Download Quadrilaterals - Gr 12 Essentials

7.2
Quadrilaterals
Try These
You will need
Circle the polygons.
• coloured pencils
• a millimetre ruler
• a protractor
quadrilateral
a polygon with
four straight sides
and four vertices
Brady installs stained-glass windows in Victoria. How can you
describe the quadrilaterals in this window?
e.g.,
square
rectangle
trapezoid
kite
parallelogram
triangle (not a
quadrilateral)
rhombus
convex
a polygon with
no interior angles
that are greater
than 1808
1
Label an example of each convex quadrilateral in Brady’s
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window. Draw an arrow to the quadrilateral.
Convex quadrilaterals
A rectangle has four 908 angles.
A square is a rectangle with four equal sides.
A parallelogram has opposite sides that are parallel and equal.
A rhombus is a parallelogram with four equal sides.
A trapezoid has only one pair of parallel sides.
A kite has two pairs of equal sides that are not opposite sides. If all four sides are
equal, the quadrilateral is a rhombus.
2 Some polygons have more than one name. What are three other
AW12 names for a square? rectangle, parallelogram, and rhombus
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Figure Number
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Label a polygon
in the window that is not a quadrilateral. How
MPS
do you know that it is not a quadrilateral?
Technical
3
Company
AW12SB
0176519637
168
Pass
1st pass
e.g., A quadrilateral
has four sides and four vertices.
Approved
This triangle has three sides and three vertices.
Not Approved
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Workplace 12
FN
CrowleArt Group
CO
Technical
3rd pass
Pass
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Approved
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Example 1
Elena is a pastry chef in Fort Qu’Appelle. She cut these square
pastries along a diagonal. This makes two congruent triangles.
What other quadrilaterals have this property?
Solution
A. Draw diagonals to form triangles. Use a different colour for
each diagonal in each quadrilateral.
parallelogram
rectangle
irregular
quadrilateral
kite
rhombus
square
trapezoid
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concave
quadrilateral
1
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diagonal
a line segment
joining vertices
that are not next
to each other
concave
quadrilateral 2
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FN
CrowleArt Group
CO
Technical
2nd pass
Pass
ApprovedQuadrilaterals
Not Approved
B. Name quadrilaterals with each property.
Property
Both diagonals make congruent triangles.
irregular
quadrilateral
a quadrilateral
with different
side lengths and
different angle
measures
concave
a polygon with an
interior angle that
is greater than 1808
rectangle, parallelogram,
rhombus, and square
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One diagonal makes congruent triangles. kite and
concave
Reflecting
quadrilateral 1
No diagonals make congruent triangles.
trapezoid, irregular
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quadrilateral, and concave
quadrilateral 2
Do you think all
trapezoids have
the same property
as in your answer
from Part B? Use
side lengths to
explain.
Example 2
Tessa is a carpenter in Whitehorse. She needs to check that
a window frame is a rectangle. She only has a tape measure.
How can she use the properties of a rectangle to check?
20
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30 mm
mm 36 mm
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Solution
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Figure Number
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umber
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A. Figure
Measure
the side
lengths and diagonals in each quadrilateral.
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Figure
Number
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Company
MPS
Number
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MPS
Sy
Record
them
on
the
Company
MPSdiagrams.
Technical
Company
MPS
9637
al
Technical
Pass AW12
1st
pass
Technical
AW12
12
1st pass
pass
Pass can Tessa 1st
1st the
pass properties of a rectangle?
B. Pass
How
use
Approved
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0176519637 pass
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d
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Approved
Figure
Number
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Figure Number
If Approved
the window
frame is a C07-F36-AW12.ai
rectangle, the opposite sides
ure
NumberNot Approved
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Sroved
Not
Approved
Not
Approved
Company
MPS
Company
MPS
mpany
MPS
are equal and the diagonals are equal .
Technical
Technical
hnical
pass
NEL
Pass
1st
pass
Pass
1st pass
s
1st pass
roved
Approved
Approved
Approved
Not Approved
Not Approved
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36 mm 20 mm
30 mm
30 mm
30 mm
20 mm
20 mm
41 mm
30 mm
Chapter 7 Polygons
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Practice
1. Which convex quadrilaterals have each property?
a) two pairs of equal sides: rectangle, square, parallelogram,
and rhombus
b) four right angles: rectangle and square
c) equal diagonals: rectangle and square
d) equal angles at opposite vertices: rectangle, square,
parallelogram, and rhombus
2. How can you use properties to show that a rectangle is a
parallelogram? A rectangle is a parallelogram because it has
Hint
Use diagrams
from Example 1
to help you with
Question 3.
3. The diagonals in a square are perpendicular. The diagonals
cross at their midpoints. Record the names of other types of
quadrilaterals to complete the chart.
Property
Hint
Draw some
quadrilaterals.
Measure the
exterior angles to
test your answer
for Question 4.
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.
opposite sides that are parallel and equal
A quadrilateral with this property
A quadrilateral without
this property
Diagonals are
perpendicular.
e.g., kite
e.g., parallelogram
Diagonals cross at their
midpoints.
e.g., rhombus
e.g., kite
4. a) Show that the exterior angles of all quadrilaterals
have the same sum.
• At each vertex, the measure of the
interior angle plus the measure of the
exterior angle equals 1808 .
• There are four vertices. The total sum of all the interior
and exterior angles of a quadrilateral is 4 ( 1808 ) 5 7208 .
• The sum of the interior angles is 3608 .
• The sum of the exterior angles is 7208 2 3608 5 3608 .
• The sum of the exterior angles of any quadrilateral
is 3608 .
b) Do all quadrilaterals have diagonals that are perpendicular?
Explain.
No. e.g., The parallelogram, trapezoid, and irregular quadrilateral
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in Example 1 do not have diagonals that are perpendicular.
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5. Jay makes picture frames. The interior angles of the
square picture frame have a sum of 3608. How can you
draw a diagonal to show that all quadrilaterals have
this property?
e.g., I looked at one diagonal in each quadrilateral
in Example 1. You get two triangles. The sum of the
interior angles of each triangle is 1808. The sum of the
4(90°) 360°
interior angles of both triangles is 2(1808) 5 3608.
6. An isosceles trapezoid is cut from an isosceles triangle.
It has two equal sides and two parallel sides. The parallel
sides are called bases.
a) Use the isosceles triangle at the right. Draw a line parallel
to the base to make an isosceles triangle. Measure to
check that there are two equal side lengths in the
trapezoid.
Hint
b) What are two properties of your isosceles trapezoid?
Think about
•side lengths
•angle measures
•diagonals
e.g., one pair of equal sides and two pairs of equal
angles OR equal diagonals
c) Compare your isosceles trapezoid with isosceles
trapezoids drawn by your classmates. What is one
property that is shared
by all the isosceles trapezoids?
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e.g., one pair of equal0176519637
sides OR two pairs of congruent
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FN
angles OR two different
angles that
add to 1808 OR two
CrowleArt Group
CO
isosceles triangles andTechnical
two congruent triangles that are
Pass
formed where the diagonals
meet 2nd pass
7. e.g.,
Approved
Not Approved
7. When you join the midpoints
of all four sides of a quadrilateral,
you always get the same type of polygon. Test some
quadrilaterals. What is the polygon? parallelogram
8. Darcy is cutting glass for a window that is a rhombus. She
knows that the sides are 10 in. long.
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Does she have all the information
she needs to cut the glass?
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Explain. Include a diagram.
Figure Number
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8. e.g.,
No. e.g., She also needs to
know the angleMPS
measures. There are
Company
10 in.
10 in.
many rhombuses such as Technical
a square with the same side lengths.
Pass
1st pass
Approved
NEL
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Not Approved
ChapterC07-F43
7 Polygons
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