Question

in the figure below, $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ are parallel.

complete the following.
(a) find the slope of $overleftrightarrow{ab}$ and the slope of $overleftrightarrow{cd}$.
slope of $overleftrightarrow{ab}$: $\bigcirc \frac{ea}{eb}$ $\bigcirc -\frac{eb}{ea}$ $\bigcirc \frac{eb}{ea}$ $\bigcirc -\frac{ea}{eb}$
slope of $overleftrightarrow{cd}$: $\bigcirc \frac{ed}{ec}$ $\bigcirc \frac{ec}{ed}$ $\bigcirc -\frac{ec}{ed}$ $\bigcirc -\frac{ed}{ec}$
(b) it can be shown that $\triangle eab sim \triangle ecd$.
based on this, choose the ratio that is equal to $\frac{ea}{eb}$.
$\bigcirc \frac{cd}{ed}$ $\bigcirc \frac{ec}{ed}$ $\bigcirc \frac{ed}{ec}$ $\bigcirc \frac{ec}{cd}$
(c) using the results above, choose the correct statement below.
$\bigcirc$ slope of $overleftrightarrow{ab}$ = -slope of $overleftrightarrow{cd}$
$\bigcirc$ slope of $overleftrightarrow{ab}$ $\times$ slope of $overleftrightarrow{cd}$ = -1
$\bigcirc$ slope of $overleftrightarrow{ab}$ = slope of $overleftrightarrow{cd}$
$\bigcirc$ slope of $overleftrightarrow{ab}$ $\times$ slope of $overleftrightarrow{cd}$ = 1
(d) the result in part (c) is an example of the following rule for any two non-vertical parallel lines.
$\bigcirc$ the slopes of the two lines are the same.
$\bigcirc$ the slopes of the two lines are reciprocals.
$\bigcirc$ the slopes of the two lines are negative reciprocals.
$\bigcirc$ the slopes of the two lines are opposites.

Explanation:

Step1: Define slope formula

Slope = $\frac{\text{vertical change}}{\text{horizontal change}}$. For left-down lines, slope is negative.

Slope of $\overleftrightarrow{AB}$:

Vertical change = $-EA$, horizontal change = $EB$.
Slope = $\frac{-EA}{EB}$

Slope of $\overleftrightarrow{CD}$:

Vertical change = $-EC$, horizontal change = $ED$.
Slope = $\frac{-EC}{ED}$

Step2: Use similar triangles property

For $\Delta EAB \sim \Delta ECD$, corresponding sides are proportional: $\frac{EA}{EC} = \frac{EB}{ED}$. Rearrange to get $\frac{EA}{EB} = \frac{EC}{ED}$.

Step3: Compare slopes of parallel lines

From Step1, slope of $\overleftrightarrow{AB} = \frac{-EA}{EB}$, slope of $\overleftrightarrow{CD} = \frac{-EC}{ED}$. From Step2, $\frac{EA}{EB} = \frac{EC}{ED}$, so $\frac{-EA}{EB} = \frac{-EC}{ED}$. Thus, slopes are equal.

Step4: Identify parallel lines slope rule

Parallel non-vertical lines have identical slopes.

Answer:

(a)
slope of $\overleftrightarrow{AB}$: $\boldsymbol{-\frac{EA}{EB}}$
slope of $\overleftrightarrow{CD}$: $\boldsymbol{-\frac{EC}{ED}}$

(b)
$\boldsymbol{\frac{EC}{ED}}$

(c)
$\boldsymbol{\text{slope of } \overleftrightarrow{AB} = \text{slope of } \overleftrightarrow{CD}}$

(d)
$\boldsymbol{\text{The slopes of the two lines are the same.}}$