Question

in $\triangle abc$ shown below, $\frac{bd}{ba} = \frac{be}{bc}$.

the following flowchart proof with missing statements and reasons proves that if a line intersects two sides of a triangle and divides these sides proportionally, the line is parallel to the third side.

which reason can be used to fill in the numbered blank space? (2 points)

\\(\bigcirc\\) 1 $\triangle abc \sim \triangle bed$
2 side - angle - side similarity postulate
\\(\bigcirc\\) 1 $\triangle abc \sim \triangle bed$
2 side - side - side similarity theorem
\\(\bigcirc\\) 1 $\triangle abc \sim \triangle dbe$
2 side - angle - side similarity postulate
\\(\bigcirc\\) 1 $\triangle abc \sim \triangle dbe$

Explanation:

Step1: Check the similarity of triangles

We know that \(\frac{BD}{BA}=\frac{BE}{BC}\) (given) and \(\angle B=\angle B\) (common angle).

Step2: Apply the similarity postulate

By the Side - Angle - Side (SAS) similarity postulate, if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar. So, \(\triangle ABC\sim\triangle DBE\)

Answer:

1 \(\triangle ABC\sim\triangle DBE\)
2 Side - Angle - Side Similarity Postulate