What states if a line divides two sides of a triangle proportionally then it is parallel to the third side?

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 △𝐴𝐷𝐸 and the smaller triangle △𝐴𝐵𝐶.

Since the two pairs of corresponding angles are equal, triangle 𝐴𝐷𝐸 is similar to triangle 𝐴𝐵𝐶: △𝐴𝐷𝐸∼△𝐴𝐵𝐶.

Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In other words, we have 𝐴𝐵𝐴𝐷=𝐴𝐶𝐴𝐸=𝐵𝐶𝐷𝐸.

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.

Using the diagram, which of the following is equal to 𝐴𝐵𝐴𝐷?

1. 𝐴𝐶𝐸𝐶
2. 𝐴𝐵𝐷𝐵
3. 𝐴𝐷𝐷𝐵
4. 𝐴𝐶𝐴𝐸
5. 𝐴𝐸𝐸𝐶

Answer

The diagram indicates that 𝐸𝐷 is parallel to 𝐶𝐵. Since corresponding angles are equal, that is, ∠𝐷𝐸𝐴=∠𝐵𝐶𝐴 and ∠𝐸𝐷𝐴=∠𝐶𝐵𝐴,𝐸𝐷 creates triangle 𝐴𝐷𝐸 that is similar to the larger triangle 𝐴𝐵𝐶.

Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In particular, 𝐴𝐸𝐴𝐶=𝐴𝐷𝐴𝐵.

To find the fraction that is equivalent to 𝐴𝐵𝐴𝐷, we can find the reciprocal of both sides of this equation: 𝐴𝐶𝐴𝐸=𝐴𝐵𝐴𝐷.

𝐴𝐶𝐴𝐸 is equal to 𝐴𝐵𝐴𝐷.

Find the value of 𝑥.

Answer

𝐴𝐶 and 𝐴𝐵 are transversals that intersect parallel lines ⃖⃗𝐷𝐸 and ⃖⃗𝐵𝐶. Since the two pairs of corresponding angles created by this intersection are equal, that is, ∠𝐷𝐸𝐴=∠𝐵𝐶𝐴,∠𝐸𝐷𝐴=∠𝐶𝐵𝐴, we can say that triangle 𝐴𝐷𝐸 is similar to triangle 𝐴𝐵𝐶: △𝐴𝐵𝐶∼△𝐴𝐷𝐸.

When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. In particular, 𝐴𝐷𝐴𝐵=𝐷𝐸𝐵𝐶.

By substituting in known values for the lengths of 𝐴𝐷, 𝐷𝐸, and 𝐴𝐵 (where we should note that 𝐴𝐵 is the sum of 𝐴𝐷 and 𝐷𝐵), we can find the value of 𝑥: 1010+11=10𝑥. Solving for 𝑥, 𝑥=21.

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 𝐴𝐵𝐶 and 𝐴𝐷𝐸 are similar, we obtain the equal proportions: 𝐴𝐵𝐴𝐷=𝐴𝐶𝐴𝐸.

From this diagram, we also note that the line segments 𝐴𝐷 and 𝐴𝐸 can be split as follows: 𝐴𝐷=𝐴𝐵+𝐵𝐷𝐴𝐸=𝐴𝐶+𝐶𝐸.and

Substituting these expressions into our earlier equation and rearranging, 𝐴𝐵𝐴𝐷=𝐴𝐶𝐴𝐸𝐴𝐵𝐴𝐵+𝐵𝐷=𝐴𝐶𝐴𝐶+𝐶𝐸𝐴𝐵(𝐴𝐶+𝐶𝐸)=𝐴𝐶(𝐴𝐵+𝐵𝐷)𝐴𝐵⋅𝐴𝐶+𝐴𝐵⋅𝐶𝐸=𝐴𝐶⋅𝐴𝐵+𝐴𝐶⋅𝐵𝐷.

We can now subtract 𝐴𝐵⋅𝐴𝐶 from both sides to find 𝐴𝐵⋅𝐶𝐸=𝐴𝐶⋅𝐵𝐷,𝐴𝐵𝐵𝐷=𝐴𝐶𝐶𝐸.

This leads us to the definition of a theorem that links the line segments created when a parallel side is added to a triangle.

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.

In the figure, 𝑋𝑌 and 𝐵𝐶 are parallel. If 𝐴𝑋=18, 𝑋𝐵=24, and 𝐴𝑌=27, what is the length of 𝑌𝐶?

Answer

We are given that 𝑋𝑌 is parallel to 𝐵𝐶. 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, 𝐴𝑌𝑌𝐶=𝐴𝑋𝑋𝐵.

Substituting 𝐴𝑋=18, 𝑋𝐵=24, and 𝐴𝑌=27 into this equation and solving for 𝑌𝐶, 27𝑌𝐶=1824𝑌𝐶27=2418𝑌𝐶=2418×27=36.

The length of 𝑌𝐶 is 36.

In our next example, we will demonstrate how to solve multistep problems involving triangles and parallel lines.

The given figure shows a triangle 𝐴𝐵𝐶.

1. Work out the value of 𝑥.
2. Work out the value of 𝑦.

Answer

Part 1

In the figure, a line parallel to side 𝐵𝐶 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 𝐷𝐸, we obtain 𝐴𝐷𝐷𝐵=𝐴𝐸𝐸𝐶.

This gives us an equation that can be solved for 𝑥: 32𝑥+3=2𝑥+53(𝑥+5)=2(2𝑥+3)3𝑥+15=4𝑥+615=𝑥+6𝑥=9.

Part 2

Now that we know the value of 𝑥, we can use this information to find the value of 𝑦. Since the two pairs of corresponding angles created by the intersection of 𝐷𝐸 are equal, triangle 𝐴𝐵𝐶 is similar to triangle 𝐴𝐷𝐸: △𝐴𝐵𝐶∼△𝐴𝐷𝐸.

In particular, 𝐴𝐷𝐴𝐵=𝐷𝐸𝐵𝐶.

The length of 𝐴𝐵 is the sum of the lengths of 𝐴𝐷 and 𝐷𝐵. We are given that 𝐴𝐷=3 and 𝐷𝐵=2𝑥+3. Since 𝑥=9, 𝐷𝐵=21. Therefore, 𝐴𝐵=3+21=24.

Substituting these values into our earlier equation and solving for 𝑦, 324=2𝑦𝑦24=23𝑦=23×24=16.

Therefore, 𝑦=16.

In our next example, we will demonstrate how to apply the side splitter theorem to a triangle which contains several pairs of parallel lines.

Find the length of 𝐶𝐵.

Answer

From the given diagram we note that 𝐷𝐹 is parallel to 𝐴𝐸 in the triangle 𝐶𝐴𝐸, and 𝐷𝐸 is parallel to 𝐴𝐵 in the triangle 𝐶𝐴𝐵. 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 𝐶𝐴𝐸 where 𝐷𝐹 is parallel to one side of the triangle, we obtain 𝐶𝐹𝐹𝐸=𝐶𝐷𝐷𝐴.

Since 𝐷𝐸 is parallel to one side of the larger triangle 𝐶𝐴𝐵, we can also obtain 𝐶𝐸𝐸𝐵=𝐶𝐷𝐷𝐴.

Both 𝐶𝐹𝐹𝐸 and 𝐶𝐸𝐸𝐵 are equal to 𝐶𝐷𝐷𝐴. This means we can set 𝐶𝐹𝐹𝐸=𝐶𝐸𝐸𝐵.

We can substitute the given values 𝐶𝐹=15, 𝐹𝐸=6, and 𝐶𝐸=15+6=21 into this equation to obtain an equation that can be solved for 𝐸𝐵: 156=21𝐸𝐵𝐸𝐵=21×615.

Therefore, 𝐸𝐵=8.4.cm

Since 𝐶𝐵=𝐶𝐹+𝐹𝐸+𝐸𝐵, 𝐶𝐵=15+6+8.4=29.4.cm

The length of 𝐶𝐵 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.

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, 𝐴𝐵𝐶 is a triangle and ⃖⃗𝐷𝐸 intersects ⃖⃗𝐴𝐵 at 𝐷 and ⃖⃗𝐴𝐶 at 𝐸.

If 𝐴𝐷𝐷𝐵=𝐴𝐸𝐸𝐶, then ⃖⃗𝐷𝐸 must be parallel to ⃖⃗𝐵𝐶.

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.

Given that 𝐴𝐵𝐶𝐷 is a parallelogram, find the length of 𝑌𝑍.

Answer

To find the length of 𝑌𝑍, we will begin by identifying relevant information about triangles 𝑋𝑌𝑍 and 𝑋𝐷𝐶. We are given that 𝑋𝑌=𝑌𝐷 and 𝑋𝑍=𝑍𝐶. 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 𝑋𝐷 and 𝑋𝐶 of the larger triangle 𝑋𝐷𝐶 have been divided into equal proportions, we can apply the converse of this theorem to deduce that 𝐷𝐶 and 𝑌𝑍 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 △𝑋𝑌𝑍∼△𝑋𝐷𝐶.

Since 𝐷𝐶 is the opposite side of 𝐴𝐵 in the parallelogram 𝐴𝐵𝐶𝐷, these two sides must have the same lengths. Hence, the length of 𝐷𝐶 is 134.9 cm. Denoting the length of 𝑋𝑌by an unknown constant 𝑥, we can draw the following diagram.

Since triangles 𝑋𝑌𝑍 and 𝑋𝐷𝐶 are similar, we can form an equation that links the lengths of the sides 𝑋𝑌, 𝑋𝐷, 𝑌𝑍, and 𝐷𝐶: 𝑋𝑌𝑋𝐷=𝑌𝑍𝐷𝐶𝑥2𝑥=𝑌𝑍134.912=𝑌𝑍134.9.

Solving for 𝑌𝑍, we find 𝑌𝑍=134.92=67.45.

The length of 𝑌𝑍 is 67.45 cm.

We will now recap the key points from this explainer.

• 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.