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. △abc - △ade 2. side - angle - side similarity postulate 1. △abc - △ade 2. side - side - side similarity theorem 1. ∠abc - ∠ade 2. side - angle - side similarity postulate 1. ∠abc - ∠ade 2. side - side - side similarity theorem

Explanation:

Step1: Recall similarity - triangle concepts

We know that if two triangles have their corresponding sides in proportion, we use the Side - Side - Side (SSS) similarity theorem. In the context of the given triangle problem where a line divides two sides of a triangle proportionally, we consider the two triangles formed (the smaller one and the larger one).

Step2: Identify the correct reason

The correct reason for showing that the two triangles are similar when the sides are in proportion is the Side - Side - Side Similarity Theorem. This is used to establish the similarity of the triangles which then helps in proving the parallel - line relationship.

Answer:

1. $\triangle ABC\sim\triangle ADE$
2. Side - Side - Side Similarity Theorem