Question

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? 1. δabc ~ δbed 2. side - angle - side similarity postulate; 1. δabc ~ δbed 2. side - side - side similarity theorem; 1. δabc ~ δdbe 2. side - angle - side similarity postulate

Explanation:

Step 1: Identify the similar triangles

We know that $\frac{BD}{BA}=\frac{BE}{BC}$ (given) and $\angle ABC=\angle DBE$ (common angle). 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, $\triangle ABC \sim \triangle DBE$.

Step 2: Determine the similarity postulate

The SAS Similarity Postulate states that if in two triangles, the ratio of two corresponding sides is equal and the included angle is equal, then the triangles are similar. Here, we have $\frac{BD}{BA}=\frac{BE}{BC}$ and $\angle ABC = \angle DBE$ (included angle), so we use the Side - Angle - Side Similarity Postulate to prove the similarity of $\triangle ABC$ and $\triangle DBE$.

Answer:

1. $\triangle ABC \sim \triangle DBE$, 2. Side - Angle - Side Similarity Postulate