Properties of Parallelograms

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6-2
Properties of Parallelograms
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Are you ready? Find the value of each
variable. 1. x 2. y 3. z
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Objectives
TSW prove and apply properties of
parallelograms. TSW use properties of
parallelograms to solve problems.
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Vocabulary
parallelogram
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Any polygon with four sides is a quadrilateral.
However, some quadrilaterals have special
properties. These special quadrilaterals are
given their own names.
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A quadrilateral with two pairs of parallel sides
is a parallelogram. To write the name of a
parallelogram, you use the symbol .
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Example 1 Properties of Parallelograms
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Example 2 Properties of Parallelograms
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Example 3 Properties of Parallelograms
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Example 4
In KLMN, LM 28 in., LN 26 in., and m?LKN
74. Find KN.
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Example 5
In KLMN, LM 28 in., LN 26 in., and m?LKN
74. Find m?NML.
Def. of angles.
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Example 6
In KLMN, LM 28 in., LN 26 in., and m?LKN
74. Find LO.
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Example 7 Using Properties of Parallelograms to
Find Measures
WXYZ is a parallelogram. Find YZ.
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Example 8 Using Properties of Parallelograms to
Find Measures
WXYZ is a parallelogram. Find m?Z .
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Example 9
EFGH is a parallelogram. Find JG.
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Example 10
EFGH is a parallelogram. Find FH.
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Example 11 Parallelograms in the Coordinate Plane
Three vertices of JKLM are J(3, 8), K(2,
2), and L(2, 6). Find the coordinates of vertex M.
Since JKLM is a parallelogram, both pairs of
opposite sides must be parallel.
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Example 12
Three vertices of PQRS are P(3, 2), Q(1,
4), and S(5, 0). Find the coordinates of vertex R.
Since PQRS is a parallelogram, both pairs of
opposite sides must be parallel.
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6.3 Are you ready? Justify each statement. 1.
2. Evaluate each expression for x 12 and y
8.5. 3. 2x 7 4. 16x 9 5. (8y 5)
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Objective
TSW prove that a given quadrilateral is a
parallelogram.
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You have learned to identify the properties of a
parallelogram. Now you will be given the
properties of a quadrilateral and will have to
tell if the quadrilateral is a parallelogram. To
do this, you can use the definition of a
parallelogram or the conditions below.
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The two theorems below can also be used to show
that a given quadrilateral is a parallelogram.
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Example 1 Verifying Figures are Parallelograms
Show that JKLM is a parallelogram for a 3 and
b 9.
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Example 2 Verifying Figures are Parallelograms
Show that PQRS is a parallelogram for x 10 and
y 6.5.
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Example 3
Show that PQRS is a parallelogram for a 2.4 and
b 9.
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Example 4 Applying Conditions for Parallelograms
Determine if the quadrilateral must be a
parallelogram. Justify your answer.
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Example 5 Applying Conditions for Parallelograms
Determine if the quadrilateral must be a
parallelogram. Justify your answer.
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Example 6
Determine if the quadrilateral must be a
parallelogram. Justify your answer.
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Example 7
Determine if each quadrilateral must be a
parallelogram. Justify your answer.
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Example 8 Proving Parallelograms in the
Coordinate Plane
Show that quadrilateral JKLM is a parallelogram
by using the definition of parallelogram. J(1,
6), K(4, 1), L(4, 5), M(7, 0).
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Example 9 Proving Parallelograms in the
Coordinate Plane
Show that quadrilateral ABCD is a parallelogram
by using Theorem 6-3-1. A(2, 3), B(6, 2), C(5,
0), D(1, 1).
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Example 10
Use the definition of a parallelogram to show
that the quadrilateral with vertices K(3, 0),
L(5, 7), M(3, 5), and N(5, 2) is a
parallelogram.
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You have learned several ways to determine
whether a quadrilateral is a parallelogram. You
can use the given information about a figure to
decide which condition is best to apply.
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Example 11 Application
The legs of a keyboard tray are connected by a
bolt at their midpoints, which allows the tray to
be raised or lowered. Why is PQRS always a
parallelogram?
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Example 12
The frame is attached to the tripod at points A
and B such that AB RS and BR SA. So ABRS is
also a parallelogram. How does this ensure that
the angle of the binoculars stays the same?
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Lesson Quiz Part I
1. Show that JKLM is a parallelogram for a 4
and b 5. 2. Determine if QWRT must be a
parallelogram. Justify your answer.
JN LN 22 KN MN 10 so JKLM is a
parallelogram by Theorem 6-3-5.
No One pair of consecutive ?s are ?, and one
pair of opposite sides are . The conditions for
a parallelogram are not met.
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Lesson Quiz Part II
3. Show that the quadrilateral with vertices
E(1, 5), F(2, 4), G(0, 3), and H(3, 2) is a
parallelogram.
Since one pair of opposite sides are and ?,
EFGH is a parallelogram by Theorem 6-3-1.
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Check It Out! Example 3 Continued
The coordinates of vertex R are (7, 6).
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Lesson Quiz Part I
In PNWL, NW 12, PM 9, and m?WLP 144.
Find each measure. 1. PW 2. m?PNW
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144
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Lesson Quiz Part II
QRST is a parallelogram. Find each measure. 2.
TQ 3. m?T
71
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Lesson Quiz Part III
5. Three vertices of ABCD are A (2, 6), B
(1, 2), and C(5, 3). Find the coordinates of
vertex D.
(8, 5)
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Lesson Quiz Part IV
6. Write a two-column proof. Given RSTU is a
parallelogram. Prove ?RSU ? ?TUS
Statements Reasons