Chapter 8: Quadrilaterals

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Chapter 8: Quadrilaterals
Section: 8.1 – Find Angle Measures in Polygons
Aim: To find angle measures in polygons.
Date: 2/17/12
What is a polygon?

It is a closed plane figure that is formed
by three or more sides.
Convex Polygons:
If you continue a line from each side
of the polygon and it does not
contain a point inside the polygon .
Concave Polygons:
If you continue a line from each side of
the polygon and it does contain a point
inside the polygon .
Polygon Interior Angles
Theorem

The sum of the measures of the interior angles of
a convex n-gon is (n - 2) • 180°.
m1  m2  ... mn  n  2 180
1
2
6
3
5
4
Find the sum of the measures
of the interior angles of a
convex octagon.
SOLUTION
An octagon has 8 sides. Use the Polygon Interior
Angles Theorem.
Substitute 8 for n.
(n – 2) 180° = (8 – 2) 180°
Subtract.
= 6 180°
= 1080°
Multiply.
EXAMPLE 2
Example 2
The sum of the measures of the interior angles of a
convex polygon is 900°. Classify the polygon by the
number of sides.
SOLUTION
(n –2) 180° = 900°
n –2 = 5
n =7
Polygon Interior Angles Theorem
Divide each side by 180°.
Add 2 to each side.
What is a Quadrilateral?

A four sided figure

With four angles that all add up to 360°

There are six most well known types of quadrilaterals:
 Parallelogram
 Rhombus
 Rectangle
 Square
 Trapezoid
 Kites
EXAMPLE 3
Find the value of x in the diagram shown.
The polygon is a quadrilateral.
Use the Corollary to the Polygon
Interior Angles Theorem to write
an equation involving x. Then
solve.
x° + 108° + 121° + 59° = 360°
x + 288 = 360
x = 72
Corollary to Theorem 8.1
Combine like terms.
Subtract 288 from each side.
Use the diagram at the right.
Find m S and m T.
103°, 103°
The measures of three of the interior angles of a
quadrilateral are 89°, 110°, and 46°. Find the
measure of the fourth interior angle.
115°
Polygon Exterior Angles
Theorem

The sum of the measures of the exterior angles of a
convex polygon, one angle at each vertex, is 360°
m1  m2  ... _ mn  360
SOLUTION
Use the Polygon Exterior Angles Theorem to write and
solve an equation.
x° + 2x° + 89° + 67° = 360° Polygon Exterior Angles Theorem
3x + 156 = 360 Combine like terms.
x = 68
Solve for x.
Chapter 8: Quadrilaterals
Section: 8.2 – Use Properties of Parallelograms
Aim: To find angle and side measures in a
parallelogram.
Date: 2/27
What is a Parallelogram?


A quadrilateral
Opposite sides are both parallel and equal


Theorem 8.3: if a quadrilateral is a parallelogram, then
its opposite sides are congruent.
Opposite angles are equal

Theorem 8.4: if a quadrilateral is a parallelogram, then
its opposite angles are congruent.
Find the values of x and y.
y-8
72°
x°
ABCD is a parallelogram by the definition
of a parallelogram. Use Theorem 8.3 to
find the value of x.
AB = CD
y - 8 = 36
x = 44
By Theorem 8.3,
Opposite sides of a
are
.
Substitute y - 8 for AB and 36 for CD.
Add 8 from each side.
A
In
36
C, or m
A=m
C. So, x ° = 72°.
ABCD, x = 72 and y = 44.
GUIDED PRACTICE
1. Find FG and m
Solve on your own:
G.
8, 60°
2. Find the values of x and y.
25, 15
Theorem 8.5

If quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
y°
x°
x°
y°
x  y  180
GUIDED
PRACTICE
Find the
indicated measure in
3. NM
ANSWER
5. m
2
4. KM
ANSWER
JML
ANSWER
6. m
4
JKLM.
70°
KML
ANSWER
40°
Chapter 8: Quadrilaterals
Section: 8.3 – Show that a Quadrilateral is a
Parallelogram
Aim: To use properties to identify a
parallelogram.
Date: 3/8/11
Theorems


If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorems


If one pair of opposite sides of a
quadrilateral are congruent and parallel,
then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
Examples
For what value of x is quadrilateral
CDEF a parallelogram?
FN = DN
5x – 8 = 3x
2x – 8 = 0
2x = 8
Set the segment lengths equal.
Substitute 5x –8 for FN and 3x for DN.
Subtract 3x from each side.
Add 8 to each side.
Divide each side by 2.
x=4
When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12.
Quadrilateral CDEF is a parallelogram when x = 4.
GUIDED PRACTICE
For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
2; The diagonals of a parallelogram bisect
each other so solve 2x = 10 – 3x for x.
Chapter 8: Quadrilaterals
Section: 8.4 – Properties of Rhombuses,
Rectangles, and Squares.
Aim: To use properties of rhombuses, rectangles,
and squares.
Date: 3/9/11
What is a Rhombus?

Has four congruent sides

Sometimes called a diamond

Diagonals are perpendicular

Diagonals bisect a pair of
opposite angles.
What is a Rectangle?

Rectangles are special
parallelograms


Two sets of parallel sides
Opposite sides are parallel and
equal

All angles are equal = 90°.

Diagonals are congruent.
What is a Square?

A square is a special
rectangle


All angles are equal


Opposite sides are
parallel
Each angle is 90°
All sides are equal
EXAMPLE
For1any rhombus QRST, decide whether the
statement is always or sometimes true.
a.
Q
S
a. By definition, a rhombus is a
parallelogram with four
congruent sides. By Theorem
8.4, opposite angles of a
parallelogram are congruent.
Q
So,
S .The statement is
always true.
EXAMPLE 1
For any rhombus QRST, decide whether the
statement is always or sometimes true.
b.
Q
R
b. If rhombus QRST is a square, then
all four angles are congruent right
R if QRST is a
angles. So, Q
square. Because not all rhombuses
are also squares, the statement is
sometimes true.
EXAMPLE 2
Classify the special quadrilateral.
The quadrilateral has four congruent sides. One
of the angles is not a right angle, so the
rhombus is not also a square.
The quadrilateral is a rhombus.
1.
For any rectangle EFGH, is it always or
sometimes true that FG GH ? Explain your
reasoning.
Sometimes; this is only true if EFGH is a square.
2. A quadrilateral has four congruent sides and four
congruent angles.
Sketch the quadrilateral and classify it.
Square
Chapter 8: Quadrilaterals
Section: 8.5 – Properties of Trapezoids and Kites.
Aim: To use properties of trapezoids.
Date: 3/11/11
What is a Trapezoid?

One set of parallel sides (which are the bases).
Special Trapezoid: Isosceles Trapezoid
•Each pair of base angles are congruent.
•Congruent Diagonals.
•Can also tell that it’s an Iso. Trap. If there is only one
pair of congruent base angles.
Midsegments


Segment that connects the midpoints of its
legs.
Theorem: parallel to each base and its length
is one half the sum of the lengths of the
bases.
In the diagram, MN is the midsegment of trapezoid
PQRS. Find MN.
Use Theorem to find MN.
MN = 1 (PQ + SR)
2
1
= 2 (12 + 28)
= 20
Apply Theorem
Substitute 12 for PQ and
28 for XU.
Simplify.
The length MN is 20 inches.
In 1 and 2, use the diagram of trapezoid EFGH.
1.
If EG = FH, is trapezoid EFGH isosceles?
Explain.
Yes; if the diagonals are congruent
then the trapezoid is isosceles.
2. If m HEF = 70o and m FGH = 110o, is
trapezoid EFGH isosceles? Explain.
Yes;
m EFG = 70° by Consecutive Interior Angles
Theorem making EFGH an isosceles trapezoid
by having one pair of base angles congruent.
GUIDED
3. InPRACTICE
trapezoid JKLM, J and M are right angles,
and JK = 9 cm. The length of the midsegment NP
of trapezoid JKLM is 12 cm. Sketch trapezoid
JKLM and its midsegment. Find ML. Explain your
reasoning.
J
9 cm
N
12 cm
M
K
P
L
1
15 cm; Solve 2 ( 9 + x ) = 12 for x to find ML.
Chapter 8: Quadrilaterals
Section: 8.5 – Properties of Trapezoids and Kites.
Aim: To use properties of kites.
Date: 3/14/11 PIE DAY!
What is a Kite?

Has two pairs of consecutive
congruent sides, but
opposite sides are not
congruent.


Exactly one pair of opposite
angles are congruent. The
congruent opposite angles are
always between the noncongruent sides.
Diagonals are perpendicular.
EXAMPLE 4
Find m
D in the kite shown at the right.
By Theorem, DEFG has exactly
one pair of congruent opposite angles.
Because E
G,
D and F must
be congruent. So, m D = m F.Write
and solve an equation to find m D.
EXAMPLE 4
m
D+m
F +124o + 80o = 360o
Corollary to Theorem 8.1
m
D+m
D +124o + 80o = 360o
Substitute m
2(m
D) +204o = 360o
m
D = 78o
D for m
Combine like terms.
Solve for m
D.
F.
GUIDED PRACTICE
1.
In a kite, the measures of the angles are 3xo, 75o,
90o, and 120o. Find the value of x. What are the
measures of the angles that are congruent?
25; 75o

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