Chapter 6 Lesson 4Objective:Objective: To use

properties of diagonals of rhombuses and

rectangles.

RhombusesRhombuses

Theorem 6-9Theorem 6-9  Each diagonal of a rhombus bisects

two angles of the rhombus.

21 43

4321

Theorem 6-10  Theorem 6-10  The diagonals of a rhombus are

perpendicular.                     

BDAC

Example 1: Finding Angle MeasuresMNPQMNPQ is a rhombus and is a rhombus and mm    NN = =

120. 120. Find the measures of the Find the measures of the

numbered angles. numbered angles.

31 mm Isosceles ∆ Theorem

18012031 mm ∆ Angle-Sum Theorem180120)1(2 m

60)1(2 m301m 4321 mmmm

Example 2: Finding Angle Measures

Find the measures of Find the measures of the numbered angles the numbered angles

in the rhombus. in the rhombus.

901m Theorem 6-10 503m Theorem 6-9 502m Theorem 6-9

1809042 mm18090450 m

1804140 m404 m

RectanglesRectangles

Theorem 6-11Theorem 6-11  The diagonals of a rectangle are congruent.

BDAC

Example 3: Finding the Lengths of Diagonals

Find the length of the diagonals of Find the length of the diagonals of rectangle rectangle GFEDGFED if if FDFD = 2 = 2yy + 4 and + 4 and

GE GE = 6= 6yy − 5. − 5.

GEFD Theorem 6-11

5642 yyy49

y49

217

449

2

GEFD

Example 4: Finding the Lengths of Diagonals

Find the length of the diagonals Find the length of the diagonals of of GFEDGFED if if FDFD == 5 5yy – 9 and – 9 and GEGE == yy

+ 5.+ 5.

GEFD Theorem 6-11

y414

y414 5.85

27

GEFD

595 yy

Is the parallelogram a rhombus or a

rectangle?

Theorem 6-12Theorem 6-12  If one diagonal of a parallelogram bisects two

angles of the parallelogram, then the parallelogram is a rhombus.

Theorem 6-13Theorem 6-13  If the diagonals of a parallelogram are

perpendicular, then the parallelogram is a rhombus.

Theorem 6-14Theorem 6-14  

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Example 5: Recognizing Special ParallelogramsDetermine whether the quadrilateral can be

a parallelogram. If not, write impossible.

                                                                                                                                                                                                                  

                                       

Example 6: Recognizing Special Parallelograms

A diagonal of a parallelogram bisects A diagonal of a parallelogram bisects two angles of the parallelogram. Is it two angles of the parallelogram. Is it

possible for the parallelogram to have possible for the parallelogram to have sides of length 5, 6, 5, and 6? sides of length 5, 6, 5, and 6?

No; if one diagonal bisects two No; if one diagonal bisects two angles, then the figure is a rhombus angles, then the figure is a rhombus and cannot have two non-congruent and cannot have two non-congruent

sides. sides.

AssignmenAssignmentt

Pg.315 #1-21(odd); 48-50; 57-60