Question

The following two-column proof with missing statements and reasons proves that if a line parallel to one side of a triangle also intersects the other two sides, the line divides the sides proportionally:

Statement Reason

1) Line segment DE is parallel to line segment AC. 1. Given
2) Line segment AB is a transversal that intersects two parallel lines. 2. Conclusion from Statement 1.
3) ΔABC ~ ΔDBE 5. Angle-Angle (AA) Similarity Postulate
4) BD over BA equals BE over BC 6. Converse of the Side-Side-Side Similarity Theorem

Which statement and reason accurately complete the proof?

a) ∠BDE ≅ ∠BAC; Corresponding Angles Postulate
b) ∠A ≅ ∠C; Isosceles Triangle Theorem
c) ∠BDE ≅ ∠BAC; Alternate Interior Angles Theorem
d) ∠A ≅ ∠C; Isosceles Triangle Theorem
e) ∠BDE ≅ ∠BAC; Corresponding Angles Postulate
f) ∠B ≅ ∠B; Reflexive Property of Equality
g) ∠BDE ≅ ∠BAC; Alternate Interior Angles Theorem
h) ∠B ≅ ∠B; Reflexive Property of Equality

Answer 1

The correct statement and reason to complete the proof is '∠BDE ≅ ∠BAC; Corresponding Angles Postulate' because DE is parallel to AC, making these angles correspond, and the proof is based on triangle similarity rules such as the AA Similarity Postulate.

Step-by-step explanation:

To prove that line segment DE is parallel to line segment AC and that DE divides the sides of triangle ABC proportionally, we examine the given information and use triangle similarity rules. Given that DE is parallel to AC, by the Corresponding Angles Postulate, we can conclude that ∠BDE ≅ ∠BAC. This information, together with the fact that angle B is shared by both triangles ABC and DBE, allows us to apply the Angle-Angle (AA) Similarity Postulate to deduce that triangle ABC is similar to triangle DBE. Consequently, the corresponding sides of these similar triangles are proportional, meaning BD/AB = BE/BC. This proportionality is justified by the definition of similar triangles and does not involve the Converse of the Side-Side-Side Similarity Theorem, which is typically used to show the similarity of triangles based on the proportionality of three pairs of corresponding sides.