Geometry: When Is a Parallelogram a Rectangle?

When Is a Parallelogram a Rectangle?

I'm thinking of a parallelogram whose diagonals are congruent. Name that parallelogram.

Not all parallelograms have congruent diagonals. Rhombuses do not have congruent diagonals. Rectangles do have congruent diagonals, and so do squares. You cannot conclude that the parallelogram that I'm thinking of is a square, though, because that would be too restrictive. When playing “Name That Quadrilateral,” your answer must be as general as possible. Because a square is a rectangle but a rectangle need not be a square, the most general quadrilateral that fits this description is a rectangle.

• Theorem 16.5: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Figure 16.5 shows parallelogram ABCD with congruent diagonals ¯AC and ¯BD. Because we are dealing with a parallelogram, you know that opposite sides are congruent. You can use the SSS Postulate to show that ΔACD ~= ΔDBA. Using CPOCTAC, we can show ∠A ~= ∠D. Because ABCD is a parallelogram, opposite angles are congruent, so ∠A ~= ∠C and ∠B ~= ∠D. By the transitive property of ~=, you have all four angles congruent. Because the measures of the interior angles of a quadrilateral add up to 360º, you can show that all four angles of our parallelogram are right angles. That's more than enough to make your parallelogram a rectangle.

	Statements	Reasons
1.	Parallelogram ABCD with ¯AC ~= ¯BCD	Given
2.	¯AB ~= ¯CD	Theorem 15.4
3.	¯AD ~= AD	Reflexive property of ~=
4.	ΔACD ~= ΔDBA	SSS Postulate
5.	∠A ~= ∠D	CPOCTAC
6.	∠A ~= ∠C and ∠B ~= ∠D	Theorem 15.5
7.	m∠A = m∠C and m∠B = m∠D	Definition of ~=
8.	m∠A + m∠B + m∠C + m∠D = 360º	The measures of the interior angles of a quadrilateral add up to 360º
9.	m∠A + m∠A + m∠A + m∠A = 360º	Substitution (steps 7 and 8)
10.	m∠A = 90º	Algebra
11.	∠A is a right angle	Definition of right angle
12.	Parallelogram ABCD is a rectangle	Definition of rectangle