1. 5.10 Properties of Rhombuses, Rectangles, and Squares

2. Vocabulary A rhombus is a parallelogram with four congruent sides.
A rectangle is a parallelogram with four right angles.
A square is a parallelogram with four congruent sides and four right angles.

3. Corollaries
Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.
Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.
Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

4. ABCD is a rhombus.
ABCD is a rectangle.
ABCD is a square.

5. EXAMPLE 1
Use properties of special quadrilaterals
For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. a. Q S
SOLUTION
a. By definition, a rhombus is a parallelogram with four congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent. So, The statement is always true. Q S

6. EXAMPLE 1
Use properties of special quadrilaterals
For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. Q R b.
SOLUTION
If rhombus QRST is a square, then all four angles are congruent right angles. So, if QRST is a square. Because not all rhombuses are also squares, the statement is sometimes true. Q R

7. Theorem 5.26: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem 5.27: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 5.28: A parallelogram is a rectangle if and only if its diagonals are congruent.