Question

consider the proof
given segment ab is parallel to line de
prove \\( \frac { a d } { d c } = \frac { b e } { e c } \\)
what is the missing statement in step 5?
\\( \bigcirc a c = b c \\)
\\( \bigcirc \frac { a c } { d c } = \frac { b c } { e c } \\)
\\( \bigcirc a d = b e \\)
\\( \frac { a d } { d c } = \frac { b e } { e c } \\)

Explanation:

Step1: Recall the definition of similar triangles

If \(\triangle ABC\sim\triangle DEC\) (from step 4, by AA similarity theorem), then the ratios of their corresponding sides are equal.

Step2: Identify the corresponding sides

For similar triangles \(\triangle ABC\) and \(\triangle DEC\), the ratio of \(AC\) to \(DC\) (where \(AC\) and \(DC\) are sides of \(\triangle ABC\) and \(\triangle DEC\) respectively) is equal to the ratio of \(BC\) to \(EC\) (where \(BC\) and \(EC\) are another pair of corresponding sides). So the statement is \(\frac{AC}{DC}=\frac{BC}{EC}\) which comes from the definition of similar triangles (step 5).

Answer:

\(\frac{AC}{DC}=\frac{BC}{EC}\)