Exercise 28 Page 589

We are given the figure below, where FD∥ BC, BF∥ CD, and AC bisects ∠ C. By definition of angle bisector we have that ∠ BCE ≅ ∠ ECD.

By definition BCDF is a parallelogram. Thus, the Parallelogram Opposite Angles Theorem tells us that ∠ B ≅ ∠ D.

Notice that △ DEC ~ △ BAC by the Angle-Angle (AA) Similarity Postulate. This, allows us to write the following proportion. DE/BA = EC/AC Finally, we multiply both sides of the equation above by BAEC and that way we get the required proportion. BA/EC* DE/BA = EC/AC * BA/EC ⇕ DE/EC = BA/AC ✓

Two-Column Proof

Given: & FD∥ BC, BF∥ CD & AC bisects∠ C Prove: & DEEC = BAAC Let's summarize the proof we did above in the following two-column table.