Question

In Δabc shown below, bd over ba equals be over bc: triangle abc with segment de intersecting sides ab and bc respectively. 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) 1. ∠bde ≅ ∠bac
2. corresponding angles postulate
2) 1. ∠bde ≅ ∠bac
2. corresponding parts of similar triangles
3) 1. ∠bde ≅ ∠bca
2. alternate exterior theorem
4) 1. ∠bde ≅ ∠bca
2. corresponding parts of similar triangles

Answer 1

In a triangle, if a line intersects two sides proportionally, it is parallel to the third side due to corresponding parts of similar triangles.

Step-by-step explanation:

The correct reason to fill in the numbered blank space would be corresponding parts of similar triangles. This is because when a line intersects two sides of a triangle and divides these sides proportionally, the triangles formed are similar. Therefore, angle ∠BDE is congruent to angle ∠BAC, as they are corresponding angles in the similar triangles. Thus, the line segment DE that divides the sides AB and BC proportionally is parallel to side AC due to the properties of similar triangles.