Download unit #7 getting started

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UNIT #7 GETTING STARTED
Definitions :
1)
2)
3)
4)
5)
Polygon : A closed figure composed of straight line segments.
Regular Polygon : A polygon with equal sides and angles.
Congruent : A congruent polygon is identical in size and shape. Congruent means equal.
Transversal : A line that intersects two or more other lines.
Adjacent : Two angles or sides next to each other.
Review of Shapes :
Ex. Draw a detailed diagram for each of the following polygons.
1) Rectangle :
2) Square :
3) Parallelogram :
4) Rhombus :
5) Trapezoid :
6) Scalene Triangle :
7) Isosceles Triangle :
8) Equilateral Triangle :
Straight Angles :
* The sum of angles that form a straight angle is 180.
a
b
a + b = 180
* Two angles whose sum is 180 are called supplementary
angles.
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Interior and Exterior Angles of a Triangle :
* The sum of the interior angles in a triangle is 180.
d
f
a
c
b
e
a + b + c = 180
* Each exterior angle equals the sum of the two interior angles opposite it.
d = b + c
and
e = a + c
and
f = a + b
* Also, each exterior angle is a supplementary angle with its adjacent interior angle.
a + d = 180
and
b + e = 180
and
c + f = 180
Angle Properties of Parallel Lines :
* When a transversal crosses two parallel lines, special angles are formed.
a b
c d
g
e f
h
transversal
* Corresponding angles form an F pattern with the transversal and the parallel lines.
Corresponding angles are equal.
d = h and c = g and f = b and e = a
* Alternate angles form a Z pattern with the transversal and the parallel lines.
Alternate angles are equal.
c = f and d = e
* The interior angles on the same side of the transversal form a C pattern.
The interior angles on the same side of the transversal are supplementary angles.
d + f = 180 and c + e = 180
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Examples :
1. Use the angle properties to determine each unknown angle.
a)
b)
x 74
41 95 x
c)
d)
50
78
x
45
x
e)
x
f)
x
y
70
w
39
82
y
z
x
z
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g)
75 z
x
y
2. Solve for x to find each missing angle measure.
a)
b)
y
3x - 6
130
x + 10
2x + 1
x + 5
x + 10
x
x
c)
x + 9
2x
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7.1
EXPLORING INTERIOR ANGLES OF POLYGONS
Definitions :
1) Vertex : A corner point of a polygon, where two sides meet. The plural form is vertices.
2) Diagonal : In a polygon, a line segment joining two vertices that are not next to each other (i.e.
not joined by one side)
Recall : The interior angles of a triangle always add up to 180.
Maybe we can use this angle property to help us determine the sum of the interior angles of other
polygons. Let’s draw non-intersecting diagonals to the divide the interior of the following polygons
into non-overlapping triangles, then count the number of triangles.
a) Quadrilateral
b) Pentagon
c) Hexagon
d) Heptagon
Ex. Use your findings to fill in the chart below.
Polygon
Number of Sides
Triangle
3
Number of
Triangles
1
Sum of Interior
Angles
180
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
* The sum of interior angles of any polygon with n number of sides is :
n  2  180 
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Examples :
1. Determine the sum of the interior angles of the following polygons.
a)
b)
2. Calculate the measure of each interior angle of a regular polygon with 16 sides.
3. The sum of the interior angles of a polygon is 1620. How many sides does the polygon have?
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7.2
ANGLE PROPERTIES OF POLYGONS
Definitions :
1) Convex Polygon : A polygon with every interior angle less than 180; any straight line through
it crosses, at most, two sides.
2) Concave Polygon : A polygon with at least one interior angle greater than 180; a straight line
through it may cross more than two sides.
Examples :
1. Determine the sum of the exterior angles of the following convex polygons.
a) A regular hexagon.
b)
120
135
x
85
50
* The sum of the exterior angles of a convex polygon is
360.
Examples :
2. Determine the measure of each exterior angle in a regular 15-sided polygon.
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3. Determine the measure of each missing angle and state the angle property used for each step.
a)
55
b)
109
93
b
a
y
z
x 30
49
125
4. An interior angle of a parallelogram is the measure of the exterior angle adjacent to it multiplied
by 2. Determine the measure of each interior angle and each exterior angle. Draw the
parallelogram.
5. In ABC, the measure of B is 8 more than 3 times the measure of A. The measure of C
is 20 less than four times the measure of A. Determine the measure of each interior and each
exterior angle of ABC. Draw the triangle.
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7.3
EXPLORING QUADRILATERAL DIAGONAL PROPERTIES
Definition :
1) Kite : A quadrilateral that has two pairs of equal sides with no parallel sides.
Diaganal Properties :
Type of
Quadrilateral
Square
Diagonal
Lengths
Equal
Diagonal Bisectors
Angles formed by the
Diagonals
All 90
(Perpendicular to each
other)
Bisect each other
Rhombus
Not equal
Bisect each other
All 90
(Perpendicular to each
other)
Rectangle
Equal
Bisect each other
Not 90
Opposite angles are equal
(OAT), and adjacent
angles add to 180 (SAT)
Parallelogram
Not equal
Bisect each other
Not 90
Opposite angles are equal
(OAT), and adjacent
angles add to 180 (SAT)
Isosceles
Trapezoid
Equal
Do not bisect each
other, but they form
two pairs of equal
line segments
Not 90
Opposite angles are equal
(OAT), and adjacent
angles add to 180 (SAT)
Kite
May or
Only one is bisected
may not be by the other
equal
Diagram
All 90
(Perpendicular to each
other)
Examples :
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1. A square has three of its vertices at A(-2, 0), B(1, 4), and C(5, 1). Use diagonal properties to find
the coordinates of the fourth vertex, D. Explain your method.
2. Hannah cut a quadrilateral from a piece of cardboard. The diagonals were congruent,
perpendicular, and bisected each other. Which type of quadrilateral did Hannah cut out?
3. Describe the possible type(s) of quadrilaterals that could be made with each set of diagonals.
Justify your answers.
a)
b)
c)
d)
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7.4
REASONING ABOUT TRIANGLE AND QUADRILATERAL
PROPERTIES
Definitions :
1)
2)
3)
4)
5)
Midpoint : The point that divides a line segment into two equal parts.
Midsegment : A line segment connecting the midpoints of two adjacent sides of a polygon.
Median : The line drawn from a vertex of a triangle to the midpoint of the opposite side.
Conjecture : A guess or prediction based on limited evidence.
Counterexample : An example that proves that a hypothesis or conjecture is false.
Examples :
1. Fill in the diagram for each of the following. (You will need a ruler.)
a) Midpoint of side PQ.
p
b) Midsegment from AB and BC.
A
c) Median from vertex G.
B
G
C
R
Q
D
I
H
You can form a conjecture about triangles and quadrilaterals and then test it using examples. You
can use properties that you already know to help you.
2. Use examples to test the following conjecture. Conjecture : The shape formed by the
midsegments of a quadrilateral has the same shape as the original quadrilateral.
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7.5
REASONING ABOUT PROPERTIES OF POLYGONS
Definitions :
1) Bimedian : The line joining the midpoints of two opposite sides in a quadrilateral.
2) Centroid : The centre of an object’s mass; the point at which it balances. It is also known as
the centre of gravity.
3) The centroid of a triangle is the intersection of its medians.
4) The centroid of a quadrilateral is the intersection of its bimedians.
Examples :
1. Draw the median lines for the given triangle. Then locate its centroid. You will need a ruler.
A
B
C
2. Draw the bimedians for the given quadrilateral. Then locate its centroid. You will need a ruler.
P
Q
S
R
3. Locate the centroid for the following quadrilaterals. Then draw the diagonals of the
quadrilateral. What do you notice about the centroid and the intersection point of the diagonals?
* The centroid of a square, rectangle, rhombus, and a parallelogram is the intersection
point of the diagonals.
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4. Draw the bimedian that connects the non-parallel sides on the following trapezoids.
a) Determine the length of the two parallel sides and of the bimedian. What do you notice?
b) Determine whether or not the bimedian is parallel to the two parallel sides.
* The bimedian of the non-parallel sides of the trapezoid is parallel to its bases. Also, its
length is the mean (average) of the bases lengths.
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