Lesson Explainer: Parallel Lines in a Triangle Mathematics

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 △𝐴𝐡𝐢.

Since the two pairs of corresponding angles are equal, triangle 𝐴𝐷𝐸 is similar to triangle 𝐴𝐡𝐢: β–³π΄π·πΈβˆΌβ–³π΄π΅πΆ.

Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In other words, we have 𝐴𝐡𝐴𝐷=𝐴𝐢𝐴𝐸=𝐡𝐢𝐷𝐸.

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 𝐴𝐡𝐴𝐷?

  1. 𝐴𝐢𝐸𝐢
  2. 𝐴𝐡𝐷𝐡
  3. 𝐴𝐷𝐷𝐡
  4. 𝐴𝐢𝐴𝐸
  5. 𝐴𝐸𝐸𝐢

Answer

The diagram indicates that 𝐸𝐷 is parallel to 𝐢𝐡. Since corresponding angles are equal, that is, ∠𝐷𝐸𝐴=∠𝐡𝐢𝐴 and ∠𝐸𝐷𝐴=∠𝐢𝐡𝐴,𝐸𝐷 creates triangle 𝐴𝐷𝐸 that is similar to the larger triangle 𝐴𝐡𝐢.

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

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

𝐴𝐢𝐴𝐸 is equal to 𝐴𝐡𝐴𝐷.

Example 2: Finding an Unknown Length in a Triangle Using Proportions

Find the value of π‘₯.

Answer

𝐴𝐢 and 𝐴𝐡 are transversals that intersect parallel lines ⃖⃗𝐷𝐸 and ⃖⃗𝐡𝐢. Since the two pairs of corresponding angles created by this intersection are equal, that is, ∠𝐷𝐸𝐴=∠𝐡𝐢𝐴,∠𝐸𝐷𝐴=∠𝐢𝐡𝐴, we can say that triangle 𝐴𝐷𝐸 is similar to triangle 𝐴𝐡𝐢: β–³π΄π΅πΆβˆΌβ–³π΄π·πΈ.

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

By substituting in known values for the lengths of 𝐴𝐷, 𝐷𝐸, and 𝐴𝐡 (where we should note that 𝐴𝐡 is the sum of 𝐴𝐷 and 𝐷𝐡), we can find the value of π‘₯: 1010+11=10π‘₯. Solving for π‘₯, π‘₯=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 𝐴𝐡𝐢 and 𝐴𝐷𝐸 are similar, we obtain the equal proportions: 𝐴𝐡𝐴𝐷=𝐴𝐢𝐴𝐸.

From this diagram, we also note that the line segments 𝐴𝐷 and 𝐴𝐸 can be split as follows: 𝐴𝐷=𝐴𝐡+𝐡𝐷𝐴𝐸=𝐴𝐢+𝐢𝐸.and

Substituting these expressions into our earlier equation and rearranging, 𝐴𝐡𝐴𝐷=𝐴𝐢𝐴𝐸𝐴𝐡𝐴𝐡+𝐡𝐷=𝐴𝐢𝐴𝐢+𝐢𝐸𝐴𝐡(𝐴𝐢+𝐢𝐸)=𝐴𝐢(𝐴𝐡+𝐡𝐷)𝐴𝐡⋅𝐴𝐢+𝐴𝐡⋅𝐢𝐸=𝐴𝐢⋅𝐴𝐡+𝐴𝐢⋅𝐡𝐷.

We can now subtract 𝐴𝐡⋅𝐴𝐢 from both sides to find 𝐴𝐡⋅𝐢𝐸=𝐴𝐢⋅𝐡𝐷,𝐴𝐡𝐡𝐷=𝐴𝐢𝐢𝐸.

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, π‘‹π‘Œ and 𝐡𝐢 are parallel. If 𝐴𝑋=18, 𝑋𝐡=24, and π΄π‘Œ=27, what is the length of π‘ŒπΆ?

Answer

We are given that π‘‹π‘Œ is parallel to 𝐡𝐢. 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, π΄π‘Œπ‘ŒπΆ=𝐴𝑋𝑋𝐡.

Substituting 𝐴𝑋=18, 𝑋𝐡=24, and π΄π‘Œ=27 into this equation and solving for π‘ŒπΆ, 27π‘ŒπΆ=1824π‘ŒπΆ27=2418π‘ŒπΆ=2418Γ—27=36.

The length of π‘ŒπΆ 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 𝐴𝐡𝐢.

  1. Work out the value of π‘₯.
  2. Work out the value of 𝑦.

Answer

Part 1

In the figure, a line parallel to side 𝐡𝐢 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 𝐴𝐷𝐷𝐡=𝐴𝐸𝐸𝐢.

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

Part 2

Now that we know the value of π‘₯, 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 𝐴𝐡𝐢 is similar to triangle 𝐴𝐷𝐸: β–³π΄π΅πΆβˆΌβ–³π΄π·πΈ.

In particular, 𝐴𝐷𝐴𝐡=𝐷𝐸𝐡𝐢.

The length of 𝐴𝐡 is the sum of the lengths of 𝐴𝐷 and 𝐷𝐡. We are given that 𝐴𝐷=3 and 𝐷𝐡=2π‘₯+3. Since π‘₯=9, 𝐷𝐡=21. Therefore, 𝐴𝐡=3+21=24.

Substituting these values into our earlier equation and solving for 𝑦, 324=2𝑦𝑦24=23𝑦=23Γ—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.

Example 5: Finding a Side Length in a Triangle Using the Relation between Parallel Lines

Find the length of 𝐢𝐡.

Answer

From the given diagram we note that 𝐷𝐹 is parallel to 𝐴𝐸 in the triangle 𝐢𝐴𝐸, and 𝐷𝐸 is parallel to 𝐴𝐡 in the triangle 𝐢𝐴𝐡. 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 𝐢𝐴𝐸 where 𝐷𝐹 is parallel to one side of the triangle, we obtain 𝐢𝐹𝐹𝐸=𝐢𝐷𝐷𝐴.

Since 𝐷𝐸 is parallel to one side of the larger triangle 𝐢𝐴𝐡, we can also obtain 𝐢𝐸𝐸𝐡=𝐢𝐷𝐷𝐴.

Both 𝐢𝐹𝐹𝐸 and 𝐢𝐸𝐸𝐡 are equal to 𝐢𝐷𝐷𝐴. This means we can set 𝐢𝐹𝐹𝐸=𝐢𝐸𝐸𝐡.

We can substitute the given values 𝐢𝐹=15, 𝐹𝐸=6, and 𝐢𝐸=15+6=21 into this equation to obtain an equation that can be solved for 𝐸𝐡: 156=21𝐸𝐡𝐸𝐡=21Γ—615.

Therefore, 𝐸𝐡=8.4.cm

Since 𝐢𝐡=𝐢𝐹+𝐹𝐸+𝐸𝐡, 𝐢𝐡=15+6+8.4=29.4.cm

The length of 𝐢𝐡 is 29.4 cm.

By applying the inverse 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 Sides’ Lengths Using the Relations of Parallel Lines

Given that 𝐴𝐡𝐢𝐷 is a parallelogram, find the length of π‘Œπ‘.

Answer

To find the length of π‘Œπ‘, we will begin by identifying relevant information about triangles π‘‹π‘Œπ‘ and 𝑋𝐷𝐢. We are given that π‘‹π‘Œ=π‘Œπ· and 𝑋𝑍=𝑍𝐢. 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 line. Since sides 𝑋𝐷 and 𝑋𝐢 of the larger triangle 𝑋𝐷𝐢 have been divided into equal proportions, we can apply the converse of this theorem to deduce that 𝐷𝐢 and π‘Œπ‘ 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 β–³π‘‹π‘Œπ‘βˆΌβ–³π‘‹π·πΆ.

Since 𝐷𝐢 is the opposite side of 𝐴𝐡 in the parallelogram 𝐴𝐡𝐢𝐷, these two sides must have the same lengths. Hence, the length of 𝐷𝐢 is 134.9 cm. Denoting the length of π‘‹π‘Œby an unknown constant π‘₯, we can draw the following diagram.

Since triangles π‘‹π‘Œπ‘ and 𝑋𝐷𝐢 are similar, we can form an equation that links the lengths of the sides π‘‹π‘Œ, 𝑋𝐷, π‘Œπ‘, and 𝐷𝐢: π‘‹π‘Œπ‘‹π·=π‘Œπ‘π·πΆπ‘₯2π‘₯=π‘Œπ‘134.912=π‘Œπ‘134.9.

Solving for π‘Œπ‘, we find π‘Œπ‘=134.92=67.45.

The length of π‘Œπ‘ 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.
  • If a line splits two sides of a triangle proportionally, then that line is parallel to the remaining side.

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