Question

In ΔABC shown below, BD over BA equals BE over 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?

1. ∠BDE ≅ ∠BAC
2. Corresponding Angles Postulate
1. ∠BDE ≅ ∠BAC
2. Corresponding Parts of Similar Triangles
1. ∠BDE ≅ ∠BCA
2. Alternate Exterior Theorem
1. ∠BDE ≅ ∠BCA
2. Corresponding Parts of Similar Triangles

Answer 1

The correct reason to fill in the numbered blank space is:

1. ∠BDE ≅ ∠BAC

2. Corresponding Parts of Similar Triangles

To prove that the line is parallel to the third side of the triangle, we need to show that the corresponding angles formed by the intersection of the line and the sides of the triangle are congruent (≅).

In this case, we are given that BD/BA = BE/BC, which means that the line dividing the sides of the triangle is dividing them proportionally. By the Corresponding Parts of Similar Triangles, we know that when two lines are parallel, the corresponding angles formed by the intersection of the parallel lines with the transversal are congruent.

Therefore, we can conclude that ∠BDE ≅ ∠BAC, and the line is parallel to the third side of the triangle.

It is important to note that the other options provided do not directly address the congruence of the corresponding angles formed by the intersection of the line and the sides of the triangle, which is essential in proving the line's parallelism to the third side.