EXAMPLE Rhombuses, Rectangles, and Squares Learn to identify each of the special parallelograms: rhombus, rectangle, and square. The Venn diagram on page 347 may help you see the relationships between parallelograms. GOAL 1 PROPERTIES OF SPECIAL PARALLELOGRAMS

Extra Example 1 Decide whether the statement is always, sometimes, or never true. a. A rectangle is a square. b. A square is a rhombus. sometimes always

Checkpoint EXAMPLE 2 Is the statement, “A rectangle is a parallelogram” always, sometimes, or never true? always

Extra Example 2 QRST is a square. What else do you know about QRST? Because it is a parallelogram, we know: 1.Opposite sides are parallel. 2.Opposite angles are congruent. 3.Consecutive angles are supplementary. 4.Diagonals bisect each other. Because it is a rhombus, we know it has four congruent sides. Because it is a rectangle, we know it has four right angles. Use the corollaries on page 348 to prove a quadrilateral is a rhombus, rectangle, or square. EXAMPLE 3

Extra Example 3 EFGH is a rectangle. K is the midpoint of If EG = 8z – 16, what is Since the diagonals bisect each other, K is the midpoint of Therefore, EK = GK = 4z – 8.

Checkpoint ABCD is a rectangle and What is the value of x? 8

GOAL 2 USING DIAGONALS OF SPECIAL PARALLELOGRAMS EXAMPLE 4EXAMPLE Rhombuses, Rectangles, and Squares Rhombuses and rectangles, and therefore squares, have special properties concerning their diagonals (see page 349). Learn them! If you do not understand these proofs, please see me!

Checkpoint the diagonals meet at point E, and AE = BE = 6. Is ABCD a rectangle? Explain. Yes. Because the diagonals of a parallelogram bisect each other, AE = CE and BE = DE. So AC = AE + CE = 12 and BD = BE + DE = 12. Because the diagonals of ABCD are congruent, it is a rectangle.

Extra Example 6 a. You cut out a parallelogram-shaped quilt piece and measure the diagonals to be congruent. What is the shape? b. An angle formed by the diagonals of the quilt piece measures 90°. Is the shape a square? yes rectangle

Checkpoint the diagonals form a pair of congruent angles at each vertex. What kind of figure is RSTV? rhombus

QUESTION: ANSWER: What is true of the diagonals of a rectangle and a square, but not of those of every rhombus? They are congruent.