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: 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 the converse of the corresponding angles postulate, line segment DE is parallel to line segment AC.

Bottom path: By the reflexive property of equality, 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 the 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?
a) δABC ~ δBED, Side-Angle-Side Similarity Postulate
b) δABC ~ δBED, Side-Side-Side Similarity Theorem
c) δABC ~ δDBE, Side-Angle-Side Similarity Postulate
d) δABC ~ δDBE, Side-Side-Side Similarity Theorem

Answer 1

The correct reason to complete the flowchart proof is option (a) which uses the Side-Angle-Side Similarity Postulate to establish the similarity of triangles ΔABC and ΔBED.

Step-by-step explanation:

The answer to fill the numbered blank space provided by the flowchart proof is (a) ΔABC ~ ΔBED, Side-Angle-Side Similarity Postulate. This reason is valid because we have two triangles, ΔABC and ΔBED, where we know that sides BD and BE are proportional to sides BA and BC respectively, and we have angle B congruent to itself by the Reflexive Property. Since proportionality of sides and congruence of an included angle are given, the Side-Angle-Side (SAS) Similarity Postulate applies, which indicates that the two triangles are similar. Consequently, angles BDE and BAC are congruent as they are corresponding angles in similar triangles. This congruence of angles leads to the conclusion that DE is parallel to AC, by the converse of the Corresponding Angles Postulate.