1. Lesson 6 – 5 Rhombi and Squares
Geometry
Lesson 6 – 5
Rhombi and Squares
Objective:
Recognize and apply the properties of rhombi and squares.
Determine whether quadrilaterals are rectangles, rhombi, or squares.

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

3. Properties of Rhombus Theorem 6.15
If a parallelogram is a rhombus, then its diagonals are perpendicular.

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

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

6. If GH = x + 9 and JH = 5x – 2, find x.
If FK = 5 and FG = 13, find KJ.
5x – 2 = x + 9
9y - 5 13
4x = 11
x + 9
x = 2.75 5
6y + 7
(FK)2 + (GK)2 = (FG)2
5x - 2
52 + (GK)2 = 132
(GK)2 = 144
GK = 12
6y + 7 = 9y - 5
12 = 3y
4 = y

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

8. Summary: Flow chart Quadrilateral Parallelogram Rectangle Rhombus
Square
Square has all of the properties of both rectangles and rhombi.

9. Summary: Venn Diagram Parallelograms Rhombi Rectangles
4 congruent sides
Rectangles
4 right angles
Squares
4 right angles
& 4 congruent sides

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

11. 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.

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

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

14. Is the figure a rectangle? Are the diagonals congruent?
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?
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

15. Is the figure a rectangle?
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?
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.

16. Homework
Pg – 6 all, 8 – 14 E, – 30 E, 48, 52 – 60 E