Question

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:

\\(\frac{bd}{ba} = \frac{be}{bc}\\)
given

\\(\delta abc \sim \delta dbe\\)
side-angle-side similarity postulate

\\(\angle bde \cong \angle bac\\)
corresponding parts of similar triangles

\\(de \parallel ac\\)
converse of the corresponding angles postulate

1. ______
2. ______

which reason can be used to fill in the numbered blank space?

Explanation:

Identify the given flowchart components

Using the Geometric Proofs knowledge point
The flowchart aims to prove that line segment \(DE\) is parallel to \(AC\) (\(DE \parallel AC\)) given that \(\frac{BD}{BA} = \frac{BE}{BC}\).
The flowchart shows:

1. A given statement: \(\frac{BD}{BA} = \frac{BE}{BC}\).
2. A second statement/reason pair labeled with "1. \(\underline{\quad\quad}\)" and "2. \(\underline{\quad\quad}\)".
3. Both lead to the similarity statement \(\triangle ABC \sim \triangle DBE\) via the SAS Similarity Postulate (Side-Angle-Side Similarity Postulate).

Determine the missing statement and reason

Using the SAS Similarity Postulate knowledge point

$$
LATEXBLOCK0
$$

Thus, the statement 1 is \(\angle B \cong \angle B\) (or \(\angle ABC \cong \angle DBE\)), and the reason 2 is the Reflexive Property of Congruence.

Answer:

The reason that can be used to fill in the numbered blank space 2 is the Refive Property of Congruence (for the statement \(\angle B \cong \angle B\)).