Transcript Geometry 1 Unit 6

Geometry 1 Unit 6
Quadrilaterals
Geometry 1 Unit 6
6.1 Polygons
Polygons
 Polygon
 A closed figure in a plane
 Formed by connecting line segments endpoint to
endpoint
 Each segment intersects exactly 2 others
 Classified by the number of sides they have
 Named by listing vertices in consecutive order
 Sides
 Line segments in a polygon
 Vertex
 Each endpoint in a polygon
Polygons
polygons
not polygons
Polygons
Pentagon ABCDE or pentagon CDEAB
A
E
B
D
C
Polygons
Sides
3
4
5
6
7
8
9
10
11
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
n-gon (a 19 sided polygon is a 19-gon)
Polygons
 Diagonals
 Line segments that connect nonconsecutive vertices.
Polygons
 Convex polygons
 Polygons with no diagonals on the
outside of the polygon
Polygons
 Concave polygons
 A polygon is concave if at least one
diagonal is outside the polygon
 These are also called nonconvex.
Polygons
 Example 1
 Identify the polygon and state whether it
is convex or concave.
Polygons
 Equilateral Polygon
 all sides the same length
 Equiangular Polygon
 all angles equal measure
 Regular Polygon
 equilateral and equiangular
Polygons
Equilateral
Equiangular
Regular
Polygons
 Example 2
 Decide whether the polygon is regular.
Polygons
 Interior Angles of a Quadrilateral
Theorem
 The sum of the measures of the interior
angles of a quadrilateral is 360°.
m1 + m2 + m3 + m4 = 360°
2
1
4
3
Polygons
 Example 3
 Find mF, mG, and mH.
H
G
x
E
55°
x
F
Polygons
 Example 4
 Use the information in the diagram to
solve for x
100° 120°
2x + 30
3x – 5
Geometry 1 Unit 6
6.2 Properties of Parallelograms
Properties of Parallelograms
Parallelogram
 Quadrilateral with two pairs of parallel
sides.
Properties of Parallelograms
Opposite Sides of a Parallelogram
Theorem
 If a quadrilateral is a parallelogram, then
its opposite sides are congruent.
Q
P
R
S
PQ  RS and SP  QR
Properties of Parallelograms
Opposite Angles in a Parallelogram
Theorem
 If a quadrilateral is a parallelogram, then
its opposite angles are congruent.
Q
R
P  R and Q  S
P
S
Properties of Parallelograms
Consecutive Angles in a Parallelogram
Theorem
 If a quadrilateral is a parallelogram, then
its consecutive angles are supplementary
R
Q
mP + mQ = 180°
mQ + mR = 180°
Add to
equal
180°
mR + mS = 180°
mS + mP = 180°
P
S
Properties of Parallelograms
Diagonals in a Parallelogram Theorem
 If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
Q
R
QM  SM and PM  RM
M
P
S
Properties of Parallelograms
 Example 1
 GHJK is a parallelogram. Find each
unknown length
 JH
 LH
K
8
J

6
L
G

H
Properties of Parallelograms
 Example 2
 In ABCD, mC = 105°. Find the
measure of each angle.
 mA
 mD
Properties of Parallelograms
 Example 3
 WXYZ is a parallelogram. Find the value
of x.
Z
3x + 18°
W

Y

4x – 9°
X
Properties of Parallelograms
 Example 4
 Given:
 ABCD is a
parallelogram.
 Prove:
 2  4
A
3
C
Reasons
ABCD is a
parallelogram
AD || BC
2  1
AB || CD
Alternate
interior
angles
theorem
B
2
1
4
D
Statement
2  4
Properties of Parallelograms
Statement
 Example 5
ACDF is a
parallelogram.
 Given:
 ACDF is a
parallelogram.
 ABDE is a
parallelogram.
 Prove:
 ∆BCD  ∆EFA
A
B
Reason
ABDE is a
parallelogram.
Opposite sides of a
parallelogram are
congruent
AC = DF
AB = DE
AC = AB + BC
C
DF = DE + EF
AC = DE + DF
AB + BC = AB + EF
BC = EF
F
Def of Congruent
E
D
∆BCD  ∆EFA
Properties of Parallelograms
 Example 6
 A four-sided concrete slab has
consecutive angle measures of 85°, 94°,
85°, and 96°. Is the slab a
parallelogram? Explain.
Geometry 1 Unit 6
6.3 Proving Quadrilaterals are
Parallelograms
Proving Quadrilaterals are
Parallelograms
 Investigating Properties of Parallelograms
 Cut 4 straws to form two congruent pairs.
 Partly unbend two paperclips, link their smaller
ends, and insert the larger ends into two cut
straws. Join the rest of the straws to form a
quadrilateral with opposite sides congruent.
 Change the angles of your quadrilateral. Is your
quadrilateral a parallelogram?
Proving Quadrilaterals are
Parallelograms
Converse of the Opposite Sides of a
Parallelogram Theorem
 If a opposite sides of a quadrilateral are
congruent, then the quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram
Proving Quadrilaterals are
Parallelograms
Converse of the Opposite Angles in a
Parallelogram Theorem
 If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
A
D
B
C
ABCD is a parallelogram.
Proving Quadrilaterals are
Parallelograms
Converse of the Consecutive Angles in a
Parallelogram Theorem
 If an angle of a quadrilateral is supplementary to both
of its consecutive angles, then the quadrilateral is a
parallelogram.
B
A
(180 – x)°
x°
ABCD is a parallelogram
x°
D
C
Proving Quadrilaterals are
Parallelograms
Converse of the Diagonals in a
Parallelogram Theorem
 If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a
parallelogram.
A
B
M
D
C
ABCD is a parallelogram
Proving Quadrilaterals are
Parallelograms
 Example 1
Statements
∆PQT  ∆RST
 Given:
CPCTC
 ∆PQT  ∆RST
 Prove:
 PQRS is a
parallelogram.
P
PT = RT
ST = QT
Def. of bisect
Q
T
S
Reasons
R
PQRS is a
parallelogram
Proving Quadrilaterals are
Parallelograms
 Example 2
 A gate is braced as shown. How do you
know that opposite sides of the gate are
congruent?
Proving Quadrilaterals are
Parallelograms
 Congruent and Parallel Sides
Theorem
 If one pair of opposite sides of a
quadrilateral are congruent and parallel,
then the quadrilateral is a parallelogram.
B
C

ABCD is a parallelogram.
A

D
Proving Quadrilaterals are
Parallelograms
 To determine if a quadrilateral is a
parallelogram, you need to know one of the
following:
 Opposite sides are parallel
 Opposite sides are congruent
 Opposite angles are congruent
 An angle is supplementary with both of its
consecutive angles
 Diagonals bisect each other
 One pair of sides is both parallel and congruent
Proving Quadrilaterals are
Parallelograms
 Example 3
 Show that A(-1,2), B(3,2), C(1,-2), and
D(-3,-2) are the vertices of a
parallelogram.
Geometry 1 Unit 6
6.4 Rhombuses, Rectangles, and
Squares
Rhombuses, Rectangles, and
Squares
Rectangle
 Parallelogram with four congruent angles
Rhombus
 Parallelogram with four congruent sides
Square
 Parallelogram with four congruent angles
and four congruent sides
Rhombuses, Rectangles, and
Squares
Example 1
 Decide if each statement is always,
sometimes or never true.
A rhombus is a rectangle
A parallelogram is a rectangle
A rectangle is a square
A square is a rhombus
Rhombuses, Rectangles, and
Squares
Example 2
 Given FROG is a rectangle, what else do
you know about FROG?
F
R
G
O
Rhombuses, Rectangles, and
Squares
Example 3
 EFGH is a rectangle. K is the midpoint of
FH. EG = 8z – 16,
What is the measure of segment EK?
What is the measure of segment GK?
Rhombuses, Rectangles, and
Squares
Rhombus Corollary
 A quadrilateral is a rhombus if and only if
it has four congruent sides.
Rhombuses, Rectangles, and
Squares
Rectangle Corollary
 A quadrilateral is a rectangle if and only if
it has four right angles.
Rhombuses, Rectangles, and
Squares
Square Corollary
 A quadrilateral is a square if and only if it
is a rhombus and a rectangle.
Rhombuses, Rectangles, and
Squares
Perpendicular Diagonals of a Rhombus
Theorem
 A parallelogram is a rhombus if and only if
its diagonals are perpendicular.
B
C

ABCD is a rhombus if and only if
AC
A

D
BD.
Rhombuses, Rectangles, and
Squares
 Diagonals Bisecting Opposite Angles
Theorem.
 A parallelogram is a rhombus if and only
if each diagonal bisects a pair of opposite
angles.
B
C
ABCD us a rhombus if and
only if
AC bisects DAB and BCD
and
BD bisects ADC and CBA
A
D
Rhombuses, Rectangles, and
Squares
 Diagonals in a Rectangle Theorem
 A parallelogram is a rectangle if and only
if its diagonals are congruent.
A
B
ABCD is a rectangle if and
only if
AC  BD.
D
C
Rhombuses, Rectangles, and
Squares
 Example 4
 You cut out a parallelogram shaped quilt
piece and measure the diagonals to be
congruent. What is the shape?
 An angle formed by the diagonals of the
quilt piece measures 90°. Is the shape a
square?
Geometry 1 Unit 6
6.5 Trapezoids and Kites
Trapezoids and Kites
 Trapezoid
 A quadrilateral with exactly one pair of parallel sides.
 Bases
 The parallel sides of a trapezoid.
 Pairs of Base Angles
 Angles in a trapezoid that share a base.
 Legs
 The nonparallel sides of a trapezoid.

 Isosceles Trapezoid
 Trapezoid with congruent legs.

Trapezoids and Kites
 Base Angles of an Isosceles Trapezoid
Theorem
 If a trapezoid is isosceles, then each pair
of base angles is congruent.
A

B
A  B, C  D
D

C
Trapezoids and Kites
 Congruent Base Angles in a Trapezoid
Theorem.
 If a trapezoid has a pair of congruent
base angles, then it is an isosceles
trapezoid.
A

B
ABCD is an isosceles trapezoid.
D

C
Trapezoids and Kites
 Diagonals in an Isosceles Trapezoid
Theorem
 A trapezoid is isosceles if and only if its
diagonals are congruent.
A

B
ABCD is isosceles if and only if
AC  BD.
D

C
Trapezoids and Kites
 Midsegment of a trapezoid
 The segment that connects the
midpoints of a trapezoids legs.
midsegment
Trapezoids and Kites
 Midsegment Theorem for Trapezoids
 The midsegment of a trapezoid is parallel
to each base and its length is one half
the sum of the lengths of the bases.
B

C
MN = ½(AD + BC)
N
M
A
MN || AD, MN || BC,

D
Trapezoids and Kites
 Kite
 A quadrilateral with two distinct pairs of
consecutive congruent sides. Opposite
sides are not congruent.
 Diagonals of a Kite Theorem
 If a quadrilateral is a kite, then its
diagonals are perpendicular.
 Opposite Angles in a Kite Theorem
 If a quadrilateral is a kite, then exactly
one pair of opposite angles are
congruent.
Trapezoids and Kites
 Example 4
 GHJK is a kite. Find HP.
H
√29
G
5
P
K
J
Trapezoids and Kites
 Example 5
 RSTU is a kite. Find mR, mS, and
mT.
S
R
x + 30°
125°
U
x°
T
Geometry 1 Unit 6
6.6 Special Quadrilaterals
Special Quadrilaterals
Special Quadrilaterals
Property
1.Both pairs
of opposite
sides are
congruent
2. Diagonals
are congruent
3. Diagonals
are
perpendicular
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
Special Quadrilaterals
Property
4. Diagonals
bisect each
other
5.
Consecutive
angles are
supplementary
6. Both pairs
of opposite
angles are
congruent
Parallelogram Rectangle
Rhombus
Square
Trapezoid
Kite
Geometry 1 Unit 6
6.7 Areas of Triangles and
Quadrilaterals
Area
 Area is the number of square units in
a figure.
Area-Rectangles
Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.
Area-Parallelograms
Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.
Rectangles and Parallelograms
 Area = length times width
 Area = base times height
 A = bh
h
b
b
Area-Triangles
Area-Triangles
 A = ½ base times height
 A = ½ bh
b
h
h
b
h
b
Area-Trapezoids
b1
A = triangle 1 + triangle 2
h
b1
A = ½ h(b1) + ½ h(b2)
h
h
b2
b2
A = ½ h(b1 + b2)
Area-Trapezoids
 A = ½ height times (base1 + base 2)
 A = ½ h(b1 + b2)
b1
b1
h
h
b2
b1
b2
h
b2
Area- Kites
 A = ½ (diagonal 1) times (diagonal 2)
 A = ½ (d1)(d2)
d1 and d2
Area-Rhombus
 A = ½ (diagonal 1) times (diagonal 2)
 A = ½ (d1)(d2)
d1 and d2
Example 1
 Find the area of ΔRST.
R
3
S
4
T
Example 2
 What is the base of a triangle that
has an area of 48 and a height of 3?
Example 3
 A rectangle has an area of 100 square
meters and a height of 25 meters.
Are all the rectangles with these
dimensions congruent?
Example 4
 Find the area of parallelogram RSTU.
R
6
S
3
U
6
T
Example 5
 What is the height of a parallelogram
that has an area of 96 square feet
and a base length of 8 feet?
Example 6
 Find the area of trapezoid EFGH.
E(-2, 3), F(2, 4), G(2, -2), H(-2, -1)
Example 7
 Find the area of kite ABCD.
A(0, 5), B(3, 6), C(6, 5), D(3, 2)
Example 8
 Use the information given in the
diagram to find the area of kite
ABCD.
B
8
A
16
8
8
D
C
Example 9
 The tray below is designed to save
space on cafeteria tables. How much
table area does the tray use?
10 in
3 in
8 in
16 in
Example 10
 Find the area of rhombus EFGH
if EG = 10 and FH = 15.