3. Proving That a Quadrilateral Is a Parallelogram
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Consider a quadrilateral with congruent opposite angles. Use the Corollary to the Polygon Interior Angles Theorem and the Consecutive Interior Angles Converse. Also, consider a quadrilateral with congruent opposite sides and use the previous case.
See solution.
To explore when a quadrilateral is a parallelogram, we will consider some different quadrilaterals.
For the first case, let's consider a quadrilateral whose opposite angles are congruent.
The latter equation implies that both ∠ A and ∠ B, as well as ∠ B and ∠ C, are supplementary. Then, by the Consecutive Interior Angles Converse we have that AD∥ BC and AB∥ CD, which proves that ABCD is a parallelogram.
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If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. |
For the second case, let's consider a quadrilateral with both pairs of opposite sides congruent.
If we draw AC, it will divide ABCD into two triangles, and by the Side-Side-Side (SSS) Congruence Theorem, the two triangles are congruent, which implies that ∠ B ≅ ∠ D.
Drawing BD and using a similar reasoning we conclude that ∠ A ≅ ∠ C.
Applying what we learned in case 1, we conclude that ABCD is a parallelogram. Thus, we can write the following statement.
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If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. |
Finally, let's consider a quadrilateral such that its diagonals bisect each other. Notice that in this case, the quadrilateral is divided into four triangles.
By the Vertical Angles Congruence Theorem, we have that ∠ CMB ≅ ∠ AMD and ∠ BMA ≅ ∠ DMC.
Next, by applying the Side-Angle-Side (SAS) Congruence Theorem, we obtain that △ AMD ≅ △ CMB and △ AMB ≅ △ CMD. These two congruences imply that AD≅ BC and AB≅ CD. Thus, by case 2, we conclude that ABCD is a parallelogram.
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If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. |