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Unit 6
Quadrilaterals
Lesson 6.1
Properties of Quadrilaterals
Lesson 6.1 Objectives
• Identify a figure to be a quadrilateral.• Use the sum of the interior angles of a
quadrilateral. (G1.4.1)
Definition of a Quadrilateral
• A quadrilateral is any four-sided figure with the following properties:
1. All sides must be line segments.
2. Each side must intersect only two other sides.• One at each of its endpoints, so that there are no:
i. Gaps that do not connect one side to another, or
ii. Tails that extend beyond another side.
Example 6.1
Determine if the figure is a quadrilateral.
1.
2.
3.
4.
5.
6.
7.
8.
Yes
Yes
Yes
No
No
No
No
NoNo curves
Too many intersecting segments
No gaps
Too many sides
No tails
Interior Angles
• Recall that the interior angles of any figure are located in the interior and are formed by the sides of the figure itself.
Review: What is the sum of the interior angles of anytriangle?
o180
Review: How manydegrees does a straightline measure?
o180 ???
Review:What do youthink the sum of the interior angles of a quadrilateral might be?
Theorem 6.1:Interior Angles of a Quadrilateral Theorem
• The sum of the measures of the interior angles of a quadrilateral is 360o.
1 2
3 4
m 1 +m 2 + m 3 + m 4 = 360o
Example 6.2
Find the missing angle.
1.
2.
3.
4.
o o o o o95 85 120 360x o o o300 360x
o60x
o o o262 360x o98x
o o o270 360x o90x
o o o253 360x o107x
Example 6.3
Find the x.
1. 2. 3.
o o o o o98 90 (17 3) 90 360x o o275 17 360x
5x
o o o o o86 94 (11 2) 94 360x
8x
o o o o o95 35 360x x
115x
o17 85x
o o272 11 360x o11 88x
o o130 2 360x o2 230x
Lesson 6.1 Homework
• Lesson 6.1 – Properties of Quadrilaterals
• Due Tomorrow
Lesson 6.2
Day 1:
Parallelograms
Lesson 6.2 Objectives
• Define a parallelogram
• Define special parallelograms• Identify properties of parallelograms (G1.4.3)
• Use properties of parallelograms to determine unknown quantities of the parallelogram (G1.4.4)
Definition of a Parallelogram
• A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Theorem 6.2:Congruent Sides of a Parallelogram
• If a quadrilateral is a parallelogram, then its opposite sides are congruent.– The converse is also true!
• Theorem 6.6
Theorem 6.3:Opposite Angles of a Parallelogram
• If a quadrilateral is a parallelogram, then its opposite angles are congruent.– The converse is also true!
• Theorem 6.7
Example 6.4
Find the missing variables in the parallelograms.1.
x = 11
y = 8
m = 101
c – 5 = 20
c = 25
d + 15 = 68d = 53
2. 3.
Theorem 6.4:Consecutive Angles of a Parallelogram
• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.– The converse is also true!
• Theorem 6.8 Q
P
R
Sm P + m S = 180o
m P + m Q = 180o
m Q + m R = 180o
m R + m S = 180o
Theorem 6.5:Diagonals of a Parallelogram
• If a quadrilateral is a parallelogram, then its diagonals bisect each other.– Remember that means to cut into two congruent segments.
• And again, the converse is also true!– Theorem 6.9
Example 6.5Find the indicated measure in HIJKa) HI
a) 16a) Theorem 6.2
b) GHb) 8
b) Theorem 6.5
c) KHc) 10
c) Theorem 6.2
d) HJd) 16
d) Theorem 6.5 & Seg Add Post
e) m KIHe) 28o
e) AIA Theorem
m JIHa) 96o
a) Theorem 6.4
a) m KJIa) 84o
b) Theorem 6.3
Theorem 6.10:Congruent Sides of a Parallelogram
• If a quadrilateral has one pair of opposite sides that are both congruent and parallel, then it is a parallelogram.
Example 6.6Is there enough information to prove the quadrilaterals to be a parallelogram.
If so, explain.1.
Yes!
Yes!
Yes!
Yes!
Yes!
Yes!
2. 3.
4. 5. 6.
One pair of parallel and congruent sides.(Theorem 6.10)
Both pairs of opposite sides are congruent.(Theorem 6.6)
Both pairs of opposite angles are congruent.
(Theorem 6.7)
The diagonals bisect each other.
(Theorem 6.9)
Both pairs of opposite sides are congruent.
(Theorem 6.6)
Both pairs of opposite angles are
congruent.(Theorem 6.7)
OROne pair of parallel and
congruent sides.(Theorem 6.10)
All consecutive angles are
supplementary.(Theorem 6.8)
Lesson 6.2a Homework
• Lesson 6.2: Day 1 – Parallelograms
• Due Tomorrow
Lesson 6.2
Day 2:
(Special) Parallelograms
Rhombus
• A rhombus is a parallelogram with four congruent sides.– The rhombus corollary states that a quadrilateral is a rhombus if
and only if it has four congruent sides.
Theorem 6.11:Perpendicular Diagonals
• A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Theorem 6.12:Opposite Angle Bisector
• A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.
Rectangle
• A rectangle is a parallelogram with four congruent angles.– The rectangle corollary states that a quadrilateral is a
rectangle iff it has four right angles.
Theorem 6.13:Four Congruent Diagonals
• A parallelogram is a rectangle iff all four segments of the diagonals are congruent.
Square
• A square is a parallelogram with four congruent sides and four congruent angles.
Square Corollary• A quadrilateral is a square iff its a rhombus
and a rectangle.
• So that means that all the properties of rhombuses and rectangles work for a square at the same time.
Example 6.7
Classify the parallelogram.Explain your reasoning.1.
RhombusDiagonals are perpendicular.
Theorem 6.11Square
Square Corollary
Must be supplementary
RectangleDiagonals are congruent.
Theorem 6.13
2. 3.
Lesson 6.2b Homework
• Lesson 6.2: Day 2 – Parallelograms
• Due Tomorrow
Lesson 6.3
Trapezoids
and
Kites
Lesson 6.3 Objectives
• Identify properties of a trapezoid. (G1.4.1)
• Recognize an isosceles trapezoid. (G1.4.1)
• Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid.
• Identify a kite. (G1.4.1)
Trapezoid
• A trapezoid is a quadrilateral with exactly one pair of parallel sides.
– The parallel sides are called the bases.
– The nonparallel sides are called legs.
– The angles formed by the bases are called the base angles.
Example 6.8
Find the indicated angle measure of the trapezoid.
1. 2.
Recall that a trapezoid has one set of parallel bases.
CIAConsecutiveInterior Angles are supplementary!
o o o62 180x o118x
CIA
o o o96 180x
o84x
Example 6.9
Find x in the trapezoid.
1.
2. CIA
ConsecutiveInterior Angles are supplementary!
o o o( 35) 135 180x
10x
o o170 180x
CIA
o o o(5 25) 115 180x
8x
o o5 140 180x
Isosceles Trapezoid
• If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Theorem 6.14:Bases Angles of a Trapezoid
• If a trapezoid is isosceles, then each pair of base angles is congruent.– That means the top base angles are congruent.– The bottom base angles are congruent.
• But they are not all congruent to each other!
Theorem 6.15:Base Angles of a Trapezoid Converse
• If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.
Theorem 6.16:Congruent Diagonals of a Trapezoid
• A trapezoid is isosceles if and only if its diagonals are congruent.– Notice this is the entire diagonal itself.
• Don’t worry about it being bisected cause it’s not!!
Example 6.10
Find the measures of the other three angles.1.
53o Supplementary
because of CIA127o
127o
Supplementarybecause of CIA
97o
83o
83o
2.
Midsegment
• The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.
Theorem 6.17:Midsegment Theorem for Trapezoids
• The midsegment of a trapezoid is1. parallel to each base and 2. its length is one half the sum of the lengths of the bases.
• It is the average of the base lengths!
A B
C D
M N
MN 2
AB CD
Example 6.11
Find the indicated length of the trapezoid.1.
2.
3.
??
?
7 13
2x
20
2x
10x 10.5x
9 12
2x
21
2x
21x
4332
2
x
64 43 x
Multiply both sides by 2.
Or essentially double the midsegment!
Kite
• A kite is a quadrilateral that has two pairs of consecutive sides that are congruent, but opposite sides are not congruent.– It looks like the kite you got for your birthday when you were 5!
• There are no sides that are parallel.
Theorem 6.18:Diagonals of a Kite
• If a quadrilateral is a kite, then its diagonals are perpendicular.
Theorem 6.19:Opposite Angles of a Kite
• If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.– The angles that are congruent are between the
two different congruent sides.• You could call those the shoulder angles.
NOT
Example 6.12
Find the missing angle measures.1.
88o
K = 88
296 + J = 360
J = 64
60 + K + 50 + M = 360
But K M
60 + M + 50 + M = 360
110 + 2M = 360
2M = 250
M = 125
K = 125
64o
125o
125o
2.
88 + 120 + 88 + J = 360
Example 6.13
Find the lengths of all the sides of the kite.Round your answer to the nearest hundredth.
Use Pythagorean Theorem!
Cause the diagonals are perpendicular!!
a2 + b2 = c2
a2 + b2 = c2
52 + 52 = c2
25 + 25 = c2
50 = c2
c = 7.07
7.07 7.07a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
c = 131313
Lesson 6.3 Homework
• Lesson 6.3 – Trapezoids and Kites
• Due Tomorrow
Lesson 6.4
Perimeter and Area
of
Quadrilaterals
Lesson 6.4 Objectives
• Find the perimeter of any type of quadrilateral. (G1.4.1)
• Find the area of any type of quadrilateral. (G1.4.3)
Postulate 22:Area of a Square Postulate
• The area of a square is the square of the length of its side.– A = s2
s
Theorem 6.20:Area of a Rectangle
• The area of a rectangle is the product of a base and its corresponding height.– Corresponding height indicates a segment
perpendicular to the base to the opposite side.
• A = bh
b
h
Example 6.14
Find the perimeter and area of the given quadrilateral.
1.
2.
Theorem 6.21:Area of a Parallelogram
• The area of a parallelogram is the product of a base and its corresponding height.– Remember the height must be perpendicular to one of the bases.
– The height will be given to you or you will need to find it.• To find it, use Pythagorean Theorem
– a2 + b2 = c2
– A = bh
b
h
Theorem 6.23:Area of a Trapezoid
• The area of a trapezoid is one half the product of the height and the sum of the bases.– The height is the perpendicular segment between the
bases of the trapezoid.
• A = ½ (b1+b2) h
b2
h
b1
Theorem 6.24:Area of a Kite
• The area of a kite is one half the product of the lengths of the diagonals.– A = ½ d1d2
d2
d1
Theorem 6.25:Area of a Rhombus
• The area of a rhombus is equal to one half the product of the lengths of the diagonals.– A = ½ d1d2
d2
d1
Example 6.15
Find the perimeter and area of the given quadrilateral.
1.
2.
3.
Area Postulates
• Postulate 23: Area Congruence Postulate– If two polygons are
congruent, then they have the same area.
• Postulate 24: Area Addition Postulate– The area of a region is
the sum of the areas of its nonoverlapping parts.
Example 6.16
Find the perimeter and area of the given figure.Assume all corners form a right angle.
Lesson 6.4 Homework
• Lesson 6.4 – Perimeter and Area of Quadrilaterals• Due Tomorrow
Lesson 6.5
Special Quadrilaterals
Lesson 6.6 Objectives
• Create a hierarchy of polygons
• Identify special quadrilaterals based on limited information
Polygon Hierarchy
Polygons
Triangles Quadrilaterals Pentagons
Rhombus Rectangle
TrapezoidParallelogram Kite
Square
Isosceles Trapezoid
How to Read the HierarchyPolygons
Triangles Quadrilaterals Pentagons
Rhombus Rectangle
TrapezoidParallelogram Kite
Square
Isosceles TrapezoidALW
AYS
SOM
ETIM
ES
So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon.
But a parallelogram is sometimes a rhombus and sometimes a square.
However, a parallelogram is never a trapezoid or a kite.
NEVER
Using the Hierarchy
• Remember that a square must fit all the properties of its “ancestors.”– That means the properties of a rhombus, rectangle,
parallelogram, quadrilateral, and polygon must all be true!
• So when asked to identify a figure as specific as possible, test the properties working your way down the hierarchy.– As soon as you find a figure that doesn’t work any
more you should be able to identify the specific name of that figure.
Homework 6.6
• HW• p367-370
– 8-35, 55-65
• Due Tomorrow
• Test Friday– March 26
