Brady installs stained-glass windows in Victoria. How can you describe the quadrilaterals in this window? square rectangle trapezoid rhombus parallelogram kite triangle (not a quadrilateral) 1 Label an example of each convex quadrilateral in Brady’s 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 names for a square? 3 Label a polygon in the window that is not a quadrilateral. How do you know that it is not a quadrilateral? Quadrilaterals 7.2 You will need • coloured pencils • a millimetre ruler • a protractor Try These Circle the polygons. quadrilateral a polygon with four straight sides and four vertices convex a polygon with no interior angles that are greater than 1808 e.g., A quadrilateral has four sides and four vertices. This triangle has three sides and three vertices. rectangle, parallelogram, and rhombus e.g., 168 Apprenticeship and Workplace 12 NEL
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Brady installs stained-glass windows in Victoria. How can you describe the quadrilaterals in this window?
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square
rectangletrapezoid
rhombus
parallelogramkite
triangle (not aquadrilateral)
1 Label an example of each convex quadrilateral in Brady’s window. Draw an arrow to the quadrilateral.
Convexquadrilaterals
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 names for a square?
3 Label a polygon in the window that is not a quadrilateral. How do you know that it is not a quadrilateral?
Quadrilaterals7.2Youwillneed• coloured pencils• a millimetre ruler • a protractor
Try These
Circle the polygons.
quadrilateral
a polygon with four straight sides and four vertices
convex
a polygon with no interior angles that are greater than 1808
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e.g., A quadrilateral has four sides and four vertices.
This triangle has three sides and three vertices.
rectangle, parallelogram, and rhombus
e.g.,
168 Apprenticeship and Workplace 12 NEL
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Example 1Elena 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?
SolutionA. Draw diagonals to form triangles. Use a different colour for
each diagonal in each quadrilateral.
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square
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rectangle
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parallelogram
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kite
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trapezoid
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irregularquadrilateral
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rhombus
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concavequadrilateral 1
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concavequadrilateral 2
B. Name quadrilaterals with each property.
Property Quadrilaterals
Both diagonals make congruent triangles.
One diagonal makes congruent triangles.
No diagonals make congruent triangles.
Example 2Tessa 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?
SolutionA. Measure the side lengths and diagonals in each quadrilateral.
Record them on the diagrams.
B. How can Tessa use the properties of a rectangle?
If the window frame is a rectangle, the opposite sides are and the diagonals are .
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30 mm
30 mm
20 mm20 mm 36 mm36 mm
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30 mm
30 mm
20 mm
30 mm
41 mm
20 mm
ReflecTinGDo you think all trapezoids have
the same property as in your answer from Part B? Use
side lengths to explain.
diagonal
a line segment joining vertices that are not next to each other
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irregularquadrilateral
a quadrilateral with different side lengths and different angle measures
concave
a polygon with an interior angle that is greater than 1808rectangle, parallelogram,
rhombus, and square
kite and concave quadrilateral 1
trapezoid, irregular quadrilateral, and concave quadrilateral 2
equal equal
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Practice 1. Which convex quadrilaterals have each property?
a) two pairs of equal sides:
b) four right angles:
c) equal diagonals:
d) equal angles at opposite vertices:
2. How can you use properties to show that a rectangle is a parallelogram? A rectangle is a parallelogram because it has
.
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.
PropertyAquadrilateralwith
thispropertyAquadrilateralwithout
thisproperty
Diagonals are perpendicular.
Diagonals cross at their midpoints.
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 .
• There are vertices. The total sum of all the interior and exterior angles of a quadrilateral is ( ) 5 .
• The sum of the interior angles is .• The sum of the exterior angles is 2 5 .• The sum of the exterior angles of any quadrilateral
is .
b) Do all quadrilaterals have diagonals that are perpendicular? Explain.
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Draw some quadrilaterals. Measure the exterior angles to test your answer for Question 4.
Hint
Use diagrams from Example 1 to help you with Question 3.
Hint
rectangle, square, parallelogram,
and rhombus
rectangle and square
rectangle and square
rectangle, square,
parallelogram, and rhombus
e.g., kite e.g., parallelogram
e.g., rhombus e.g., kite
1808
four1808 72084
3608
7208 3608 3608
3608
opposite sides that are parallel and equal
No. e.g., The parallelogram, trapezoid, and irregular quadrilateral
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?
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.
b) What are two properties of your isosceles trapezoid?
c) Compare your isosceles trapezoid with isosceles trapezoids drawn by your classmates. What is one property that is shared by all the isosceles trapezoids?
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?
8. Darcy is cutting glass for a window that is a rhombus. She knows that the sides are 10 in. long.
Does she have all the information she needs to cut the glass? Explain. Include a diagram.
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Think about • side lengths• angle measures• diagonals
Hint
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4(90°) � 360°
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10 in.10 in.
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
interior angles of both triangles is 2(1808) 5 3608.
e.g., one pair of equal sides and two pairs of equal
angles OR equal diagonals
e.g., one pair of equal sides OR two pairs of congruent
angles OR two different angles that add to 1808 OR two
isosceles triangles and two congruent triangles that are
formed where the diagonals meet
No. e.g., She also needs to know the angle measures. There are
many rhombuses such as a square with the same side lengths.
7. e.g.,
parallelogram
8. e.g.,
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