4.3: Transversals to Three Parallel Lines

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In Chapter 1 we defined a transversal to be a line which intersects two other lines, We will now extend the definition to a line which intersects three other lines. In Figure $\PageIndex{1}$, $AB$ is a transversal to three lines.

屏幕快照 2020-11-17 下午7.05.15.png — Figure $\PageIndex{1}$: $AB$ is a transversal to three lines.

If the three lines are parallel and we have two such transversals we may state the following theorem:

Theorem $\PageIndex{1}$

The line segments formed by two transversals crossing three parallel lines are proportional.

In Figure $\PageIndex{2}$, $\dfrac{a}{b} = \dfrac{c}{d}$.

屏幕快照 2020-11-17 下午7.09.40.png — Figure $\PageIndex{2}$: The line segments formed by the transversals are proportional.

Example $\PageIndex{1}$

Find $x$:

$屏幕快照 2020-11-17 下午7.12.17.png$

Solution

\[\begin{array} {rcl} {\dfrac{x}{3}} & = & {\dfrac{8}{4}} \\ {4x} & = & {24} \\ {x} & = & {6} \end{array}\]

Check:

$屏幕快照 2020-11-17 下午7.12.37.png$

Answer: $x = 6$.

Proof of Theorem $\PageIndex{1}$

Draw $GB$ and $HC$ parallel to $DF$ (Figure $\PageIndex{3}$). The corresponding angles of the parallel lines are equal and so $\triangle BCH \sim \triangle ABG$. Therefore

\[\dfrac{BC}{AB} = \dfrac{CH}{BG}.\]

Now $CH = FE = c$ and $BG = ED = d$ because they are the opposite sides of a parallelogram. Substituting, we obtain

\[\dfrac{a}{b} = \dfrac{c}{d}.\]

屏幕快照 2020-11-17 下午7.16.41.png — Figure $\PageIndex{3}$: Draw $GB$ and $HC$ parallel to $DF$.

Example $\PageIndex{2}$

Find $x$:

$屏幕快照 2020-11-17 下午7.18.35.png$

Solution

$\begin{array} {rcl} {\dfrac{x}{3}} & = & {\dfrac{2x + 2}{4x + 1}} \\ {(x)(4x + 1)} & = & {(3)(2x + 2)} \\ {4x^2 + x} & = & {6x + 6} \\ {4x^2 - 5x - 6} & = & {0} \\ {(x - 2)(4x + 3)} & = & {0} \end{array}$

$\begin{array} {rcl} {x - 2} & = & {0} \\ {x} & = & {-2} \end{array}$ or $\begin{array} {rcl} {4x + 3} & = & {0} \\ {4x} & = & {-3} \\ {x} & = & {-\dfrac{3}{4}} \end{array}$

We reject $x = -\dfrac{3}{4}$ because $BC = x$ cannot be negative.

Check, $x = 2$:

$屏幕快照 2020-11-17 下午7.27.25.png$