Exercise 28 Page 589

Use the Parallelogram Opposite Angles Theorem and the definition of angle bisector to get a pair of congruent angles. Then, use the Angle-Angle (AA) Similarity Postulate.

Statements	Reasons
1. FD∥ BC, BF∥ CD, and AC bisects ∠ C	1. Given
2. BCDF is a parallelogram	2. Definition of parallelogram
3. ∠ B ≅ ∠ D	3. Parallelogram Opposite Angles Theorem
4. ∠ BCA ≅ ∠ ECD	4. Definition of angle bisector
5. △ DEC ~ △ BAC	5. AA Similarity Postulate
6. DE/BA = EC/AC	6. Definition of similar triangles
7. BA/EC* DE/BA = EC/AC * BA/EC	7. Multiplication property of equality
8. DE/EC = BA/AC	8. Simplification

Practice makes perfect

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.

Statements	Reasons
1. FD∥ BC, BF∥ CD, and AC bisects ∠ C	1. Given
2. BCDF is a parallelogram	2. Definition of parallelogram
3. ∠ B ≅ ∠ D	3. Parallelogram Opposite Angles Theorem
4. ∠ BCA ≅ ∠ ECD	4. Definition of angle bisector
5. △ DEC ~ △ BAC	5. AA Similarity Postulate
6. DE/BA = EC/AC	6. Definition of similar triangles
7. BA/EC* DE/BA = EC/AC * BA/EC	7. Multiplication property of equality
8. DE/EC = BA/AC	8. Simplification

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Two-Column Proof