Question

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.

statement	reason
2. ab is a transversal that intersects two parallel lines	2. conclusion from statement 1
3.	3.
4. ∠b≅∠b	4. reflexive property of equality
5. △abc∼△dbe	5. angle - angle (aa) similarity postulate
6. \\(\frac{bd}{ba}=\frac{be}{bc}\\)	6. converse of the side - side - side similarity theorem

which statement and reason accurately completes the proof?

1. ∠bde≅∠abc corresponding angles postulate
2. ∠bde≅∠abc alternate interior angles theorem
3. ∠bde≅∠bac corresponding angles postulate

Explanation:

Step1: Recall parallel - line angle relationships

Since $DE\parallel AC$ and $AB$ is a transversal, corresponding angles are equal. $\angle BDE$ and $\angle BAC$ are corresponding angles.

Step2: Analyze similarity postulates

To prove $\triangle ABC\sim\triangle DBE$ using the AA (Angle - Angle) similarity postulate, we need two pairs of equal angles. We already have $\angle B=\angle B$ (reflexive property). We need another pair of equal angles.

Step3: Identify the correct angle - pair

The correct pair of equal angles is $\angle BDE=\angle BAC$ by the Corresponding Angles Postulate.

Answer:

$\angle BDE\cong\angle BAC$; Corresponding Angles Postulate