GeometryLesson 6 – 5

Rhombi and Squares

Objective:Recognize and apply the properties of rhombi and squares.

Determine whether quadrilaterals are rectangles, rhombi, or squares.

Rhombus

What is the definition of a rhombus?

A parallelogram with all four sides congruent.

Properties of RhombusTheorem 6.15 If a parallelogram is a rhombus, then its

diagonals are perpendicular.

Properties of Rhombus

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.

8298

49

49

If GH = x + 9 and JH = 5x – 2, find x.

If FK = 5 and FG = 13, find KJ.

x + 9

5x - 2

5x – 2 = x + 9

4x = 11

x = 2.75

yfindyKFGmandyJFKmif ,5976

5

13

(FK)2 + (GK)2 = (FG)2

52 + (GK)2 = 132

(GK)2 = 144 GK = 12

6y + 7

9y - 5

6y + 7 = 9y - 512 = 3y 4 = y

Square

What is the definition of a square?

A parallelogram with four congruent sides and four right angles.

Summary: Flow chart

Quadrilateral

Parallelogram

Rectangle Rhombus

Square

Square has all of the properties of both rectangles and rhombi.

Summary: Venn Diagram

Parallelograms

Rectangles

4 right angles

Rhombi

4 congruent sidesS

quares

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.

Theorem 6.19 If one pair of consecutive sides of a

parallelogram are congruent, the parallelogram is a rhombus.

Conditions for Rhombi and Squares

NEW!

Theorem 6.20 If a quadrilateral is both a rectangle and a

rhombus, then it is a square.

Conditions for Rhombi and Squares

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 rhombus?Can either check that 2 consecutive sides are congruent or that the slope of the diagonals are perpendicular.

Slope of KM = -4Slope of JL = 1/4 Parallelogram JKLM is a Rhombus

Is the figure a rectangle? Are the diagonals congruent? 5.161742722297 22 JL

2.8172684420 22 KM

The figure is not a rectangle.If its not a rectangle, then its not a square.

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? 2.3265426014035 22 JL

2.32165426031168 22 KM

The figure is a rectangle. Is the figure a square?Are the diagonals perpendicular?

Slope of JL = 7/4Slope 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.

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