Question

proof:

which reason completes the proof?

a. corresponding sides of congruent triangles are proportional.

b. corresponding parts of similar triangles are congruent.

c. corresponding parts of congruent triangles are congruent.

d. corresponding sides of similar triangles are proportional.

Explanation:

Since we have \(\triangle ABC\sim\triangle ACD\) and \(\triangle ABC\sim\triangle CBD\) (from step 3, using AA similarity criteria), for similar triangles, the ratio of their corresponding sides is equal.

• Option A is incorrect because congruent triangles have equal - length sides (not just proportional), and we are dealing with similar triangles here.
• Option B is wrong. For similar triangles, corresponding parts (sides) are proportional, not congruent. Congruent means equal in length.
• Option C is incorrect as we are working with similar triangles (\(\sim\)) in step 4, not congruent triangles (\(\cong\)).
• Option D is correct. When two triangles are similar (\(\triangle ABC\sim\triangle ACD\) and \(\triangle ABC\sim\triangle CBD\)), by the property of similar triangles, corresponding sides of similar triangles are proportional. So, from \(\triangle ABC\sim\triangle ACD\), we get \(\frac{AC}{AB}=\frac{AD}{AC}\), and from \(\triangle ABC\sim\triangle CBD\), we get \(\frac{BC}{AB}=\frac{DB}{BC}\)

Answer:

D. Corresponding sides of similar triangles are proportional.