Geometry Companion Book, Volume 2

6.1.2 Properties of Parallelograms (continued) Example 4 Using Properties of a Parallelogram in a Proof

The angles opposite each other in a parallelogram are congruent.

The two diagonals of a parallelogram bisect each other.

A proof that ∠ B is congruent to ∠ 4 is developed in this example. It is given that there are three parallelograms ABCD , DEFG , and FIHJ , and that AE , CG , JG , and EI are line segments. According to the Properties of Parallelograms, ∠ B and ∠ 1 are congruent, and ∠ 2 and ∠ 3 are congruent, because they are opposite angles. According to the Vertical Angles Theorem ∠ 1 and ∠ 2 are congruent, and ∠ 3 and ∠ 4 are congruent. Therefore, because of the Transitive Property of Congruence, ∠ B ≅ ∠ 4.

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