1

Objectives• State the properties of rectangles,

rhombuses, and squares• Solve problems involving rectangles,

rhombuses, and squares

2

Diagonals of Rhombus Bisect Angles

• A parallelogram is a rhombus if and only if each diagonal bisects two angles of the rhombus.

– ∠1 ≅ ∠2 and ∠3 ≅ ∠4– Since ∠BCD and ∠BAD are opposite angles

of a parallelogram, ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4

3

Diagonals of a Rhombus are Perpendicular

• A parallelogram is a rhombus if and only if the diagonals are perpendicular.

4

Diagonals of a Rectangle

• A parallelogram is a rectangle if and only if the diagonals are congruent.

5

Squares

• A square is a parallelogram, rectangle, and rhombus. All properties of parallelograms, rectangles, and rhombi apply to squares

6

Example: Rhombus

• Find m∠XTZWe need to solve for a before we can find

m∠XTZ. 14a + 20 = 90 (diagonals of a rhombus are perpendicular)14a = 70a = 5 5a – 5 = 20° (substituting a = 5 in order to find m∠XTZ)

7

Example: Rectangle

Find FD in rectangle FEDG if FD = 2y + 4 and GE = 6y – 5 6y – 5 = 2y + 44y – 5 = 44y = 9 y = 9/4

FD = 2y + 4 = 2(9/4) + 4 = 8.5

8

Example: Square• Show that the figure is

a square.– Strategy:

• Show that the diagonals are perpendicular (rhombus)

• Show that the diagonals are congruent (rectangle)

Since ,

Since EG = FH,