and the smaller triangle .
Since the two pairs of corresponding angles are equal, triangle is similar to triangle :
Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In other words, we have
In the first example, we will demonstrate how to use this definition of the similarity of triangles to identify which pairs of side lengths have equal proportions when a triangle is cut by a line parallel to one of its sides.
Example 1: Identifying Proportions in Triangles
Using the diagram, which of the following is equal to ?
Answer
The diagram indicates that is parallel to . Since corresponding angles are equal, that is, and creates triangle that is similar to the larger triangle .
Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In particular,
To find the fraction that is equivalent to , we can find the reciprocal of both sides of this equation:
is equal to .
Example 2: Finding an Unknown Length in a Triangle Using Proportions
Find the value of .
Answer
and are transversals that intersect parallel lines and . Since the two pairs of corresponding angles created by this intersection are equal, that is, we can say that triangle is similar to triangle :
When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. In particular,
By substituting in known values for the lengths of , , and (where we should note that is the sum of and ), we can find the value of : Solving for ,
In the previous two examples, we noted that, if a line intersecting two sides of a triangle is parallel to the third side, then the smaller triangle created by the parallel line is similar to the original triangle. We recall the diagram we presented earlier.
Since triangles and are similar, we obtain the equal proportions:
From this diagram, we also note that the line segments and can be split as follows:
Substituting these expressions into our earlier equation and rearranging,
We can now subtract from both sides to find
This leads us to the definition of a theorem that links the line segments created when a parallel side is added to a triangle.
Theorem: Side Splitter Theorem
If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
Note:
The side splitter theorem can be extended to include parallel lines that lie outside of the triangle. When a straight line lies outside of a triangle and is parallel to one side of the triangle, it forms another triangle that is similar to the first one. This is demonstrated in the following diagram. In this case, an analog of the side splitter theorem can be deduced directly from the similar triangles.
In our next example, we will see how to use this theorem to identify proportional segments of triangles to calculate a missing length.
Example 3: Using Proportions in a Triangle to Calculate an Unknown Length
In the figure, and are parallel. If , , and , what is the length of ?
Answer
We are given that is parallel to . The side splitter theorem says that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
In particular,
Substituting , , and into this equation and solving for ,
The length of is 36.
In our next example, we will demonstrate how to solve multistep problems involving triangles and parallel lines.
Example 4: Finding Unknowns in an Applied Problem
The given figure shows a triangle .
- Work out the value of .
- Work out the value of .
Answer
Part 1
In the figure, a line parallel to side is intersecting the other two sides of the triangle. The side splitter theorem tells us that this line divides those sides proportionally.
Labelling this line segment as , we obtain
This gives us an equation that can be solved for :
Part 2
Now that we know the value of , we can use this information to find the value of . Since the two pairs of corresponding angles created by the intersection of are equal, triangle is similar to triangle :
In particular,
The length of is the sum of the lengths of and . We are given that and . Since , . Therefore,
Substituting these values into our earlier equation and solving for ,
Therefore,
In our next example, we will demonstrate how to apply the side splitter theorem to a triangle which contains several pairs of parallel lines.
Example 5: Finding a Side Length in a Triangle Using the Relation between Parallel Lines
Find the length of .
Answer
From the given diagram we note that is parallel to in the triangle , and is parallel to in the triangle . The side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Applying this theorem to triangle where is parallel to one side of the triangle, we obtain
Since is parallel to one side of the larger triangle , we can also obtain
Both and are equal to . This means we can set
We can substitute the given values , , and into this equation to obtain an equation that can be solved for :
Therefore,
Since ,
The length of is 29.4 cm.
Recall that the side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Moreover, we have learned that this theorem can be extended to include parallel lines that lie outside of the triangle. It turns out that the converse of this result is also true, which proves very useful when solving problems of this type.
Theorem: The Converse of the Side Splitter Theorem
If a line intersects two sides of a triangle and splits those sides in equal proportions, then that line must be parallel to the third side of the triangle.
In all three diagrams above, is a triangle and intersects at and at .
If , then must be parallel to .
By applying the converse of the side splitter theorem, we are able to prove that a straight line is parallel to one side of a triangle due to having proportional parts. In our final example, we will demonstrate this process.
Example 6: Finding the Unknown Lengths in a Triangle given the Other Sidesโ Lengths Using the Relations of Parallel Lines
Given that is a parallelogram, find the length of .
Answer
To find the length of , we will begin by identifying relevant information about triangles and . We are given that and . We also recall that the side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Conversely, if a line splits two sides of a triangle into equal proportions, then that line must be parallel to the third side. Since sides and of the larger triangle have been divided into equal proportions, we can apply the converse of this theorem to deduce that and must be parallel.
We also recall that if a line parallel to a side of a triangle intersects two other sides, then the smaller triangle created by the parallel line is similar to the original triangle. Hence, we obtain
Since is the opposite side of in the parallelogram , these two sides must have the same lengths. Hence, the length of is 134.9 cm. Denoting the length of by an unknown constant , we can draw the following diagram.
Since triangles and are similar, we can form an equation that links the lengths of the sides , , , and :
Solving for , we find
The length of is 67.45 cm.
We will now recap the key points from this explainer.
Key Points
- If a line intersecting two sides of a triangle is parallel to the remaining side, then the smaller triangle created by the parallel line is similar to the larger, original triangle.
- The side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
- The side splitter theorem can be extended to include parallel lines that lie outside a triangle. If a line lying outside a triangle is parallel to one side of the triangle and intersects the extensions of the other two sides of the triangle, then the line divides the extensions of those sides proportionally.
- The converse of the side splitter theorem states that if a line splits two sides of a triangle proportionally, then that line is parallel to the remaining side.
