6.2.2 Conditions for Special Parallelograms Key Objectives • Prove that a given quadrilateral is a rectangle, rhombus, or square. Theorems, Postulates, Corollaries, and Properties • Theorem If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
• Theorem If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. • Theorem If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. • Theorem If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. • Theorem If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. Example 1 Carpentry Application This is a condition for a rectangle. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
This is a condition for a rhombus. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
This is a condition for a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
A quadrilateral ABCD is proved to be a rectangle in this application example. It is given that opposite sides of the quadrilateral are congruent, as are the diagonals. By the conditions for parallelograms, ABCD is a parallelogram because its opposite sides are congruent. By the conditions for rectangles, ABCD is a rectangle because the diagonals are congruent. In carpentry, the way to ensure a frame is “square,” meaning it has four right angles, is to make sure the lengths of the two diagonals are the same.
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