Question

given: \\( \triangle abc \\) with \\( \frac{bd}{ba}=\frac{be}{bc} \\) prove: \\( \overline{de}\parallel\overline{ac} \\) what is the missing step in this proof? a. \\( \angle abc\cong\angle dbe \\) b. \\( \angle bca\cong\angle bde \\) c. \\( \angle acb\cong\angle deb \\) d. \\( \angle bde\cong\angle ade \\) e. \\( \angle cab\cong\angle dac \\)

Explanation:

Step1: Use the Side - Angle - Side (SAS) similarity criterion

If \(\frac{BD}{BA}=\frac{BE}{BC}\) (given) and \(\angle ABC=\angle DBE\) (common angle). Then \(\triangle ABC\sim\triangle DBE\) by SAS similarity criterion.

Step2: Use the property of similar triangles

If \(\triangle ABC\sim\triangle DBE\), then corresponding angles are equal. So \(\angle ACB=\angle DEB\) (corresponding angles of similar triangles \(\triangle ABC\) and \(\triangle DBE\)). By the converse of the basic proportionality theorem (Thales' theorem), if the corresponding angles are equal, then \(DE\parallel AC\).

Answer:

C. \(\angle ACB\cong\angle DEB\)