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: top path, by given the ratio of line segments bd to ba is equal to the ratio of line segments be to bc. by space labeled by 2, space labeled by 1 occurs. by corresponding parts of similar triangles, angle bde is congruent to angle bac. by converse of the corresponding angles postulate, line segment de is parallel to line segment ac. bottom path, by reflexive property of quality, angle b is congruent to angle b. by space labeled by 2, space labeled by 1 occurs. by corresponding parts of similar triangles, angle bde is congruent to angle bac. by converse of the corresponding angles postulate, line segment de is parallel to line segment ac. which reason can be used to fill in the numbered blank space?

1) 1. Δabc Δbed
2. side-angle-side similarity postulate
2) 1. Δabc Δbed
2. side-side-side similarity theorem
3) 1. Δabc Δdbe
2. side-angle-side similarity postulate
4) 1. Δabc Δdbe
2. side-side-side similarity theorem

Answer 1

The correct statements to fill in the blank spaces are '1. △ABC △DBE' and '2. Side-Angle-Side Similarity Postulate' as they establish the necessary similarity of triangles by angles and proportional sides.

Step-by-step explanation:

The proof involves showing that a line divides two sides of a triangle proportionally and is therefore parallel to the third side. From the given, we have a ratio of line segments BD to BA equal to the ratio of line segments BE to BC. Using the properties of similar triangles, we can assert that △ABC is similar to △DBE (not △BED as incorrectly stated).

Since the angles at B are the same (by the Reflexive Property of Equality), and the given ratios prove sides are proportional, the Side-Angle-Side (SAS) Similarity Postulate applies here. Thus, statement 1 should correctly identify the similar triangles as △ABC and △DBE, and reason 2 should be the Side-Angle-Side Similarity Postulate. This similarity leads to congruent angles which, by the Converse of the Corresponding Angles Postulate, confirms that line segment DE is parallel to line segment AC.