1. Properties of Special Parallelograms
Lesson 7.4
Properties of Special Parallelograms

2. Essential Question: What are the properties of the diagonals of rectangles, rhombuses, and squares?

3. Parallelogram
A quadrilateral with both pairs of opposite sides parallel.
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4. Rhombus
Parallelogram with four congruent sides.
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5. Rectangle
A parallelogram with four right angles.
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6. Square
A parallelogram with four congruent sides and four right angles.
A rectangle with four congruent sides.
A rhombus with four right angles.
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7. How these relate Polygons Quadrilaterals Parallelograms Squares
Rectangles
Rhombuses
Squares
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8. Learning the relations
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Some Quadrilaterals are Parallelograms.
Parallelograms
All Parallelograms are quadrilaterals.
Rhombuses
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9. Learning the relations
Some Quadrilaterals are Rectangles.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
All Rectangles are Parallelograms…
Parallelograms
Rhombuses
…and quadrilaterals, too.
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10. Learning the relations
Some Quadrilaterals are Rhombuses.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Parallelograms
All Rhombuses are Parallelograms…
Rectangles
…and quadrilaterals, too.
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11. Sometimes, Always, Never
A Parallelogram is a Rhombus.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Sometimes
Parallelograms
Rhombuses
Squares
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12. Sometimes, Always, Never
A Square is a Rectangle.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Parallelograms
Always
Rhombuses
Squares
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13. Sometimes, Always, Never
A Parallelogram is a Square.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Sometimes
Parallelograms
Rhombuses
Squares
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14. Sometimes, Always, Never
A Rhombus is a Rectangle.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Sometimes – if it’s a square.
Parallelograms
Rhombuses
Squares
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15. Sometimes, Always, Never
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
A Square is a Parallelogram.
Always
Parallelograms
Rhombuses
Squares
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16. This chart should be memorized.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
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17. Example True or False? The diagonals of a rectangle bisect each other.
What we know: the diagonals of a parallelogram bisect each other.
A rectangle is a parallelogram.
The statement is TRUE.
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18. Another Example True or False?
Opposite sides of a square are parallel.
What we know: All squares are parallelograms.
Opposite sides of a parallelogram are parallel.
Therefore, opposite sides of a square are parallel. The statement is TRUE.
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19. Rhombus Properties: Th. 7.11: Rhombus Diagonals Theorem
The Diagonals of a rhombus are perpendicular.
The converse is also true:
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
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20. Theorem 7.12 The diagonals of a rhombus bisect the angles.
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21. Finding Angle Measures in a Rhombus
Find the measures of the numbered angles in rhombus ABCD. m1= 90 m2 = 61 m3 = 61 m1 + m3 + m4 = 180 90 + 61 + m4 = 180 151 + m4 = 180 m4 = 29

22. Theorem 7.13 The diagonals of a rectangle are congruent.
The converse is: if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
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23. Do you know properties of a rectangle, rhombus, and square?
Which have congruent sides?
Rhombus & Square
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24. Do you know properties of a rectangle, rhombus, and square?
Congruent diagonals?
Rectangle & Square
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25. Do you know properties of a rectangle, rhombus, and square?
Which have four right angles?
Rectangle & Square
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26. Do you know properties of a rectangle, rhombus, and square?
Which have diagonals that bisect each other?
Rectangle & Rhombus & Square
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27. True or False? The Diagonals of a square are perpendicular. TRUE!
Reason: the diagonals of a rhombus are perpendicular, and a square is a rhombus.
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28. Diagonals Tell Us a Lot. rectangle rhombus square
If the diagonals of a parallelogram are congruent, the parallelogram is a ______________.
If diagonals of a parallelogram are perpendicular, the parallelogram is a ___________.
If the diagonals of a parallelogram are perpendicular and congruent, the parallelogram is a _____________.
rectangle
rhombus
square
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29. Figures inherit properties as we read down the chart.
Polygons
Quadrilaterals
Parallelograms
Rhombuses
Rectangles
Squares
Parallelograms
Rhombuses
Rectangles
Diagonals 
Diagonals 
Squares
Diagonals  and 
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30. Coordinate Geometry Given a parallelogram is formed by the points
Q(1, 4)
R(1, -1)
S(-3, -4)
Graph these points.
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31. This is the graph: Q(1, 4) P(-3, 1) R(1, -1) S(-3, -4)
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32. What shape is it? To prove it is a rhombus you could:
Q(1, 4)
R(1, -1)
S(-3, -4)
To prove it is a rhombus you could:
Show all sides congruent (we know opposite sides are parallel).
Or, show the diagonals are perpendicular.
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33. Find slopes of diagonals.
Q(1, 4)
R(1, -1)
S(-3, -4)
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34. Recall: lines are perpendicular if the product of slopes is -1.
Q(1, 4)
R(1, -1)
S(-3, -4)
Diagonals are perpendicular so the parallelogram is a rhombus.
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35. Another view Quadrilaterals Parallelograms Rhombuses Rectangles
Squares
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