Transcript 6.6 Special Quadrilaterals Geometry Mrs. Spitz Spring 2005 Objectives: Identify special quadrilaterals based on limited information.  Prove that a quadrilateral is a special type of quadrilateral, such.

6.6 Special
Quadrilaterals
Geometry
Mrs. Spitz
Spring 2005
Objectives:
Identify special quadrilaterals based
on limited information.
 Prove that a quadrilateral is a special
type of quadrilateral, such as a
rhombus or trapezoid.

Assignment

pp. 367-369 #2-35
Summarizing Properties of
Quadrilaterals

Quadrilateral
In this chapter, you have
studied the seven special
Trapezoid
Kite
Parallelogram
types of quadrilaterals
shown at the right. Notice
Isosceles
Rhombus
Rectangle
that each shape has all the
trapezoid
Square
properties of the shapes
linked above it. For
instance, squares have the
properties of rhombuses,
rectangles, parallelograms,
and quadrilaterals.
Ex. 1: Identifying
Quadrilaterals

Parallelogram
Quadrilateral ABCD has at least one
pair of opposite sides congruent.
What kinds of quadrilaterals meet this
condition? Rhombus
Opposites
sides are ≅.
Opposite sides
are congruent.
All sides are
congruent.
All sides are
congruent.
Legs are
congruent.
Ex. 2: Connecting midpoints
of sides
A

When you join the
midpoints of the
sides of any
quadrilateral, what
special
quadrilateral is
formed? Why?
F
E
B
D
G
H
C
Ex. 2: Connecting midpoints
of sides



A
Solution: Let E, F, G, and H
be the midpoints of the sides
of any quadrilateral, ABCD E
as shown.
D
If you draw AC, the
Midsegment Theorem for
H
triangles says that FG║AC
and EG║AC, so FG║EH.
C
Similar reasoning shows that
EF║HG.
So by definition, EFGH is a
parallelogram.
F
B
G
Proof with Special
Quadrilaterals

When you want to prove that a
quadrilateral has a specific shape, you
can use either the definition of the
shape as in example 2 or you can use
a theorem.
Proving Quadrilaterals are
Rhombuses
You have learned 3 ways to prove that a quadrilateral is a
rhombus.

You can use the definition and show that the
quadrilateral is a parallelogram that has four
congruent sides. It is easier, however, to use the
Rhombus Corollary and simply show that all four
sides of the quadrilateral are congruent.

Show that the quadrilateral is a parallelogram and
that the diagonals are perpendicular (Thm. 6.11)

Show that the quadrilateral is a parallelogram and
that each diagonal bisects a pair of opposite angles.
(Thm 6.12)
Ex. 3: Proving a
quadrilateral is a rhombus
8




K(2, 5)N(6, 3) = 4.47 cm
Show KLMN is a rhombus
M(2, 1)N(6, 3) = 4.47 cm
Solution: You can use any of the L(-2, 3)M(2, 1) = 4.47 cm
three ways described in the
6
L(-2, 3)K(2, 5) = 4.47 cm
concept summary above. For
instance, you could show that
opposite sides have the same
4
slope and that the diagonals are
perpendicular. Another way
L(-2, 3)
shown in the next slide is to prove
that all four sides have the same
2
length.
AHA – DISTANCE FORMULA If
you want, look on pg. 365 for the
whole explanation of the distance
formula
So, because LM=NK=MN=KL,
KLMN is a rhombus.
-2
K(2, 5)
N(6, 3)
M(2, 1)
5
Ex. 4: Identifying a
quadrilateral
A

What type of quadrilateral is
ABCD? Explain your
reasoning.
D
60°
120°
120°
60°
B
C
Ex. 4: Identifying a
quadrilateral
A

What type of quadrilateral is
ABCD? Explain your reasoning.
D
60°
120°
120°

Solution: A and D are
supplementary, but A and B
are not. So, AB║DC, but AD is
not parallel to BC. By definition,
ABCD is a trapezoid. Because base
angles are congruent, ABCD is an
isosceles trapezoid
60°
B
C
Ex. 5: Identifying a Quadrilateral


The diagonals of quadrilateral ABCD
intersect at point N to produce four
congruent segments: AN ≅ BN ≅ CN ≅
DN. What type of quadrilateral is ABCD?
Prove that your answer is correct.
First Step: Draw a diagram. Draw the
diagonals as described. Then connect the
endpoints to draw quadrilateral ABCD.
Ex. 5: Identifying a Quadrilateral
B


First Step: Draw a diagram.
Draw the diagonals as
described. Then connect the
endpoints to draw
quadrilateral ABCD.
2nd Step: Make a conjecture:


Quadrilateral ABCD looks like
a rectangle.
3rd step: Prove your
conjecture


Given: AN ≅ BN ≅ CN ≅ DN
Prove ABCD is a rectangle.
C
N
A
D
Given: AN ≅ BN ≅ CN ≅ DN
Prove: ABCD is a rectangle.


Because you are given information about diagonals, show
that ABCD is a parallelogram with congruent diagonals.
First prove that ABCD is a parallelogram.


Because BN ≅ DN and AN ≅ CN, BD and AC bisect each
other. Because the diagonals of ABCD bisect each other,
ABCD is a parallelogram.
Then prove that the diagonals of ABCD are congruent.


From the given you can write BN = AN and DN = CN so, by
the addition property of Equality, BN + DN = AN + CN. By
the Segment Addition Postulate, BD = BN + DN and AC =
AN + CN so, by substitution, BD = AC.
So, BD ≅ AC.
ABCD is a parallelogram with congruent diagonals, so
ABCD is a rectangle.