Question

given (\frac{ad}{dc}=\frac{be}{ec}) prove (overline{ab}paralleloverline{de}) complete the steps of the proof

Explanation:

Step1: Analyze step 8

We have \(\frac{AC}{DC}=\frac{BC}{EC}\) (from step 6) and \(\angle3\cong\angle3\) (from step 7). In the SAS (Side - Angle - Side) similarity criterion, if the ratio of two pairs of corresponding sides of two triangles is equal and the included angles are congruent, then the triangles are similar. Here, for \(\triangle ABC\) and \(\triangle DEC\), \(\frac{AC}{DC}=\frac{BC}{EC}\) and the included angle \(\angle C\) (which is \(\angle3\)) is common. So, \(\triangle ABC\sim\triangle DEC\) by the SAS similarity criterion.

Step2: Analyze step 9

Since \(\triangle ABC\sim\triangle DEC\) (from step 8), when two triangles are similar, their corresponding angles are congruent. \(\angle1\) in \(\triangle ABC\) and \(\angle2\) in \(\triangle DEC\) are corresponding angles. So, \(\angle1\cong\angle2\) because corresponding angles of similar triangles are congruent.

Answer:

1. SAS (Side - Angle - Side) similarity criterion; 9. Corresponding angles of similar triangles are congruent