6.2.2 Conditions for Special Parallelograms (continued) Example 2 Applying Conditions for Special Parallelograms
This is a condition for a rectangle. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
This is a condition for a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
It is determined that a quadrilateral satisfies the conditions to be a square in this example. It is given that two pairs of opposite sides are congruent, that two sides are perpendicular, and that the diagonals are perpendicular. PQRS is a parallelogram by the conditions for parallelograms, because the opposite sides are congruent. PQRS is a rectangle by the conditions for rectangles, because two of the sides form a right angle. PQRS is a rhombus by the conditions for a rhombus, because the diagonals are perpendicular. PQRS is a rectangle, so it has four right angles, and a rhombus, so it has four congruent sides. Therefore, by definition, PQRS is also a square. It is determined whether a quadrilateral satisfies the conditions to be a rhombus in this example.
It is given that the diagonals are perpendicular.
There is insufficient information to determine if the quadrilateral is a rhombus. For example, a kite has perpendicular diagonals like the given figure. But a kite is not a rhombus because it does not have four congruent sides.
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