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*}? After completing this Concept, you'll be able to solve problems like this one.

### Watch This

CK-12 Foundation: Chapter7ParallelLinesandTransversalsA

Brightstorm: Proportional Segments Between Parallel Lines

### Guidance

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.

#### Example A

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*}

#### Example B

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*}

#### Example C

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*}

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7ParallelLinesandTransversalsB

#### Concept 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.

\begin{align*}\frac{2640}{x}=\frac{1320}{2380}=\frac{1980}{y}\end{align*}

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*}

### Vocabulary

Pairs of numbers are ** proportional** if they are in the same ratio. Two lines are

**if they have the same slope and thus never intersect. A**

*parallel***is a line intersecting a system of lines.**

*transversal*### Guided Practice

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.

### Practice

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.

- What is the distance between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}?
- What is the distance between points \begin{align*}C\end{align*} and \begin{align*}D\end{align*}?
- What is the distance between points \begin{align*}A\end{align*} and \begin{align*}D\end{align*}?

Using the diagram, answer the questions.

- What is the value of \begin{align*}w\end{align*}?
- What is the value of \begin{align*}x\end{align*}?
- What is the value of \begin{align*}y\end{align*}?
- What is the length of \begin{align*}\overline{AB}\end{align*}?
- What is the length of \begin{align*}\overline{AC}\end{align*}?

Using the diagram, fill in the blank.

- If \begin{align*}b\end{align*} is one-third \begin{align*}d\end{align*}, then \begin{align*}a\end{align*} is ____________________.
- If \begin{align*}c\end{align*} is two times \begin{align*}a\end{align*}, then \begin{align*}b\end{align*} is ____________________.
- \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.