1. 6.6 Special Quadrilaterals
Geometry
6.6 Special Quadrilaterals

2. Geometry 6.6 Special Quadrilaterals
Goals:
Identify special quadrilaterals based on limited information.
Prove that a quadrilateral is a special type, such as a rhombus or trapezoid.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

3. Geometry 6.6 Special Quadrilaterals
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid
Each shape has all the properties of all of the shapes above it.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

4. Geometry 6.6 Special Quadrilaterals
Example 1
Which of these have at least one pair of congruent sides?
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid      
February 17, 2019
Geometry 6.6 Special Quadrilaterals

5. Geometry 6.6 Special Quadrilaterals
Example 2
Which of these have at least one pair of congruent angles?
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid      
February 17, 2019
Geometry 6.6 Special Quadrilaterals

6. Geometry 6.6 Special Quadrilaterals
Example 3
Which of these have two pairs of congruent angles?
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid     
February 17, 2019
Geometry 6.6 Special Quadrilaterals

7. Geometry 6.6 Special Quadrilaterals
Example 4
In which of these figures are diagonals perpendicular?
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid   
February 17, 2019
Geometry 6.6 Special Quadrilaterals

8. Geometry 6.6 Special Quadrilaterals
Example 5
In which of these is the sum of the angles 360? 
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kite
Isosceles Trapezoid       
All of these have angle sums of 360 -- they are all quadrilaterals.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

9. Identifying Quadrilaterals
Be as specific as possible.

10. Geometry 6.6 Special Quadrilaterals
What is this?
What we know:
Diagonals bisect each other.
Diagonals congruent.
Must be a…
Rectangle.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

11. Geometry 6.6 Special Quadrilaterals
What is this?
What we know:
Diagonals bisect each other.
Diagonals congruent.
Diagonals perpendicular.
Must be a…
Square.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

12. Geometry 6.6 Special Quadrilaterals
What is this?
What we know:
One pair of opposite sides parallel.
Base angles congruent.
Must be an…
Isosceles Trapezoid.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

13. Geometry 6.6 Special Quadrilaterals
What is this?
What we know:
Both pair of opposite angles congruent.
And that’s all.
Must be a…
Parallelogram.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

14. Geometry 6.6 Special Quadrilaterals
Why?
1 + 2 + 3 + 4 = 360
1  3; 2  4
2 1 + 2 2 = 360
1 + 2 = 180
1 and 2 are supplementary.
The same is true for all consecutive pairs.
Opposite sides are parallel. 1 2 3 4
February 17, 2019
Geometry 6.6 Special Quadrilaterals

15. Geometry 6.6 Special Quadrilaterals
What is this?
What we know:
Four angles congruent
Must be a…
Rectangle.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

16. What kinds of quadrilaterals meet the conditions shown?
Rectangle
Square
February 17, 2019
Geometry 6.6 Special Quadrilaterals

17. Geometry 6.6 Special Quadrilaterals
What kind of quadrilateral has at least one pair of opposite sides congruent?
Parallelogram
Rhombus
Rectangle
Square
Isosceles Trapezoid
February 17, 2019
Geometry 6.6 Special Quadrilaterals

18. Geometry 6.6 Special Quadrilaterals
In the following examples, which two segments or angles must be congruent to enable you to prove ABCD is the given quadrilateral?
February 17, 2019
Geometry 6.6 Special Quadrilaterals

19. Example 1: show ABCD is a rectangle.
Which two segments or angles must be congruent to enable you to prove ABCD is a rectangle?
Possible Solutions:
Show A  B
B  C
C  D
D  A
or AC  BD A B C D
February 17, 2019
Geometry 6.6 Special Quadrilaterals

20. Example 2: show ABCD is a parallelogram.
Which two segments or angles must be congruent to enable you to prove ABCD is a parallelogram?
Possible Solutions:
AD  BC
A C & B  D A B D C
February 17, 2019
Geometry 6.6 Special Quadrilaterals

21. Your Turn. Show ABCD is an Isosceles Trapezoid.
Possible Answers
AD  BC
A  B
D  C A B D C
February 17, 2019
Geometry 6.6 Special Quadrilaterals

22. Geometry 6.6 Special Quadrilaterals
Example 3
Mark these points on the graph and connect them to form a quadrilateral.
P(-2, 4)
Q(3, 5)
R(4, 2)
S(-1, 1)
February 17, 2019
Geometry 6.6 Special Quadrilaterals

23. Geometry 6.6 Special Quadrilaterals
What kind of quadrilateral is PQRS? Prove your conjecture.
P(-2, 4)
R(4, 2)
S(-1, 1)
Probably a parallelogram.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

24. Geometry 6.6 Special Quadrilaterals
To prove PQRS is a parallelogram:
Show opposite sides parallel, or
Show opposite sides congruent, or
Show one pair of opposite sides congruent and parallel.
Q(3, 5)
P(-2, 4)
R(4, 2)
S(-1, 1)
February 17, 2019
Geometry 6.6 Special Quadrilaterals

25. Geometry 6.6 Special Quadrilaterals
1) Show opposite sides parallel.
P(-2, 4)
Q(3, 5)
R(4, 2)
S(-1, 1)
Find slopes of PQ and SR.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

26. Geometry 6.6 Special Quadrilaterals
1) Show opposite sides parallel.
P(-2, 4)
Q(3, 5)
R(4, 2)
S(-1, 1)
Find slopes of PS and QR.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

27. Geometry 6.6 Special Quadrilaterals
1) Show opposite sides parallel.
P(-2, 4)
Q(3, 5)
R(4, 2)
S(-1, 1)
We know that PQ || SR
and PS || QR.
PQRS is a parallelogram by definition.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

28. Geometry 6.6 Special Quadrilaterals
Another Example
Graph the points
P(-1, 2)
Q(4, 4)
R(2, -1)
S(-1, -1)
and connect them.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

29. Geometry 6.6 Special Quadrilaterals
Conjecture:
PQRS is a kite.
Q(4, 4)
To prove this:
Would showing SQ  PR be enough?
NO. Why?
The diagonals of a rhombus are perpendicular, also.
P(-1, 2)
R(2, -1)
S(-1, -1)
February 17, 2019
Geometry 6.6 Special Quadrilaterals

30. Geometry 6.6 Special Quadrilaterals
Conjecture:
PQRS is a kite.
Q(4, 4)
P(-1, 2)
R(2, -1)
S(-1, -1)
A kite has two pairs of consecutive & congruent sides.
Show PQ  RQ and PS  RS.
February 17, 2019
Geometry 6.6 Special Quadrilaterals

31. Geometry 6.6 Special Quadrilaterals
Conjecture:
PQRS is a kite.
Find PQ, RQ, PS, and RS.
KITE
February 17, 2019
Geometry 6.6 Special Quadrilaterals

32. Geometry 6.6 Special Quadrilaterals
As you are doing these problems tonight, be sure to have enough information to justify your conjecture.
February 17, 2019
Geometry 6.6 Special Quadrilaterals