Question

In Δabc shown below, segment de is parallel to segment ac: the following two-column proof proves that if a line parallel to one side of a triangle also intersects the other two sides, the line divides the sides proportionally. Which statement and reason accurately completes the proof?

1) Δbde ≅ Δbac; angle-angle (aa) similarity postulate
2) ∠bde ≅ ∠bac; angle-angle (aa) similarity postulate
3) ∠bde ≅ ∠bac; side-angle-side (sas) similarity postulate
4) Δbde ≅ Δbac; side-angle-side (sas) similarity postulate

Answer 1

The correct option to complete the proof about proportional division by a parallel line in a triangle is statement 2, based on the AA similarity postulate.

Step-by-step explanation:

The student's question pertains to similarity in triangles and specifically to a theorem in geometry that states if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. The correct statement and reason to complete the proof is statement 2) ∆bde ≅ ∆bac; angle-angle (AA) similarity postulate. This is because two triangles are similar if two angles of one triangle are congruent to two angles of another triangle, which is the basis of the AA similarity postulate.

Answer 2

The statement that accurately completes the proof is: 4) Δbde ≅ Δbac; side-angle-side (SAS) similarity postulate.

Step-by-step explanation:

When a line is parallel to one side of a triangle and intersects the other two sides, it forms similar triangles. In this case, segment DE is parallel to segment AC, so we have two pairs of angles that are congruent: ∠BDE ≅ ∠BAC (corresponding angles) and ∠BED ≅ ∠BCA (alternate interior angles). Additionally, we have the shared side BD. According to the side-angle-side (SAS) similarity postulate, when we have two triangles with a pair of congruent angles and the included side in proportion, the triangles are similar. Hence, ΔBDE is similar to ΔBAC, ensuring that the line divides the sides proportionally.

This proof relies on the principles of geometry where parallel lines create congruent angles and side proportions within triangles. The angle-angle (AA) similarity postulate is not applicable here since we're establishing similarity based on a combination of angles and sides. By demonstrating the equality of two angles and the proportionality of the included sides, the side-angle-side (SAS) similarity postulate stands as the suitable reasoning for concluding the similarity of triangles ΔBDE and ΔBAC. Therefore, this proves that when a line is parallel to one side of a triangle and intersects the other two sides, the line divides the sides proportionally.