Exercise 25 Page 533

We are asked to prove that quadrilateral EBFD is a parallelogram. In addition to the given segment congruence, let's also mark two congruent opposite sides and opposite angles of parallelogram ABCD.

As a first step, let's concentrate on triangles △ ABE and △ CDF.

Congruence	Justification
AE≅CF	Given
∠ EAB≅∠ FCD	Opposite angles of a parallelogram (Theorem 6.4)
AB≅CD	Opposite sides of a parallelogram (Theorem 6.3)

We can see that two sides and the included angle of triangle △ ABE are congruent to the corresponding sides and angle of triangle △ CDF. According to the Side-Angle-Side (SAS) Congruence Postulate, this means that the triangles are congruent. △ ABE≅ △ CDF

Corresponding angles of congruent triangles are congruent. ∠ BEA≅∠ DFC Let's mark this congruence on the diagram.

Let's focus now on the parallel opposite sides of parallelogram ABCD and the transversal BE.

Since angles ∠ BEA and ∠ EBF are alternate interior angles, the Alternate Interior Angles Theorem guarantees that they are congruent. ∠ BEA≅∠ EBF Since angle ∠ BEA is congruent to both ∠ EBF and angle ∠ DFC, according to the Transitive Property of Congruence, we know that these two angles are also congruent. ∠ DFC≅∠ EBF Notice that these two angles are corresponding angles along transversal BF of segments BE and FD.

According to the Converse of the Corresponding Angles Postulate, we can conclude that segments BE and FD are parallel.

We know now that opposite sides of quadrilateral EBFD are parallel, so by definition it is a parallelogram. We can summarize the steps above in a two-column proof.

Completed Proof

2 &Given:&& ABCD is a parallelogram & && AE≅CF &Prove:&& EBFD is a parallelogram Proof:

Statements	Reasons
1. ABCD is a parallelogram.	1. Given
2. AB≅CD	2. Opposite sides of a parallelogram (Theorem 6.3)
3. ∠ EAB≅∠ FCD	3. Opposite angles of a parallelogram (Theorem 6.4)
4. AE≅CF	4. Given.
5. △ ABE≅△ CDF	5. SAS
6. ∠ BEA≅∠ DFC	6. Corresponding angles of congruent triangles
7. BC∥AD	7. Opposite sides of a parallelogram
8. ∠ BEA≅∠ EBF	8. Alternate Interior Angles Theorem
9. ∠ DFC≅∠ EBF	9. Transitive property of congruence
10. BE∥FD	10. Converse of the Corresponding Angles Postulate
11. EBFD is a parallelogram.	11. Definition

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