Transcript Chapter 2 Geometry Section 2.1 Lines and Angles The Foundation A Point: Usually named using a capital letter Line: (1-dimensional) Usually named using two points included on the.

Chapter 2
Geometry
1
Section 2.1
Lines and Angles
2
The Foundation
A
Point: Usually named using a capital letter
Line: (1-dimensional) Usually named using two points
included on the line
B
A
Plane: (2-dimensional)
Ray: (half-line) Has one endpoint and extends infinitely in
one direction
B
A
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Angles
An angle is formed by two ___________ with a common
_____________, called the ____________ of the angle.
A
B
C
The amount of ________________ of the terminal ray is
called the angle.
4
Types of Angles
1.
Right Angle
________________
2.
Straight Angle
________________
3.
Acute Angle
________________
4.
Obtuse Angle
_________________
5
Types of Lines
1.
Lines in the same plane that do not intersect are called
____________________ lines.
B
A
AB
CD
D
C
2.
Lines that intersect at right angles are called
_______________________ lines.
1
1

2
6
2
More on Angles
1.
________________________ angles are two angles
whose sum is 90°.
2.
________________________ angles are two angles
whose sum is 180°.
3.
________________________ angles share a common
vertex and a common side.
H
E
F
G
7
When two lines intersect, the angles that are formed on
opposite sides of the point of intersection are called
___________________ angles.
2
1
3
4
_______________ angles are equal in measure (congruent).
8
If a line intersects two or more lines in a plane, it is called a
_____________________________.
M
N
MN
P
PQ
Q
When two parallel lines are cut by a transversal, the

corresponding angles are equal (congruent)

alternate interior angles are equal (congruent)

alternate exterior angles are equal (congruent)
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M
1
3
P
N
2
MN
4
5
1.
Q
6
7
PQ
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Pairs of corresponding angles:
________&_________; ________&_________; ________&_________; ________&_________
2.
Pairs of alternate interior angles:
________&_________; ________&_________
3.
Pairs of alternate exterior angles:
________&_________; ________&_________
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M
1
3
P
N
2
MN
4
5
Q
6
7
PQ
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Example
If 4 = 41°, find the measures of the remaining angles.
1 =______
2 = ______
3 = ______
5 =______
6 = ______
7 = ______
8 =______
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When more than two parallel lines are cut by two transversals,
the segments of the transversals between the same two
parallel lines are called corresponding segments.
A
B
W
X
C
AB CD
EF
D
Y
Z
E
F
Ratios of corresponding segments of the transversal are equal.
W Y

X Z
OR
W X

Y
Z
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Example p 52 # 23-28
Find the measures of the angles.
F
1) BDF =
D
E
2)  ABE =
3)  DEB =
A
44°
B
C
4)  DBE =
5) DFE =
6) ADE =
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Section 2.2
Triangles
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Polygons
A polygon is a closed figure with straight sides.
Some common polygons:
3 sides: __________________
4 sides: __________________
5 sides: __________________
6 sides: __________________
7 sides: __________________
8 sides: __________________
9 sides: __________________
10 sides: __________________
12 sides: __________________
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Polygons
Perimeter:
_____________________________________________
_____________________________________________
_____________________________________________
Units used for perimeter:
Area:
_____________________________________________
_____________________________________________
Units used for area:
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Triangles
Triangles are polygons with three sides and
three interior angles.
The sum of the measures of the three
angles of a triangle is __________.
Triangles are often classified according to the types of
sides or the types of angles they contain.
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Types of Triangles
(classified by sides)
1. Scalene:
2. Isosceles:
3. Equilateral:
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Right Triangles
hypotenuse
leg
leg
Pythagorean Theorem:
In a right triangle, the square of the length of the hypotenuse
equals the sum of the squares of the lengths of the other
two sides (legs).
c
Example:
If a = 5 and c = 11, find the exact value of b.
a
b
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Area of a Triangle
The area of a triangle is one-half the product of a
base and its height.
The height (or altitude) of a triangle is the line segment drawn
from a vertex perpendicular to the opposite side (base).
h
b
Area of a triangle:
1
A  bh
2
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Find the perimeter and area of the triangle shown below
(all numbers are approximate).
6.5 cm
1.3 cm
2.2 cm
4.8 cm
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Find the perimeter and area (to 2 s.d.) of an isosceles
triangle whose two equal sides measure 35 mm and
height from the vertex angle to the base measures 28 mm.
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A Super Hero!
If we know the three lengths of the sides of a triangle, but
not the height, we can use Hero’s Formula (Alternate:
Heron’s Formula) to calculate the area.
Hero’s Formula
For a triangle with sides of length a, b, & c :
1
A  s  s  a  s  b  s  c  , where s   a  b  c 
2
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Example
A triangular-shaped park is bounded by three streets. The
lengths of the three sides of the park are found to be 358 ft,
437 ft, and 509 ft. What is the area of the park (to 3 sig dig)?
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Congruent Triangles
Two or more triangles are congruent if the measures of
each of the ___________ and the measures of each of
the ____________ are the same.
A
5.8 cm
P
84
84
4.6 cm
C
41
55
7.0 cm
B
5.8 cm
4.6 cm
Q 55
41
7.0 cm
R
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Similar Triangles
Similar triangles have congruent ________________ but do
not necessarily have congruent _______________.
Properties of Similar Triangles
1.
Corresponding angles of similar triangles are equal.
2.
Corresponding sides of similar triangles are proportional.
ABC  ADE
B
D
Corresponding angles:
Corresponding sides:
C
E
A
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Example
A woman is standing next to a building on level ground. The woman,
who is 5’6” tall, casts a shadow that is 3.0 feet long. At the same
time, a building casts a shadow that is 92 feet long. How tall is the
building? Round ans to 2 sig digits.
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Section 2.3
Quadrilaterals
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A quadrilateral is a polygon with ______ sides and _____
interior angles.
The sum of the measures of the interior angles of a
quadrilateral is ___________.
A ________________ of a polygon is a line segment joining
any two non-adjacent vertices.
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Special Types of Quadrilaterals
 Parallelogram
 Rhombus
 Rectangle
 Square
 Trapezoid
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Determine if each statement is true sometimes, always, or never.
1.
A rectangle is a parallelogram.
_______________
2.
A rhombus is a square.
_______________
3.
A square is a rhombus.
_______________
4.
A trapezoid is a parallelogram.
_______________
5.
A square is a rectangle.
_______________
6.
A parallelogram is a rectangle.
_______________
7.
A square is a trapezoid.
_______________
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Perimeter
The perimeter of a quadrilateral is the sum of the lengths
of the four sides. (The distance around the figure.)
Some Special Formulas:
1.
Perimeter of a Rectangle:
P = ________________
2.
Perimeter of a Square:
P = ____________
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Area Formulas
1.
Area of a Parallelogram:
A = ________________
2.
Area of a Rectangle:
A = ________________
3.
Area of a Square:
A = ________________
4.
Area of a Trapezoid:
A = ________________
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Examples
1) Text p. 62 # 30
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Examples
2) Text p. 62 # 32
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Examples
3) Text p. 63 # 38
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Section 2.4
Circles
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Some Circle Vocab
1.
________________: The set of all points equidistant
from a fixed point, called the __________.
2.
________________: A line segment with endpoints on
the circle that passes through its center.
3.
________________: A line segment from the center to
a point on the circle.
4.
________________: A line segment with endpoints on
the circle.
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More Circle Vocab
5.
________________: A line that intersects (touches)
the circle at EXACTLY ONE POINT.
o
6.
The radius is perpendicular to the tangent at the
point of tangency.
________________: A line that passes through two
points of the circle.
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An example
Let O be the center of the circle.
N
MN is tangent to the circle.
o
P
M
If MNO = 14°, find MOP.
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Circumference
 (pi) is an irrational number that is approximately equal to ______
 is the ratio of the circumference (perimeter) of a circle to its diameter.
C

d
Circumference of a Circle:
C d
or
C  2 r
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Area
Area of a Circle:
A r
2
Example
Find the area of a circle with diameter 18.5 cm. Give answer
to three significant digits.
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An example
A circular walkway, with a uniform width of 3.0 ft, is to be installed
around a traffic circle that has a diameter of 18 ft. Find the area of
the walkway to two significant digits.
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Another example
What is the area of the largest circle that can be cut from a
rectangular plate 21.2 cm by 15.8 cm? How much waste is there?
Give answers to 3 sig digits.
15.8 cm
21.2 cm
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One More Example
Neglecting waste, how much would it cost to lay down a hardwood
floor on the indoor rink if flooring costs $22.50 per square meter?
Round to nearest dollar.
Note: Ends are semicircular.
14 m
20 m
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Arcs & Angles
1.
A _________ __________ is an angle formed by two
radii (its vertex is the center of the circle.)
A
B
o
2.
An _________ is a part of the outside of the circle (a
curved segment)
The measure of an arc is the same as the measure of
the central angle that forms it. (Measured in degrees.)
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Regions of a Circle
1.
A ___________________ is a region bounded by two
radii and the arc they intercept.
A
B
o
2.
A ____________ is a region bounded by a chord and
its arc.
A
B
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More Angles
An ____________ ___________ has its vertex on the circle.
A
Intercepted arc
B
C
Inscribed angle
The measure of an inscribed angle is half of its
intercepted arc.
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An Example
If the measure of an inscribed angle is 45°, find the length
(to 2 sig dig) of its intercepted arc, given that the radius
of the circle is 8.0 cm.
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Section 2.6
Solid Geometric Figures
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Volume & Surface Area
Volume is the measure of space occupied by a solid
geometric figure.
Units used for volume:
Surface Area is the total area of all of the faces of a solid
geometric figure.
Units used for surface area:
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Formulas for Volume & Surface Area
V = Volume
A = Total Surface Area
S = Lateral Surface Area (does not include area of the bases)
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Solid Geometric Figures
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Formulas for Volume & Surface Area
Solid Geometric Figure
Volume Formula
Surface Area Formula
Rectangular Solid
V  lwh
A  2lw  2lh  2wh
Cube
V  e3
A  6e2
Right Circular Cylinder
V r h
A  2 r 2  2 rh
Right Circular Cone
1 2
V  r h
3
A   r 2   rs
1
V  Bh
3
4 3
V  r
3
1
A  ps
2
Regular Pyramid
Sphere
2
A  4 r 2
e is length of edge of cube
s is the lateral height
B is area of the base
p is the perimeter of the base
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Example 1
P 74 # 26
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Example 2
P 74 # 28
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Example 3
P 74 # 34
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