Question

to prove that the slope of line a is the opposite reciprocal of the slope of line b, mark wants to first prove that △lnm and △knl are both similar to △klm. which similarity theorem can mark use in his proof?
side - side - side similarity theorem
side - angle - side similarity theorem
angle - angle similarity theorem
mark knows that corresponding sides of similar triangles are proportional. which of the following true statements should mark use in his proof that the slope of line a is equal to the opposite reciprocal of the slope of line b?
$\frac{mn}{ln}=\frac{ln}{kn}$ $\frac{lm}{ln}=\frac{kl}{kn}$ $\frac{mn}{lm}=\frac{ln}{lk}$

Explanation:

For the first part, when dealing with right - angled triangles formed by lines (which is likely the case here as we are working with slopes), it is common to use the Angle - Angle similarity theorem since we can easily identify pairs of equal angles (such as right angles and other complementary angles). For the second part, when relating slopes to side - length ratios in right - angled triangles, we know that the slope is the ratio of the vertical change to the horizontal change. If we consider the triangles formed by the lines, the correct proportion for showing the relationship between the slopes of perpendicular lines is $\frac{MN}{LM}=\frac{LN}{LK}$ as it relates the vertical and horizontal side - lengths of the relevant triangles.

Answer:

1. C. Angle - Angle similarity theorem
2. $\frac{MN}{LM}=\frac{LN}{LK}$