Question

18 complete both statements. \\(\frac{ad}{ac} = \frac{\boxed{\quad}}{bc}\\) \\(\frac{ad}{\boxed{\quad}} = \frac{be}{ec}\\)

Explanation:

Step1: Identify similar triangles

Since $AB \parallel DE$, $\triangle ADB \sim \triangle CDE$ (by AA similarity, as corresponding angles are equal due to parallel lines and a transversal). For similar triangles, corresponding sides are proportional.

Step2: Fill first proportion

From the similarity, the side corresponding to $AD$ on the smaller triangle is $DC$, and $AC = AD + DC$. The proportionality gives $\frac{AD}{AC} = \frac{BE}{BC}$.

Step3: Fill second proportion

Using the same similarity of triangles, the side corresponding to $BE$ is $EC$, and the side corresponding to $AD$ is $DC$. This gives $\frac{AD}{DC} = \frac{BE}{EC}$.

Answer:

First blank: $BE$
Second blank: $DC$