Quadrilaterals

1
Quadrilaterals

• 8.1 Quadrilaterals

• 8.2 Parallelograms

• 8.3 Tests for Parallelograms

• 8.4 Rectangles, Rhombi, and Squares

• 8.5 Trapezoids

2
Quadrilaterals
What You'll Learn
You will learn to identify parts of
quadrilaterals and find thesum of the measures
of the interior angles of a quadrilateral.
1) Quadrilateral 2) Consecutive 3)
Nonconsecutive 4) Diagonal
3
Quadrilaterals
four
four
A quadrilateral is a closed geometric figure with
____ sides and ____ vertices.
Quadrilaterals Not Quadrilaterals

The segments of a quadrilateral intersect only at
their endpoints.
Special types of quadrilaterals include squares
and rectangles.
4
Quadrilaterals
Quadrilaterals are named by listing their
vertices in order.
There are several names for the quadrilateral
below.
Some examples
quadrilateral ABCD
B
quadrilateral BCDA
A
quadrilateral CDAB or
quadrilateral DABC
D
C
5
Quadrilaterals
Any two _______ of a quadrilateral are either
__________ or _____________.
consecutive
sides
vertices
angles
nonconsecutive
6
Quadrilaterals
Segments that join nonconsecutive vertices of a
quadrilateral are called________.
diagonals
R and P arenonconsecutivevertices.
S and Q arenonconsecutivevertices.
7
Quadrilaterals
Name all pairs of consecutive sides
Name all pairs of nonconsecutive angles
Name the diagonals
8
Quadrilaterals
Considering the quadrilateral to the right.
1
What shapes are formed if a diagonal is drawn?
___________
two triangles
2
3
5
4
Use the Angle Sum Theorem (Section 5-2)to find
m?1 m?2 m?3
180
Use the Angle Sum Theorem (Section 5-2)to find
m?4 m?5 m?6
180
6
180
Find m?1 m?2 m?3 m?4
m?5 m?6
180
This leads to the following theorem.
9
Quadrilaterals
Theorem 8-1 The sum of the measures of the angles of a quadrilateral is ____.
360
360
a b c d
10
Quadrilaterals
Find the measure of ?B in quadrilateral ABCD if
?A x, ?B 2x,?C x 10, and ?D 50.
m?A m?B m?C m?D 360
x 2x x 10 50 360
4x 40 360
4x 320
x 80
?B 2x
?B 2(80)
?B 160
11
Quadrilaterals
End of Section 8.1
12
Parallelograms
What You'll Learn
You will learn to identify and use the properties
of parallelograms.
1) Parallelogram
13
Parallelograms
parallel sides
A parallelogram is a quadrilateral with two pairs
of ____________.
congruent
Also, the parallel sides are _________.
Knowledge gained about parallels (chapter
4)will now be used in the following theorems.
14
Parallelograms
Theorem8-2
Theorem8-3
Theorem8-4
Opposite angles of a parallelogram are ________.
congruent
?A ? ?C and ?B ? ?D
Opposite sides of a parallelogram are ________.
congruent
The consecutive angles of a parallelogram are
____________.
supplementary
m?A m?B 180m?D m?C 180
15
Parallelograms
Find RU ____
70
Theorem 8-3
45
UT _____
Theorem 8-3
68
Theorem 8-2
m?S _____
112
Theorem 8-4
m?T _____
16
Parallelograms
Theorem 8-5 The diagonals of a parallelogram ______ each other.
bisect
E
RE 28
17
Parallelograms
In the figure below, ABCD is a parallelogram.
Since AD BC and diagonal DB is a
transversal, then ?ADB ? ?CBD.
(Alternate Interior angles)
Since AB DC and diagonal DB is a
transversal, then ?BDC ? ?DBA.
(Alternate Interior angles)
DB ? BD
ASA Theorem
18
Parallelograms
Theorem 8-6 A diagonal of a parallelogram separates it into two_________________.
congruent triangles
19
Parallelograms
The Escher design below is based ona
_____________.
parallelogram
You can use a parallelogram to make a
simple Escher-like drawing.
Change one side of the parallelogram and
then translate (slide) the change to the opposite
side.
The resulting figure is used to make a
design with different colors and textures.
20
Parallelograms
End of Section 8.2
21
Tests for Parallelograms
What You'll Learn
You will learn to identify and use tests to show
that a quadrilateral is a parallelogram.
Nothing New!
22
Tests for Parallelograms
Theorem 8-7 If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram.
congruent
23
Tests for Parallelograms
You can use the properties of congruent triangles
and Theorem 8-7 to find other ways to show that a
quadrilateral is a parallelogram.
Show that PQRS is a parallelogram by providing a
reason for each step.
Definition of segment bisector
Vertical angles are congruent
SAS
Corresp. parts of Congruent Triangles are
Congruent
Theorem 8-7
24
Tests for Parallelograms
Theorem 8-8 If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram.
parallel
congruent
25
Tests for Parallelograms
Theorem 8-9 If the diagonals of a quadrilateral ________________,then the quadrilateral is a parallelogram.
bisect each other
26
Tests for Parallelograms
Determine whether each quadrilateral is a
parallelogram.If the figure is a parallelogram,
give a reason for your answer.
Given
Alt. Int. Angles
Therefore, quadrilateral ABCD is a parallelogram.
Theorem 8-8
27
Tests for Parallelograms
End of Section 8.3
28
Rectangles, Rhombi, and Squares
What You'll Learn
You will learn to identify and use the properties
of rectangles, rhombi, and squares.
1) Rectangle 2) Rhombus 3) Square
29
Rectangles, Rhombi, and Squares
A closed figure, 4 sides 4 vertices
Quadrilateral
Opposite sides parallel opposite sides congruent
Parallelogram
Parallelogram with 4 congruent sides
Parallelogram with 4 right angles
Rhombus
Rectangle
Parallelogram with 4 congruent sides and 4
right angles
Square
30
Rectangles, Rhombi, and Squares
Identify the parallelogram below.
Identify the parallelogram below.
rhombus
Parallelogram ABCD has 4 right angles, but the
four sides are not congruent.
Therefore, it is a _________
rectangle
31
Rectangles, Rhombi, and Squares
Theorem 8-10 The diagonals of a rectangle are _________.
congruent
32
Rectangles, Rhombi, and Squares
Theorem 8-11 The diagonals of a rhombus are ____________.
perpendicular
33
Rectangles, Rhombi, and Squares
Theorem 8-12 Each diagonal of a rhombus _______ a pair of opposite angles.
bisects
A
1
2
D
8
7
6
5
C
3
4
B
34
Rectangles, Rhombi, and Squares
Use square XYZW to answer the following questions
14
1) If YW 14, XZ ____
A square has all the properties of a rectangle,
and the diagonals of a rectangle are congruent.
90
2) m?YOX ____
A square has all the properties of a rhombus,
and the diagonals of a rhombus are perpendicular.
The diagonals are congruent and they bisect each
other.
35
Rectangles, Rhombi, and Squares
Use the Venn diagram to answer the following
questions T or F
1) Every square is a rhombus ___
T
5) All rhombi are parallelograms ___
T
F
2) Every rhombus is a square ___
F
3) Every rectangle is a square ___
6) Every parallelogram is a
rectangle ___
F
4) Every square is a rectangle ___
T
36
Rectangles, Rhombi, and Squares
End of Section 8.4
37
Trapezoids
What You'll Learn
You will learn to identify and use the properties
of trapezoidsand isosceles trapezoids.
1) Trapezoid
38
Trapezoids
parallel sides
A trapezoid is a ____________ with exactly one
pair of ____________.
quadrilateral
The parallel sides are called ______.
bases
legs
The non parallel sides are called _____.
base
Each trapezoid has two pair of base angles.
leg
leg
base angles
?T and ?R are one pair of base angles.
?P and ?A are the other pair of base angles.
base
39
Trapezoids
Theorem 8-13 The median of a trapezoid is parallel to the _____,
bases
and the length of the median equals
_______________ of the lengths of the bases.
one-half the sum
40
Trapezoids
Find the length of median MN in trapezoid ABCD
if AB 16 and DC 20
16
18
20
41
Trapezoids
If the legs of a trapezoid are congruent, the
trapezoid is an _________________.
isosceles trapezoid
In lesson 6 4, you learned that the base angles
of an isosceles triangle are congruent.
There is a similar property of isosceles
trapezoids.
42
Trapezoids
Theorem 8-14 Each pair of __________ in an isosceles trapezoid is congruent.
base angles
43
Trapezoids
Find the missing angle measures in isosceles
trapezoid TRAP.
Theorem 8 14
?P ? ?A
120
120
m?P m?A
60 m?A
Theorem 8 14
?T ? ?R
60
?P ?A 2(?T) 360
60 60 2(?T) 360
120 2(?T) 360
2(?T) 240
?T 120
?R 120
44
Trapezoids
End of Section 8.5