Section 2: Summary – Properties of ... - Willets Geometry

Section 2: Summary – Properties of Quadrilaterals
Kite
Family
Trapezoid
Family
Parallelogram
Family
Kite
1. Diagonals are perpendicular
2. Has 1 symmetry diagonal
Special Kite:
3. Diagonals are congruent to each other
Trapezoid
1. Parallel sides are called BASES
2. Non-parallel sides are called LEGS
3. IASSOTS on the left and right sides
Isosceles Trapezoid
4. Legs are congruent
5. Diagonals are congruent
6. Both pairs of base angles are congruent
7. Opposite angles are supplementary
8. Has 1 symmetry line
Parallelograms
1. Both pairs of opposite sides are parallel
2. A diagonal separates it into two congruent triangles
3. Both pairs of opposite sides are congruent
4. Both pairs of opposite angles are congruent
5. All 4 pairs of consecutive angles are supplementary
6. Diagonals bisect each other.
Rectangle:
7. has 4 right angles (equiangular)
8. Diagonals are congruent
9. Has 2 symmetry lines
Square:
Has all 13 properties
Rhombus:
7. all 4 sides are congruent (equilateral)
8. diagonals are perpendicular
9. has 2 symmetry diagonals
10. diagonals bisect the angles they connect

Note: Within the Parallelogram Family, there are “sub-families”
Rectangle
Family
Rhombus
Family
Rectangle
Square
Square
Rhombus

Quadrilaterals
Trapezoid Isosceles Trapezoid
Kite
Special Kite
Parallelogram Rectangle
Rhombus
Square

Assignment: Section 2
Part 1: Refer to the quadrilaterals on page 140.
Use a protractor and ruler as necessary to measure.
1. Make a list of all quadrilaterals that are equilateral:
2. Make a list of all parallelograms with congruent diagonals:
3. Make a list of all quadrilaterals with perpendicular diagonals:
4. Make a list of all equiangular quadrilaterals:
5. Make a list of all parallelograms with perpendicular diagonals:
6. Make a list of all quadrilaterals with at least one symmetry diagonal:
7. Make a list of all quadrilaterals whose diagonals bisect each other:
8. Make a list of all quadrilaterals with congruent diagonals:
9. Make a list of all quadrilaterals with both pairs of opposite sides congruent:
10. Make a list of all quadrilaterals with both pairs of opposite angles supplementary:
11. Make a list of all quadrilaterals with all four pairs of consecutive angles supplementary:
12. Make a list of all quadrilaterals with only one symmetry diagonal:
13. Make a list of all parallelograms whose diagonals are congruent and perpendicular:
14. Make a list of all quadrilaterals with both pairs of opposite angles congruent:
15. Make a list of all quadrilaterals with all four pairs of consecutive angles congruent:
16. Make a list of all quadrilaterals where both diagonals bisect the angles they connect:
17. Make a list of any regular quadrilaterals:
18. Make a list of all quadrilaterals whose diagonals are congruent and bisect each other:
19. Make a list of all quadrilaterals with exactly one pair of opposite sides parallel

Part 2: Families of Quadrilaterals
1. (a) The trapezoid and the isosceles trapezoid are members of the __________family.
(b) The kite and the special kite are members of the __________ family.
(c) The parallelogram, rectangle, rhombus and square are members of
the __________ family
(d) The rectangle and the square are members of the ____________ family
(e) The rhombus and the square are members of the ___________ family
2. Determine which shapes have the given property and then fill in the blank
with the FAMILY name:
(a) If the diagonals of a quadrilateral bisect each other, then the quadrilateral must be a
_______________
(b) If the diagonals of a parallelogram are perpendicular, then the parallelogram must be a
_______________
(c) If the diagonals of a parallelogram are congruent, then the parallelogram must be a
_______________.
(d) If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral
must be a __________________.
(e) If a quadrilateral is equiangular, then the quadrilateral must be a __________________
(f) If a quadrilateral has both pairs of opposite angles congruent, then the quadrilateral
must be a ________________
(g) If a quadrilateral is equiangular, then the quadrilateral must be a __________
(h) If the diagonals of a parallelogram are congruent and perpendicular, then the parallelogram must be
a ___________
(i) If a quadrilateral only has one symmetry diagonal, then the quadrilateral must be a __________
(j) If a quadrilateral is regular, then it must be a ___________
(k) If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral must be a _____
(l) If both pairs of opposite angles of a parallelogram are supplementary, then the parallelogram must
be a _______
(m) If exactly two sides of a quadrilateral are parallel and the diagonals are congruent, then the
quadrilateral must be a(an) _______________________

Part 3- Name or sketch your answer to the following:
1. Can you name a quadrilateral that is not a parallelogram?
2. Can you name a quadrilateral, other than a rhombus, which has perpendicular diagonals?
3. Can you name a quadrilateral whose diagonals bisect each other? Can you name more than one?
What general name can you give to all the quadrilaterals whose diagonals bisect each other?
4. Can you name a quadrilateral, other than a rectangle, where the diagonals are congruent?
5. Can you name a quadrilateral, other than a square, where the diagonals are congruent and
perpendicular?
6. Can you name a parallelogram, other than a square, which has perpendicular diagonals?
7. Can you name a parallelogram, other than a square, which has congruent diagonals?
8. Can you name a parallelogram whose diagonals are both congruent and perpendicular?
9. Can you name a parallelogram, other than a rectangle, rhombus or square, whose diagonals bisect
each other?
10. Can you name an equilateral quadrilateral, other than a square?
11. Can you name an equiangular quadrilateral, other than a square?
l2. Can you name a quadrilateral, other than a square, where the diagonals are perpendicular and bisect
each other?
l3. Can you name a parallelogram, other than a square, which contains at least one right angle?
l4. Can you name a quadrilateral whose opposite sides are congruent?
l5. Can you name a quadrilateral whose opposite sides are parallel? Can you name more than one?
What general name can you give to all the quadrilaterals whose opposite sides are parallel?

Part 4: Decide whether each of the following clues describes the members of one specific family of
quadrilaterals. If the clue is not specific enough to guarantee membership in one particular family,
then write “not enough information”
1. Diagonals are perpendicular and has 1 symmetry diagonal.
2. Exactly two sides are parallel
3. Has 2 symmetry diagonals
4. Exactly two sides are parallel and diagonals are congruent.
5. Exactly two sides are parallel and both pairs of opposite angles are supplementary
6. Both pairs of opposite sides are parallel
7. Diagonals are congruent and has 1 symmetry diagonal.
8. Equiangular quadrilateral
9. Diagonals bisect each other and diagonals are congruent
10. Quadrilateral with congruent diagonals
11. An equilateral quadrilateral
12. Quadrilateral with perpendicular diagonals
13. Diagonals bisect each other and diagonals are congruent and diagonals are perpendicular
14. Both pairs of opposite sides are parallel and diagonals are congruent
15. Has 2 symmetry diagonals and 4 right angles
16. Quadrilateral with congruent and perpendicular diagonals
17. Both pairs of opposite angles are supplementary
18. Parallelogram with perpendicular diagonals
19. Equilateral and Equiangular
20. All 4 pairs of consecutive angles are supplementary and diagonals are perpendicular
21. Both pairs of opposite angles are congruent
22. Has 4 right angles
23. Both pairs of opposite sides are congruent and diagonals are perpendicular
24. All 4 pairs of consecutive angles supplementary and congruent
25. Diagonals bisect each other
26. Both pairs of opposite sides are congruent and diagonals are congruent
27. Both diagonals bisect the angles they connect
28. Both pairs of opposite angles are congruent and diagonals are perpendicular

29. All 4 pairs of consecutive angles are supplementary
30. Both pairs of opposite sides are parallel and diagonals are perpendicular
31. Parallelogram with congruent diagonals
32. Both pairs of opposite sides are congruent
33. Parallelogram with congruent and perpendicular diagonals
Part 5 – Review Questions
1. A polygon with four sides is called a_________.
2. If the diagonals of a quadrilateral bisect each other, then the quadrilateral must be a _____.
3. If the diagonals of a parallelogram are congruent, then the parallelogram must be a ______.
4. If the diagonals of a parallelogram are perpendicular, then the parallelogram must be a _____.
5. If the diagonals of a parallelogram are congruent and perpendicular, then the parallelogram
must be a ____.
6. In any parallelogram, consecutive angles are ________.
7. If the opposite angles of a parallelogram are supplementary, then the parallelogram
must be a _______
8. (TF) A parallelogram is a quadrilateral where both pairs of opposite sides are parallel.
9. (TF) If the diagonals of a parallelogram are congruent and perpendicular, then the
parallelogram is always a square.
10. (TF) An equiangular, equilateral quadrilateral must be a square.
11. (TF) If a quadrilateral is equilateral, then it must be a square.
12. (TF) Every rhombus is equiangular.
13. (TF) The diagonals of a trapezoid are always congruent.
14. (TF) The diagonals of an isosceles trapezoid bisect each other.
15. (TF) A square is a rhombus.
16. (TF) If the diagonals of a parallelogram are congruent, then the parallelogram must be a
rectangle.
Fill in the blanks with “Always”, “Sometimes” or “Never”
17. The opposite angles of a parallelogram are congruent.
18. The opposite angles of a parallelogram are supplementary.
19. The diagonals of a parallelogram are congruent.
20. The diagonals of a rhombus are congruent.

21. The diagonals of a parallelogram are perpendicular.
22. A parallelogram whose diagonals are congruent and perpendicular is a square.
23. If the diagonals of a quadrilateral are perpendicular , then the quadrilateral is a
rhombus.
24. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
25. A parallelogram whose angles are congruent is a square.
26. If two sides of a quadrilateral are parallel and the other two sides are congruent, then the quadrilateral is
a parallelogram.
27. The diagonals of a trapezoid are congruent.
Multiple Choice:
28. Which of the following statements is false?
(a) a parallelogram is a quadrilateral
(b) a rectangle is a square
(c) a rectangle is a parallelogram
29. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is always a
(a) rhombus (b) rectangle (c) parallelogram
30. The diagonals of a kite always (a) bisect each other (b) are congruent (c) are perpendicular
31. All quadrilaterals whose diagonals bisect each other are
(a) rectangles (b) rhombi (c) parallelograms
32. The diagonals of a rectangle are always (a) congruent and perpendicular
(b) congruent and bisect each other (c) perpendicular and bisect each other.
33. The consecutive angles of a parallelogram are always
(a) right angles (b) supplementary (c) congruent
34. The diagonals of a rectangle are always
(a) congruent (b) bisectors of the angles they connect (c) perpendicular
35. The opposite angles of a parallelogram are always
(a) right angles (b) congruent (c) supplementary
36. The diagonals of an isosceles trapezoid are always (a) congruent
(b) perpendicular (c) bisectors of each other
37. If the angles of a parallelogram are right angles then the parallelogram must be a
(a) rhombus (b) rectangle (c) square