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IMA 101 Basic Mathematics
1
LECTURE 22
IMA101 2/2010 – Lecture 22
February 3, 2011
Geometry
2
 Angles
 Lines
 Planes
 Polygons
 Triangle properties
 Circles
IMA101 2/2010 – Lecture 22
February 3, 2011
Definitions
3
SECTIONS 9.1 -9.3 T&G
IMA101 2/2010 – Lecture 22
February 3, 2011
Basic Definitions
4
A point is an exact location in space
A point has no dimension.
A line is a collection of points along a
straight path.
A line extends forever in both
directions.
A line segment is a part of a line
having two endpoints.
A line segment has only one
dimension-its length.
IMA101 2/2010 – Lecture 22
A
(read “point A”)
C
B
(read “line CB”)
A
B
(read “line segment AB”)
AB
February 3, 2011
Basic Definitions
5
A ray is a part of a line having only
one endpoint.
C
D
(read “ray CD”)
An angle consists of two rays that
have a common endpoint called the
vertex of the angle.
C
B
Vertex
A
ABC (read “angle ABC”)
A plane is a flat surface that extends
endlessly in two dimentions.
IMA101 2/2010 – Lecture 22
February 3, 2011
Angles
6
 Angles are measured in Degrees, where a full
revolution is 360°.
 We can use a protractor to measure degrees.
B
A
C
D
V
The measure of AVC is m(AVC )
IMA101 2/2010 – Lecture 22
February 3, 2011
NOTE: Angles are measured in degrees (°).
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ABC is a straight angle if it
A
measures 180°.
B
C
ABC is an obtuse angle if it
measures less than 180° and more
than 90°.
A
C
B
ABC is an right angle if it
measures 90°.
A
C
B
ABC is an acute angle if it
measures less than 90°.
IMA101 2/2010 – Lecture 22
A
B
C
February 3, 2011
Basic Definitions
8
Complementary angles are
two angles whose measures
sum to 90°.
Supplementary angles are
two angles whose measures
sum to 180°.
IMA101 2/2010 – Lecture 22
65°
25°
H
J
40°
K
140°
L
February 3, 2011
Exercises
9
 Find x. Is x acute, obtuse, right, or straight?
100°
90°
x°
x°
60°
J
35°
K
135°
20°
x°
L
IMA101 2/2010 – Lecture 22
14°
x°
45°
L
February 3, 2011
Basic Definitions
10
Vertical angles are two angles
formed by two intersecting
lines.
Vertical angles have the same
measurement
120°
60°
60°
120°
Two angles are congruent when their measurement is the
same.
A is congruent to B when
IMA101 2/2010 – Lecture 22
A  B
February 3, 2011
Exercises
11
 Find x. Is x acute, obtuse, right, or straight?
x°
100°
(2x)°
(x+30)°
(4x+15)°
(7x-60)°
IMA101 2/2010 – Lecture 22
February 3, 2011
Basic Definitions
12
Intersecting lines are two lines
that meet.
l2
l1
Parallel lines are two lines in the
same plane that do not intersect.
l2
l1 || l2  l2 || l1
Perpendicular lines are two
lines that intersect to form right
angles.
l1  l2  l2  l1
IMA101 2/2010 – Lecture 22
l1
l2
l1
February 3, 2011
Transversal
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 A line that intersects 2 or more co-planar lines is
called a transversal.
 l3 is a transversal intersecting l1 and l2
l3
l1
l2
IMA101 2/2010 – Lecture 22
February 3, 2011
Interior Angles
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 Angles 3,4, 5, and 6 are interior angles
 Specifically the pairs 3 and 6 ; as well as 4 and 5 are
both pairs of alternate interior angles
l3
8
7
5
3
1
IMA101 2/2010 – Lecture 22
4
l1
6
l2
2
February 3, 2011
Parallel lines
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 If two parallel lines are cut by a transversal then
their alternate interior angles are congruent.
 if l1
|| l2 then
4  5 and 3  6
l3
8
7
5
3
1
IMA101 2/2010 – Lecture 22
4
l1
6
l2
2
February 3, 2011
Corresponding Angles
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 The following pairs of angles are corresponding
angles
1 and 5
2 and 6
7 and 3
8 and 4
l3
5
3
1
IMA101 2/2010 – Lecture 22
8
7
4
l1
6
l2
2
February 3, 2011
Parallel lines- alternate interior angle property
17
 If two parallel lines are cut by a transversal then
their alternate interior angles are congruent.
 if l1
|| l2 then
4  5 and 3  6
l3
8
7
5
3
1
IMA101 2/2010 – Lecture 22
4
l1
6
l2
2
February 3, 2011
Parallel lines- corresponding angle property
18
 If two parallel lines are cut by a transversal then
their corresponding angles are congruent.
 if l1
|| l2 then
1  5 and 2  6
3  7 and 4  8
l3
5
3
1
IMA101 2/2010 – Lecture 22
8
7
4
l1
6
l2
2
February 3, 2011
Parallel lines- supplementary angle property
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 If two parallel lines are cut by a transversal then the
interior angles on the same side of the transversal
are supplementary.
l3
 if l1
|| l2 then
3 and 5 are supplements
4 and 6 are supplements
5
3
1
IMA101 2/2010 – Lecture 22
8
7
4
l1
6
l2
2
February 3, 2011
Parallel lines- perpendicular transversal property
20
 If a transversal is perpendicular to on of two parallel
lines, then it is also perpendicular to the other line
|| l2 and l3  l1
then l3  l2
 if l1
l3
8
7
5
3
1
IMA101 2/2010 – Lecture 22
4
l1
6
l2
2
February 3, 2011
Exercise
21
 If
l1 || l2 and m(3)  120 find the measures of
the other angles.
l3
8
7
5
3
1
IMA101 2/2010 – Lecture 22
4
l1
6
l2
2
February 3, 2011
Exercise
22
 Find x if
AB || CD
B
C
(7x-2)°
(2x+33)°
A
IMA101 2/2010 – Lecture 22
D
February 3, 2011
Exercise
23
 Find x if
AB || DE
E
(9x-38)°
B
C
D
(6x-2)°
A
IMA101 2/2010 – Lecture 22
February 3, 2011
Polygons
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IMA101 2/2010 – Lecture 22
February 3, 2011
Polygon
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 Is a closed geometric figure that has line segments
for its sides.
 Is a closed figure, which means that if you start at
any point on the figure, you can trace completely
around it and return to the starting point.
IMA101 2/2010 – Lecture 22
February 3, 2011
Triangle
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 Is a polygon with three sides.
 It also has 3 vertices, and is called by its vertices
IMA101 2/2010 – Lecture 22
February 3, 2011
Types of Triangles
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An equilateral triangle is a triangle
with three sides equal in length.
All angles are equal
An isosceles triangle is a triangle
with two sides equal in length.
2 base angles are equal
An scalene triangle is a triangle with
no sides equal in length.
IMA101 2/2010 – Lecture 22
February 3, 2011
Types of Triangles
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An acute triangle is a triangle with
three acute angle.
An right triangle is a triangle with
one right angle.
An obtuse triangle is a triangle with
one obtuse angle.
In all triangles, the sum of the measures of all three angles is 180°
The sum of any two sides is greater than the third
IMA101 2/2010 – Lecture 22
February 3, 2011
Exercises
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 If the one base angle is 70° then how large is the
vertex angle?
 What is x?
x°
40°
IMA101 2/2010 – Lecture 22
February 3, 2011
Exercise
30
 Find
ACB if AB || DE
E
(9x-36)°
B
C
D
(6x-6)°
A
IMA101 2/2010 – Lecture 22
February 3, 2011
Quadrilateral
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 Is a polygon with four sides.
IMA101 2/2010 – Lecture 22
February 3, 2011
Rectangle: properties
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 All angles are right angles
 Opposite sides are parallel
 Opposite sides are of equal length
 Diagonals are equal length
IMA101 2/2010 – Lecture 22
February 3, 2011
Exercise
33
 Find
ADB if AB || CD
B
C
(7x-2)°
(3x+22)°
A
IMA101 2/2010 – Lecture 22
D
February 3, 2011
Pentagon
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 Is a polygon with five sides.
IMA101 2/2010 – Lecture 22
February 3, 2011
Hexagon
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 Is a polygon with six sides, and so forth.
IMA101 2/2010 – Lecture 22
February 3, 2011
Sum of the angles of a polygon
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 the sum of the measures of the angles of a polygon
with n sides is given by
S = (n – 2) 180°
IMA101 2/2010 – Lecture 22
February 3, 2011
Triangle Properties
37
IMA101 2/2010 – Lecture 22
February 3, 2011
Congruent triangles
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 Triangles with the same area and the same shape are
congruent
 Same side lengths and the same angle measures
 But we don’t need to know all these pieces of
information to know whether two triangles are
congruent
IMA101 2/2010 – Lecture 22
February 3, 2011
SSS
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 Side-Side-Side Property
 If three sides of one triangle are congruent to three
sides of another triangle, the triangles are congruent
4
3
5
IMA101 2/2010 – Lecture 22
4
3
5
February 3, 2011
SAS
40
 Side-Angle-Side Property
 If two sides and the angle between them in one
triangle are congruent respectively to two sides and
the angle between them in a second triangle, the
triangles are congruent
5
80°
IMA101 2/2010 – Lecture 22
80°
3
3
5
February 3, 2011
ASA
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 Angle-Side-Angle Property
 If two angles and the side between them in one
triangle are congruent, respectively, to two angles
and the side between them in a second triangle, the
triangles are congruent
80°
5
80°
10°
IMA101 2/2010 – Lecture 22
5
10°
February 3, 2011
THERE IS NO SSA
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 Even if two sides and the angle beside them in one
triangle are congruent, respectively, to two sides and
the angle beside them in a second triangle, the
triangles may NOT be congruent
2
5
5
2
10°
IMA101 2/2010 – Lecture 22
10°
February 3, 2011
Are the congruent triangles?
43
 Our properties are :
SSS SAS ASA
5
6
6
5
IMA101 2/2010 – Lecture 22
February 3, 2011
Are the congruent triangles?
44
 Our properties are :
SSS SAS ASA
5
5
6
6
IMA101 2/2010 – Lecture 22
February 3, 2011
Are the congruent triangles?
45
 Our properties are :
SSS SAS ASA
6
5
5
IMA101 2/2010 – Lecture 22
6
February 3, 2011
Which of the following are congruent triangles?
46
 Our properties are :
SSS SAS ASA
4
50°
50°
4
4
2
2
4
IMA101 2/2010 – Lecture 22
February 3, 2011
Which of the following are congruent triangles?
47
 Our properties are :
50°
IMA101 2/2010 – Lecture 22
SSS SAS ASA
50°
February 3, 2011
Similar Triangles
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 If two angles of one triangle are congruent to two
angles of a second triangle, the triangles will have the
same shape. (similar triangles)
 If two triangles are similar then all pairs of
corresponding sides are proportional
IMA101 2/2010 – Lecture 22
February 3, 2011
Similar Triangles
49
H
1m
4m
IMA101 2/2010 – Lecture 22
0.5 m
February 3, 2011
The Pythagorean Theorem (right triangles)
50
 For right triangles:
 Hypotenuse is the side opposite the right angle
 c2 = a2 + b2
This is only true for right triangles!
a
c
b
IMA101 2/2010 – Lecture 22
February 3, 2011