Question

prove:
a line parallel to one side of a triangle divides the other
two proportionally: \\( \frac { a } { b } = \frac { c } { d } \\).

opposite sides of similar triangles are in proportion.
corresponding sides of similar triangles are in proportion.
adjacent sides of similar triangles are in proportion.
interior sides of similar triangles are in proportion.

Explanation:

Step1: Recall the property of similar triangles

When two triangles are similar, their corresponding sides are in proportion. In the proof, we first show that \(\triangle LMN\sim\triangle OPN\) using the AA (Angle - Angle) similarity postulate.

Step2: Analyze the proportion \(\frac{LN}{b}=\frac{MN}{d}\)

Since \(LN=a + b\) and \(MN=c + d\) (by segment addition postulate), substituting these into \(\frac{LN}{b}=\frac{MN}{d}\) gives \(\frac{a + b}{b}=\frac{c + d}{d}\). After simplification (\(1+\frac{a}{b}=1+\frac{c}{d}\)) and subtracting 1 from both sides, we get \(\frac{a}{b}=\frac{c}{d}\). The key idea throughout is that for similar triangles \(\triangle LMN\) and \(\triangle OPN\), the ratios of their corresponding sides are equal.

Answer:

B. Corresponding sides of similar triangles are in proportion.