## Proving that Figures Are Parallelograms

Many times you will be asked to prove that a figure is a parallelogram. The following theorems are tests that determine whether a quadrilateral is a parallelogram:

*Theorem 46:* If both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram.

*Theorem 47:* If both pairs of opposite angles of a quadrilateral are equal, then it is a parallelogram.

*Theorem 48:* If all pairs of consecutive angles of a quadrilateral are supplementary, then it is a parallelogram.

*Theorem 49:* If one pair of opposite sides of a quadrilateral is both equal and parallel, then it is a parallelogram.

*Theorem 50:* If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Quadrilateral *QRST* in Figure 1 is a parallelogram if:

**Figure 1** A quadrilateral with its diagonals.

*QR* = *ST* and *QT* = *RS,* by *Theorem 46.*

*m* ∠ *Q* = *m* ∠ *S* and *m* ∠ *T* = *m* ∠ *R,* by *Theorem 47.*

- ∠
*Q* and ∠ *R,* ∠ *R* and ∠ *S,* ∠ *S* and ∠ *T,* and ∠ *Q* and ∠ *T* are all supplementary pairs, by *Theorem 48.*

*QR* = *ST* and *QR *∥ *ST *or *QT* = *RS* and *QT *∥ *RS *, by *Theorem 49.*

*QP* = *PS* and *RP* = *PT,* by *Theorem 50.*