Geometry Concepts
Chapter 8 Quadrilaterals
Identify parts of quadrilaterals
Find the sum of interior angles
Identify properties of parallelograms
Use properties of parallelograms
Identify and use properties of rectangles
Identify and use properties of rhombi
Identify and use properties of squares
Identify and use properties of trapezoids
Identify and use properties of isosceles trapezoids
Section 8.1 Quadrilaterals
Questions to think about:
•
Definition
Characteristics
QUADRILATERAL
Example
Nonexample
Definition
Example
Characteristics
CONSECUTIVE/NONCONSECTIVE SIDES
Nonexample
Definition
Characteristics
DIAGONALS
Example
Nonexample
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EXAMPLES…
Use this diagram to answer the next three
questions.
1)
Name all the pairs of consecutive angles.
2)
Name all pairs of nonconsecutive vertices.
3)
Name the diagonals.
B
E
A
L
THEOREM
8.1
b°
The sum of the measure of the angles of a quadrilateral
is 360 degrees.
c°
a°
a + b + c + d = 360
d°
EXAMPLES…
4)
Find the missing measure
in quadrilateral WXYZ.
5)
Find the missing measure if three of the four angle
measures in quadrilateral ABCD are 50, 60 and 150.
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6)
Find the measure if ∠U in quadrilateral KDUC
if m∠K = 2x, m∠D = 40, m∠U = 2x and m∠C = 40.
7)
Find the measure if ∠B in quadrilateral ABCD
if m∠A = x, m∠B = 2x, m∠C = x - 10 and m∠D = 50.
Section 8.2 Parallelograms
Questions to think about:
•
Definition
Characteristics
PARALLELOGRAM
Example
Nonexample
THEOREM
8.2
Opposite angles of parallelogram are congruent.
ABCD is a parallelogram therefore
∠A ≅ ∠C
∠D ≅ ∠B
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THEOREM
Opposite sides of parallelogram are congruent.
8.3
THEOREM
The consecutive angles of a parallelogram are
supplementary.
8.4
EXAMPLES…
8)
In parallelogram PQRS, PQ = 20, QR = 15, and m∠S
= 70. Find SR.
10) In parallelogram PQRS, PQ = 20, QR = 15, and m∠S
= 70. Find m∠Q.
9)
In parallelogram PQRS, PQ = 20, QR = 15, and m∠S =
70. Find SP.
11) In parallelogram PQRS, PQ = 20, QR = 15, and m∠S =
70. Find m∠S.
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12) In parallelogram DEFG, DE = 70, EF = 45, and m∠G =
68. Find GF.
13) In parallelogram PQRS, PQ = 20, QR = 15, and m∠S =
70. Find m∠E.
14) In parallelogram ABCD, if AC = 56, find AE.
15) If DE = 11, find DB.
THEOREM
The diagonals of a parallelogram bisects each
other.
8.5
Section 8.3 Test for Parallelograms
Questions to think about:
•
THEOREM
8.7
If both pairs of opposite sides of a quadrilateral are congruent, then
the quadrilateral is a
parallelogram.
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EXAMPLES…
16) In quadrilateral ABCD, with diagonal BD, AB||CD and AB≅CD. Show that ABCD is a
parallelogram.
a.
∠ABD ≅ ∠CDB
b.
BD ≅ BD
c.
AB ≅ CD
d.
△ABD ≅ △CDB
e.
AD ≅ CB
f.
ABCD is parallelogram
17) In quadrilateral PQRS, PR and QS bisect each other at T. Show that PQRS is a
parallelogram by providing a reason for each step.
a.
PT ≅ TR and QT ≅ TS
b.
∠PTQ ≅ ∠RTS and ∠STP ≅ ∠QTR
c.
△PTQ ≅ △RST and △PTS ≅ △QTR
d.
PQ ≅ RS and PS ≅ RQ
e.
PQRS is a parallelogram
THEOREM
8.8
If one pair of opposite sides of a quadrilateral is
parallel and congruent, then the quadrilateral is a
parallelogram.
THEOREM
8.9
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
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EXAMPLES… Determine whether each quadrilateral is a parallelogram. If the figure is a parallelogram, give a
reason for your answer.
18)
19)
20)
21)
Section 8.4 Rectangles, Rhombi, and Squares
Questions to think about:
•
Definition
Characteristics
RECTANGLE
Example
Nonexample
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Definition
Characteristics
RHOMBI
Example
Nonexample
Definition
Characteristics
SQUARE
Example
Nonexample
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EXAMPLES… identify the quadrilateral
22)
23)
24)
THEOREM
8.10
The diagonals of a rectangle are congruent.
THEOREM
8.11
The diagonals of a rhombus are perpendicular.
THEOREM
8.12
Each diagonal of a rhombus bisects a pair of opposite
angles.
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EXAMPLES…
25) Find XZ in square XYZW if
YW = 14.
26) Find m∠YOX in square XYZW.
27) Name all segments that are
congruent to WO. Why?
28) Name all the angles that are
congruent to ∠XYO. Why?
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