Question

statement reason
1 \\( \frac { a d } { d b } = \frac { c e } { e b } \\) given
2
3 \\( \frac { a d + d b } { d b } = \frac { c e + e b } { e b } \\) using common denominators
4 \\( \

$$\begin{array} { l } { a b = a d + d b } \\ { c b = c e + e b } \\end{array}$$

\\) segment addition
5 \\( \frac { a b } { d b } = \frac { c b } { e b } \\) substitution property of equality
6 \\( \angle a b c = \angle d b e \\) reflexive property of congruence
7 \\( \triangle a b c \sim \triangle d b e \\) sas criterion for similarity
8 \\( \angle b a c = \angle b d e \\) corresponding angles of similar trangles are congruent
9 \\( \overline { d e } \parallel \overline { a c } \\) if the corresponding angles formed by two lines intersected by a transversal are
congruent, then the lines are parallel.

Explanation:

Step1: Identify the missing step

We need to find a step that logically follows the given first step \(\frac{AD}{DB}=\frac{CE}{EB}\) and precedes step 3 \(\frac{AD + DB}{DB}=\frac{CE+EB}{EB}\).

Step2: Apply the property of equality

If \(\frac{a}{b}=\frac{c}{d}\), then by the property of equality (adding 1 to both sides of the proportion \(\frac{a}{b}=\frac{c}{d}\), which is equivalent to \(\frac{a}{b}+1=\frac{c}{d}+1\)). In fraction form, \(1=\frac{b}{b}=\frac{d}{d}\). So, \(\frac{AD}{DB}+ 1=\frac{CE}{EB}+1\).

Answer:

1. \(\frac{AD}{DB}+1=\frac{CE}{EB}+1\); Addition Property of Equality