In triangle ABC shown below, DE is parallel to AC. 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: Which statement and reason accurately completes the proof? A. 3. ∠BDE ≅ ∠BAC; Corresponding Angles Postulate 4. ∠A ≅ ∠C; Isosceles Triangle Theorem B. 3. ∠BDE ≅ ∠BAC; Alternate Interior Angles Theorem 4. ∠A ≅ ∠C; Isosceles Triangle Theorem C. 3. ∠BDE ≅ ∠BAC; Corresponding Angles Postulate 4. ∠B ≅ ∠B; Reflexive Property of Equality D. 3. ∠BDE ≅ ∠BAC; Alternate Interior Angles Theorem 4. ∠B ≅ ∠B; Reflexive Property of Equality

The correct option is;

C. 3. ∠BDE ≅∠BAC, Corresponding Angles Postulate 4. ∠B ≅ ∠B Reflexive Property of Equality

Step-by-step explanation:

The two column proof can be written as follows;

Statement,                         Reason

1. ,                           Given

2. is a transversal ,         Conclusion from statement 1.

We note that ∠BDE  and ∠BAC are on the same side of the transversal relative to the parallel lines, and are therefore, corresponding angles.

Therefore, we have;

3. ∠BDE ≅∠BAC,                 Corresponding Angles Postulate

Also

4. ∠B ≅ ∠B,                           Reflexive Property of Equality

In the two triangles, ΔABC and ΔDBE, we have ∠BDE ≅∠BAC and ∠B ≅ ∠B,

From ∠BDE + ∠B + ∠BED = 180°

∠BAC + ∠B +  ∠BCA = 180°

Therefore, ∠BED = ∠BCA Substitution property of equality

Which gives;

5, ΔABC ~ ΔDBE,          Angle Angle Similarity Postulate

6. BD/BA = BE/BC,      Converse of the Side-Side-Side Similarity Theorem