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QuadrilateralsQuadrilateralsQuadrilateralsQuadrilaterals
§§ 8.1 Quadrilaterals 8.1 Quadrilaterals
§§ 8.4 Rectangles, Rhombi, and Squares 8.4 Rectangles, Rhombi, and Squares
§§ 8.3 Tests for Parallelograms 8.3 Tests for Parallelograms
§§ 8.2 Parallelograms 8.2 Parallelograms
§§ 8.5 Trapezoids 8.5 Trapezoids
QuadrilateralsQuadrilaterals
You will learn to identify parts of quadrilaterals and find thesum of the measures of the interior angles of a quadrilateral.
1) Quadrilateral2) Consecutive3) Nonconsecutive4) Diagonal
QuadrilateralsQuadrilaterals
A quadrilateral is a closed geometric figure with ____ sides and ____ vertices.four four
The segments of a quadrilateral intersect only at their endpoints.
Quadrilaterals Not Quadrilaterals
Special types of quadrilaterals include squares and rectangles.
QuadrilateralsQuadrilaterals
A
B
CD
Quadrilaterals are named by listing their vertices in order.
There are several names for the quadrilateral below.
Some examples:
quadrilateral ABCD
quadrilateral BCDA
quadrilateral CDAB or
quadrilateral DABC
S R
Q
P
QuadrilateralsQuadrilaterals
Any two _______ of a quadrilateral are either __________ or _____________.
consecutivenonconsecutive
sidesverticesangles
QuadrilateralsQuadrilaterals
S R
Q
P
Segments that join nonconsecutive vertices of a quadrilateral are called________.diagonals
S and Q arenonconsecutivevertices. diagonal a is SQ
R and P arenonconsecutivevertices. diagonal a is RP
QuadrilateralsQuadrilaterals
Q
T
S
R
Name all pairs of consecutive sides:
Name all pairs of nonconsecutive angles:
Name the diagonals:
RS and QR ST and RS
TQ and ST QR and TQ
S and Q R and T
and QS TR
QuadrilateralsQuadrilaterals
D
C
B
AConsidering the quadrilateral to the right.
What shapes are formed if a diagonal is drawn? ___________two triangles
1
23
45
6
Use the Angle Sum Theorem (Section 5-2)to find m1 + m2 + m3 180
Use the Angle Sum Theorem (Section 5-2)to find m4 + m5 + m6 180
Find m1 + m2 + m3 + m4 + m5 + m6
180+ 180
360
This leads to the following theorem.
QuadrilateralsQuadrilaterals
Theorem
8-1
The sum of the measures of the angles of a quadrilateral is
____.360
a°
d°
c°
b°
a + b + c + d = 360
QuadrilateralsQuadrilaterals
A
D
C
BmA + mB + mC + mD = 360
x + 2x + x – 10 + 50 = 360
Find the measure of B in quadrilateral ABCD if A = x, B = 2x,C = x – 10, and D = 50.
4x + 40 = 360
4x = 320
x = 80
B = 2x
B = 2(80)
B = 160
QuadrilateralsQuadrilaterals
ParallelogramsParallelograms
You will learn to identify and use the properties of parallelograms.
1) Parallelogram
ParallelogramsParallelograms
A parallelogram is a quadrilateral with two pairs of ____________.parallel sides
A B
D C
In parallelogram ABCD below, and CBDA || DCAB ||
Also, the parallel sides are _________.congruent
Knowledge gained about “parallels” (chapter 4)will now be used in the following theorems.
Theorem8-2
Theorem8-3
Theorem8-4
ParallelogramsParallelograms
Opposite angles of a parallelogram are ________.
Opposite sides of a parallelogram are ________.
The consecutive angles of a parallelogram are ____________.
A B
D C
A B
D C
A B
D C
A C and B D
DCABCBDA || and ,||
mA + mB = 180mD + mC = 180
congruent
congruent
supplementary
ParallelogramsParallelograms
In RSTU, RS = 45, ST = 70, and U = 68.
R
U
S
T
45
70
68°
Find:
RU = ____
UT = _____
mS = _____
mT = _____
70 Theorem 8-3
45 Theorem 8-3
68° Theorem 8-2
112° Theorem 8-4
ParallelogramsParallelograms
Theorem
8-5
The diagonals of a parallelogram ______ each other.
A
D
B
C
E
bisect
ECAE
EBDE
In RSTU, if RT = 56, find RE. R
U
S
T
E
RE = 28
RTRE2
1
56RE2
1
A
D
B
C
ParallelogramsParallelograms
In the figure below, ABCD is a parallelogram.
DB BD
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)
BDCDBA
ASA Theorem
ParallelogramsParallelograms
Theorem
8-6
A diagonal of a parallelogram separates it into two_________________.
A
D
B
C
congruent triangles
BDCDBA
ParallelogramsParallelograms
The Escher design below is based ona _____________.
You can use a parallelogram to make a simpleEscher-like drawing.
Change one side of the parallelogram and thentranslate (slide) the change to the opposite side.
The resulting figure is used to make a designwith different colors and textures.
parallelogram
ParallelogramsParallelograms
Tests for ParallelogramsTests for Parallelograms
You will learn to identify and use tests to show that a quadrilateral is a parallelogram.
Nothing New!
Tests for ParallelogramsTests for Parallelograms
Theorem
8-7
If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram.
A
D C
B
congruent
BCAD
DCAB
and
Tests for ParallelogramsTests for Parallelograms
You can use the properties of congruent triangles and Theorem 8-7 to findother ways to show that a quadrilateral is a parallelogram.
In quadrilateral PQRS, PR and QS bisect eachother at T.
Show that PQRS is a parallelogram by providing a reason for each step.
TSQTTRPT and Definition of segment bisector
QTRSTPRTSPTQ and Vertical angles are congruent
RTQPTSRSTPQT and
RQPSRSPQ and
ramparallelog a is PQRS
SAS
Corresp. parts of Congruent Triangles are Congruent
Theorem 8-7
T
P
S R
Q
Tests for ParallelogramsTests for Parallelograms
Theorem
8-8
If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram.
A
D C
B
congruent
DCAB DCAB ||
parallel
Tests for ParallelogramsTests for Parallelograms
Theorem
8-9
If the diagonals of a quadrilateral ________________,then the quadrilateral is a parallelogram.
EBDE ECAE
bisect each other
A
D C
B
E
Tests for ParallelogramsTests for Parallelograms
Determine whether each quadrilateral is a parallelogram.If the figure is a parallelogram, give a reason for your answer.
A
D C
B
DCAB ||
DCAB Given
Alt. Int. Angles
Therefore, quadrilateral ABCD is a parallelogram. Theorem 8-8
Tests for ParallelogramsTests for Parallelograms
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
You will learn to identify and use the properties of rectangles,rhombi, and squares.
1) Rectangle2) Rhombus3) Square
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
A closed figure,4 sides & 4 verticesQuadrilateral
Opposite sides parallelopposite sides congruent
Parallelogram
Parallelogram with4 right angles
RectangleParallelogram with4 congruent sidesRhombus
Parallelogram with
4 congruent sides and
4 right angles
Square
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Identify the parallelogram below.
D C
BA
Parallelogram ABCD has4 right angles, but the foursides are not congruent.
Therefore, it is a _________rectangle
Identify the parallelogram below.
rhombus
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Theorem
8-10
The diagonals of a rectangle are _________.congruent
A
D
B
C
DBAC
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Theorem
8-11
The diagonals of a rhombus are ____________.perpendicular
A
B
D
C
DBAC
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Theorem
8-12
Each diagonal of a rhombus _______ a pair of opposite angles.bisects
43
8
7
6
5
43
21
21
65
87 D
C
B
A
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Use square XYZW to answer the following questions:
W Z
YX
O
1) If YW = 14, XZ = ____
2) mYOX = ____
A square has all the properties of a rectangle, and the diagonals of a rectangle are congruent.
14
A square has all the properties of a rhombus, and the diagonals of a rhombus are perpendicular.
90
3) Name all segments that are congruent to WO. Explain your reasoning.
The diagonals are congruent and they bisecteach other.
OY, XO, and OZ
Quadrilaterals
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
Parallelograms
Rhombi RectanglesSquares
Use the Venn diagram to answer the following questions: T or F
1) Every square is a rhombus: ___
2) Every rhombus is a square: ___
3) Every rectangle is a square: ___
4) Every square is a rectangle: ___
5) All rhombi are parallelograms: ___
6) Every parallelogram is a rectangle: ___
T
T
TF
F
F
Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares
TrapezoidsTrapezoids
You will learn to identify and use the properties of trapezoidsand isosceles trapezoids.
1) Trapezoid
Trapezoids Trapezoids
A trapezoid is a ____________ with exactly one pair of ____________.quadrilateral parallel sides
T
P A
R
The parallel sides are called ______.bases
base
baseThe non parallel sides are called _____.legs
leg leg
Each trapezoid has two pair ofbase angles.
base anglesT and R are one pairof base angles.
P and A are the other pair of base angles.
Trapezoids Trapezoids
Theorem
8-13
The median of a trapezoid is parallel to the _____, basesand the length of the median equals _______________ of the lengths of the bases.
one-half the sum
C
N
B
D
M
A
MNDCMNAB || ,||
DCABMN 2
1
Trapezoids Trapezoids
C
N
D
M
BA
Find the length of median MN in trapezoid ABCD if AB = 16 and DC = 20
16
20
DCABMN 2
1
20162
1MN
362
1MN
18MN
18
Trapezoids 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 arecongruent.
There is a similar property of isosceles trapezoids.
Trapezoids Trapezoids
Theorem
8-14
Each pair of __________ in an isosceles trapezoid is congruent.base angles
Z Y
XW
XW
YZ
Trapezoids Trapezoids
T
P A
R
60°
Find the missing angle measures in isosceles trapezoid TRAP.
P A
mP = mA
60 = mA
60°
Theorem 8 – 14
T R
P + A + 2(T) = 360
Theorem 8 – 14
60 + 60 + 2(T) = 360
120 + 2(T) = 360
2(T) = 240
T = 120
120°
R = 120
120°
Trapezoids Trapezoids