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Special Parallelograms Definitions Name Definition Picture Rhombus Rectangle Square Theorems and Corollaries Name Theorem/Corollary Picture Rhombus Corollary A quadrilateral is a rhombus if and only if Rectangle Corollary A quadrilateral is a rectangle if and only if Square Corollary A quadrilateral is a square if and only if Perpendicular Diagonals Theorem A parallelogram is a rhombus if and only if Bisecting Diagonals Theorem A parallelogram is a rhombus if and only if Congruent Diagonals Theorem A parallelogram is a rectangle if and only if
Ways to Prove a Quadrilateral is a Parallelogram Use the information in the diagram above to answer the following questions.
- For what value of x is quadrilateral CDEF a parallelogram? 2. For what value of x is quadrilateral MNPQ a parallelogram? Determine whether each quadrilateral is a parallelogram (use the table above). Write YES if it is.
- 1 pair of opposite sides is parallel and it has 2 consecutive right angles.
- It has 4 right angles.
- 1 pair of opposite sides is parallel and congruent.
- 1 pair of opposite sides is parallel. The other pair of opposite sides is congruent.
- Consecutive angles are supplementary.
- The diagonals are perpendicular.
Review
Special Parallelograms Parallelogram Rectangle Rhombus Square Trapezoid Sketch Properties 1 Both pair of opposite sides || 2 Both pair of consecutive sides ⊥ 3 Diagonals are ≅ 4 Diagonals are ⊥ 5 Diagonals bisect each other 6 Both diagonals are angle bisectors 7 All sides are ≅ 8 Exactly one pair of opposite sides are || 9 All angels are 90° 10 Both pair of opposite angles are ≅ 11 Consecutive angles are supplementary Examples: Name each quadrilateral – parallelogram, rectangle, rhombus, and square – for which the statement is true.
- It is equiangular.
- It is equilangular and equilateral.
- Its diagonals are perpendicular.
- Opposite angles are congruent.
- The diagonals bisect each other.
- The diagonals bisect opposite angles. Additional Examples: Name each quadrilateral and then find the values of x and y.
