Objective:
Recognize and apply the properties of rhombi and squares.
Determine whether quadrilaterals are rectangles, rhombi, or squares.

What is the definition of a rhombus?
A parallelogram with all four sides congruent.

Theorem 6.15
 If a parallelogram is a rhombus, then its diagonals are perpendicular.

Theorem 6.16
 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

The diagonals of rhombus FGHJ intersect at K. Use the given info to find each value.
49
82
98
49

If GH = x + 9 and JH = 5x – 2, find x.
5x – 2 = x + 9
9y - 5
4x = 11
13 x = 2.75
x + 9
5
If FK = 5 and FG = 13, find KJ.
(FK) 2 + (GK) 2 = (FG) 2
5 2 + (GK) 2 = 13 2
(GK) 2 = 144 GK = 12
6y + 7
5x - 2 if m

JFK

6 y

7 and m

KFG

9 y

5 , find y
6y + 7 = 9y - 5
12 = 3y
4 = y

What is the definition of a square?
A parallelogram with four congruent sides and four right angles.

Parallelogram
Rectangle
Rhombus
Square
Square has all of the properties of both rectangles and rhombi.

Parallelograms
Rectangles
Rhombi
4 right angles
4 congruent sides
4 right angles
& 4 congruent sides

Conditions for Rhombi and Squares
Theorem 6.17
 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Conditions for Rhombi and Squares
Theorem 6.18
 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Conditions for Rhombi and Squares
Theorem 6.19
NEW!
 If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

Conditions for Rhombi and Squares
Theorem 6.20
 If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Determine whether parallelogram JKLM with vertices
J (-7, -2) K(0, 4) L (9, 2) and M (2, -4) is a rhombus, a rectangle, or a square. List all that apply. Explain.
Is the figure a rectangle? Are the diagonals congruent?
JL
KM





7
0


9
2
 
 

4
2


2
4

2

2 

272
68


2
4 17
17


16 .
5
8 .
2
The figure is not a rectangle.
If its not a rectangle, then its not a square.
Is the figure a rhombus?
Can either check that 2 consecutive sides are congruent or that the slope of the diagonals are perpendicular.
Slope of KM = -4
Slope of JL = 1/4 Parallelogram JKLM is a Rhombus

Given J (5, 0) L (-3, -14) K (8, -11) M (-6, -3), determine whether parallelogram JKLM is a rhombus, rectangle, or square. List all that apply. Explain.
Is the figure a rectangle?
JL
KM




8
5


6
3
2
 
 
0

11
14

3

2

2


260

4
260

4
65

32 .
2
165

32 .
2
The figure is a rectangle. Is the figure a square?
Are the diagonals perpendicular?
Slope of JL = 7/4
Slope of KM = -4/7
The figure is a square. Since it’s a square it is also a rhombus.
The figure is a rhombus, rectangle, and a square.

Pg. 431 1 – 6 all, 8 – 14 E,
22 – 30 E, 48, 52 – 60 E