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Parallel Lines and Transversals

Three or more parallel lines cut by two transversals divide them proportionally.

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Parallel Lines and Transversals

What if you were given the street map, below, of Washington DC and told to find the missing street lengths? \begin{align*}R\end{align*} Street, \begin{align*}Q\end{align*} Street, and \begin{align*}O\end{align*} Street are parallel and \begin{align*}7^{th}\end{align*} Street is perpendicular to all three. \begin{align*}R\end{align*} and \begin{align*}Q\end{align*} are one “city block” (usually \begin{align*}\frac{1}{4}\end{align*} mile or 1320 feet) apart. The other given measurements are on the map. What are \begin{align*}x\end{align*} and \begin{align*}y\end{align*}?

Parallel Transverals 

The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. We can extend this theorem to a situation outside of triangles where we have multiple parallel lines cut by transverals.

Theorem: If three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

Solving for an Unknown Value 

1. Find \begin{align*}a\end{align*}.

The three lines are marked parallel, so you can set up a proportion.

\begin{align*}\frac{a}{20} &= \frac{9}{15}\\ 180 &= 15a\\ a &= 12\end{align*}

2. Find \begin{align*}b\end{align*}.

To solve, set up a proportion.

\begin{align*}\frac{12}{9.6} &= \frac{b}{24}\\ 288 &= 9.6b\\ b &= 30\end{align*}

Solving for an Unknown Value that Makes Lines Paralell 

Find the value of \begin{align*}x\end{align*} that makes the lines parallel.

To solve, set up a proportion and solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{5}{8} = \frac{3.75}{2x-4} \longrightarrow \ 5(2x-4) &= 8(3.75)\\ 10x-20 &= 30\\ 10x &= 50\\ x &= 5\end{align*}

Earlier Problem Revisited

To find \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, you need to set up a proportion using parallel the parallel lines.


From this, \begin{align*}x = 4760 \ ft\end{align*} and \begin{align*}y = 3570 \ ft\end{align*}.

To find \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*}, use the Pythagorean Theorem.

\begin{align*}2640^2+a^2 &= 4760^2\\ 3960^2+b^2 &= 7140^2\\ 5940^2+c^2 &= 10710^2\end{align*}

\begin{align*}a = 3960.81, \ b = 5941.21, \ c = 8911.82\end{align*}


Example 1 

Find \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*}.

Look at the corresponding segments. Only the segment marked “2” is opposite a number, all the other segments are opposite variables. That means we will be using this ratio, 2:3 in all of our proportions.

\begin{align*}\frac{a}{2} &= \frac{9}{3} && \ \ \frac{2}{4}=\frac{3}{b} && \ \frac{2}{3}=\frac{3}{c}\\ 3a &= 18 && \ 2b=12 && 2c=9\\ a &= 6 && \ \ \ b=6&& \ \ c=4.5\end{align*}

There are several ratios you can use to solve this example. To solve for \begin{align*}b\end{align*}, you could have used the proportion \begin{align*}\frac{6}{4}=\frac{9}{b}\end{align*}, which will still give you the same answer.


Find the value of each variable in the pictures below.

The street map shows part of New Orleans. Burgundy St., Dauphine St. and Royal St. are parallel to each other. If Spain St. is perpendicular to all three, find the indicated distances.

  1. What is the distance between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}?
  2. What is the distance between points \begin{align*}C\end{align*} and \begin{align*}D\end{align*}?
  3. What is the distance between points \begin{align*}A\end{align*} and \begin{align*}D\end{align*}?

Using the diagram, answer the questions.

  1. What is the value of \begin{align*}w\end{align*}?
  2. What is the value of \begin{align*}x\end{align*}?
  3. What is the value of \begin{align*}y\end{align*}?
  4. What is the length of \begin{align*}\overline{AB}\end{align*}?
  5. What is the length of \begin{align*}\overline{AC}\end{align*}?

Using the diagram, fill in the blank.

  1. If \begin{align*}b\end{align*} is one-third \begin{align*}d\end{align*}, then \begin{align*}a\end{align*} is ____________________.
  2. If \begin{align*}c\end{align*} is two times \begin{align*}a\end{align*}, then \begin{align*}b\end{align*} is ____________________.
  3. \begin{align*}{\;}\end{align*} This is a map of lake front properties. Find \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, the length of the edge of Lot 1 and Lot 2 that is adjacent to the lake.

Review (Answers)

To view the Review answers, open this PDF file and look for section 7.9. 

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Coordinate Plane The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.
Perpendicular Perpendicular lines are lines that intersect at a 90^{\circ} angle. The product of the slopes of two perpendicular lines is -1.
Proportion A proportion is an equation that shows two equivalent ratios.
Quadrilateral A quadrilateral is a closed figure with four sides and four vertices.
transversal A transversal is a line that intersects two other lines.
Triangle Proportionality Theorem The Triangle Proportionality Theorem states that if a line is parallel to one side of a triangle and it intersects the other two sides, then it divides those sides proportionally.

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