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Chapter 6
Quadrilaterals
Types of Polygons
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
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
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Triangle – 3 sides
Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon – 6 sides
Heptagon – 7 sides
Octagon – 8 sides
Nonagon – 9 sides
Decagon – 10 sides
Dodecagon – 12 sides
All other polygons = n-gon
Lesson 6.1 : Angles of Polygons

Interior Angle Sum
Theorem
 The
sum of the
measures of the
interior angles of a
polygon is found by
S=180(n-2)

Ex: Hexagon

Exterior Angle Sum
Theorem
 The
sum of the
measures of the
exterior angles of a
polygon is 360 no
matter how many
sides.
Lesson 6.1 : Angles of Polygons

Find the measure of
an interior and an
exterior angle for
each polygon.
 24-gon
 3x-gon

Find the measure of
an exterior angle
given the number of
sides of a polygon
 260
sides
Lesson 6.1: Angles of Polygons

The measure of an
interior angle of a
polygon is given. Find
the number of sides.
 175
 168.75
180-175=5
360/5=
72

A pentagon has
angles (4x+5), (5x-5),
(6x+10), (4x+10), and
7x. Find x.
A. Find the value of x in the diagram.
Lesson 6.2: Parallelograms
Opposite sides of a
parallelogram are
congruent
Opposite angles in a
parallelogram are
congruent
Consecutive angles in
a parallelogram are
supplementary
Properties of
Parallelograms
A parallelogram is a
quadrilateral with both
pairs of opposite sides
parallel
If a parallelogram has 1
right angle, it has 4 right
angles.
The diagonals of a
parallelogram bisect
each other
The diagonals of a
parallelogram split it into 2
congruent triangles
?
____
?
?
A. ABCD is a parallelogram. Find AB.
B. ABCD is a parallelogram. Find mC.
C. ABCD is a parallelogram. Find mD.
A. If WXYZ is a parallelogram, find the value of r, s
and t.
What are the coordinates of the
intersection of the diagonals of
parallelogram MNPR, with vertices
M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?
Lesson 6.3 : Tests for
Parallelograms

If…
 Both
pairs of opposite sides are parallel
 Both pairs of opposite sides are congruent
 Both pairs of opposite angles are congruent
 The diagonals bisect each other
 One pair of opposite sides is congruent and
parallel

Then the quadrilateral is a parallelogram
Determine whether the quadrilateral is a
parallelogram. Justify your answer.
Which method would prove the
quadrilateral is a parallelogram?
Determine whether the
quadrilateral is a parallelogram.
Determine whether the
quadrilateral is a parallelogram.
Find x and y so that the quadrilateral is a
parallelogram.
COORDINATE GEOMETRY Graph
quadrilateral QRST with vertices Q(–1, 3),
R(3, 1), S(2, –3), and T(–2, –1). Determine
whether the quadrilateral is a
parallelogram. Justify your answer by
using the Slope Formula.
Given
quadrilateral EFGH with
vertices E(–2, 2), F(2, 0), G(1, –5), and
H(–3, –2). Determine whether the
quadrilateral is a parallelogram.
(The graph does not determine for you)
6.4-6.6 Foldable
Fold the construction paper in half both
length and width wise
 Unfold the paper and hold width wise
 Fold the edges in to meet at the center
crease
 Cut the creases on the tabs to make 4
flaps

Lesson 6.4 : Rectangles

Characteristics of a rectangle:






Both sets of opp. Sides are
congruent and parallel
Both sets opp. angles are
congruent
Diagonals bisect each other
Diagonals split it into 2
congruent triangles
Consecutive angles are
supplementary
If one angle is a right angle
then all 4 are right angles

In a rectangle the diagonals
are congruent.

If diagonals of a parallelogram
are congruent, then it is a
rectangle.
Quadrilateral EFGH is a rectangle. If GH = 6 feet
and FH = 15 feet, find GJ.
Quadrilateral RSTU is a rectangle. If mRTU =
8x + 4 and mSUR = 3x – 2, find x.
Quadrilateral EFGH is a rectangle. If mFGE =
6x – 5 and mHFE = 4x – 5, find x.
Quadrilateral JKLM has vertices J(–2, 3),
K(1, 4), L(3, –2), and M(0, –3). Determine
whether JKLM is a rectangle using the
Distance Formula.
6.5: Squares (special type of parallelogram)

A quadrilateral with 4 congruent sides

Characteristics of a square:
 Both sets of opp. sides are
congruent and parallel
 Both sets of opp. angles are
congruent
 Diagonals bisect each other
 Diagonals split it into 2 congruent
triangles
 Consecutive angles are
supplementary
 If an angle is a right angle then all
4 angles are right angles
 Diagonals bisect the pairs of
opposite angles
 Diagonals are perpendicular

A square is a rhombus and a
rectangle.
Lesson 6.5 : Rhombi


A quadrilateral with 4 congruent
sides
Characteristics of a rhombus:






Both sets of opp. sides are
congruent and parallel
Both sets of opp. angles are
congruent
Diagonals bisect each other
Diagonals split it into 2 congruent
triangles
Consecutive angles are
supplementary
If an angle is a right angle then all
4 angles are right angles

(special type of parallelogram)
In a rhombus:


Diagonals are perpendicular
Diagonals bisect the pairs of
opposite angles
A. The diagonals of rhombus WXYZ intersect at V.
If mWZX = 39.5, find mZYX.
B. The diagonals of
rhombus WXYZ intersect at
V. If WX = 8x – 5 and WZ =
6x + 3, find x.
A. ABCD is a rhombus. Find
mCDB if mABC = 126.
B. ABCD is a rhombus. If
BC = 4x – 5 and CD = 2x + 7,
find x.
QRST is a square. Find n if mTQR = 8n + 8.
QRST is a square. Find QU if QS = 16t – 14 and
QU = 6t + 11.
Determine whether parallelogram ABCD
is a rhombus, a rectangle, or a square for
A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2).
List all that apply. Explain.
Kite
Two sets of
consecutive sides
are congruent
 Diagonals are
perpendicular

6.6: Trapezoids

A quadrilateral with exactly 1 pair of
opposite parallel sides (bases), 2 pairs
of base angles, and 1 pair of nonparallel sides (legs)

Diagonals of an isosceles
trapezoid are congruent
A
B
AC = BD
base
D
leg
leg
Base angle
base


Base angle
C
Median (of a trapezoid):

The segment that connects
the midpoints of the legs
Isosceles Trapezoid:

A trapezoid with congruent legs
and congruent base angles

The median is parallel to the
bases
Median = ½ (base + base)
A. Each side of the basket
shown is an isosceles
trapezoid. If mJML = 130,
KN = 6.7 feet, and LN = 3.6
feet, find mMJK.
B. Each side of the basket shown
is an isosceles trapezoid. If
mJML = 130, KN = 6.7 feet, and
JL is 10.3 feet, find MN.
In the figure, MN is the midsegment of trapezoid
FGJK. What is the value of x.
WXYZ is an isosceles trapezoid with median
Find XY if JK = 18 and WZ = 25.
A. If WXYZ is a kite, find mXYZ.
A. If WXYZ is a kite, find mXYZ.