1. 6.4 Rhombuses, Rectangles, and Squares

2. Warm up:
ABCD is a parallelogram. Find each measure.
A. CD B. mC
104°

3. Objective
Use properties of special types of parallelograms.

4. Rhombus A rhombus is a with four congruent sides.
The diagonals are perpendicular
The diagonals bisect opposite angles
Looks like a parallelogram but has more

5. Rhombus

6. Rectangle A rectangle is a with four right angles
The diagonals of a rectangle are congruent
The adjacent sides are perpendicular (right angle)

7. Square A square is a with four congruent sides AND four right angles
The diagonals of a square are congruent
The diagonals are perpendicular (right angles)

8. 1) In the diagram, ABCD is a rectangle.
a. Find AD and AB.
b. Find m∠A, m∠B, m∠C, and m∠D. 8 5
All angles are 90 degrees.

9. 2) In the diagram, PQRS is a rhombus. Find QR, RS, and SP.6 6 6

10. Corollaries
A corollary- to a theorem is a statement that can be proved easily using the theorem.

11. Rhombus corollary
If a quadrilateral has four congruent sides, then it is a rhombus.

12. Rectangle Corollary
If a quadrilateral has four right angles, then it is a rectangle.

13. Square Corollary
If a quadrilateral has four congruent sides AND right angles, then it is a square.

14. Sides are not congruent
Use the information in the diagram to name the special quadrilateral.
Four right angles
Sides are not congruent
Must be a rectangle

15. Sides are all congruent
2. Use the information in the diagram to name the special quadrilateral.
No info on angles
Sides are all congruent
Must be a rhombus

16. Four right angles Four congruent sides Must be a square
3. Use the information in the diagram to name the special quadrilateral.
Four right angles
Four congruent sides
Must be a square

17. Remember: We wrote this before but it is important to remember that….
…the diagonals of a rhombus are perpendicular

18. ABCD is a rhombus. Find the value of x.1)
We know that the diagonals are perpendicular, so lets add a box by E.

19. ABCD is a rhombus. Find the value of x.2)
We know that the diagonals are perpendicular, so lets add a box in the center .
X = 90 degrees

20. Remember: We wrote this before but it is important to remember that….
…the diagonals of a rectangle are congruent

21. Opposite sides are congruent
1a) You nail four pieces of wood together to build a four-sided frame, as shown. What is the shape of the frame?
No info on angles
Opposite sides are congruent
Must be a parallelogram

22. 1b) The diagonals measure 7 ft 4 in. and 7 ft 2 in
1b) The diagonals measure 7 ft 4 in. and 7 ft 2 in. Is the frame a rectangle?
PYTHAGOREAN THEOREM
Notice the triangle is a right triangle
What formula do we use to solve for missing sides of a RIGHT triangle?

23. 2) rectangle EFGH . Find the value of x.
The pieces of the diagonals of a rectangle are all all congruent
x=12

24. 3) square JKLM. Find the value of x.
The diagonals of a square are perpendicular