Question

the table contains the proof of the theorem of the relationship between slopes of parallel lines. what is the missing statement in step 2?
a. $\frac{bc}{ab}=\frac{de}{ef}$
b. $\frac{bc}{ef}=\frac{de}{ab}$
c. $bc = ef, ab = de$
d. $\frac{bc}{ab}=\frac{ef}{ed}$
e. $bc\times ab=-(ef\times de)$

Explanation:

To determine the missing statement in step 2, we use the property of similar triangles (ΔABC ~ ΔDEF). For similar triangles, the ratios of corresponding sides are equal.

• In ΔABC, the vertical side is \( BC \) and the horizontal side is \( AB \).
• In ΔDEF, the vertical side is \( DE \) and the horizontal side is \( EF \).

The slope of a line is \( \frac{\text{vertical change}}{\text{horizontal change}} \). For similar triangles, the ratio of vertical sides to horizontal sides should be equal. So, \( \frac{BC}{AB} = \frac{DE}{EF} \), which matches option A.

Answer:

A. \(\boldsymbol{\frac{BC}{AB} = \frac{DE}{EF}}\)