Geometry

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Chapter 2

• Geometry

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Section 2.1 Lines and Angles
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The Foundation
Point Usually named using a capital letter
Line (1-dimensional) Usually named using two
points included on the line Plane
(2-dimensional) Extends indefinitely in all
directions on it. Ray (half-line) Has one
endpoint and extends infinitely in one direction
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Angles
An angle is formed by two ___________ with a
common _____________, called the ____________
of the angle. The amount of ________________
of the terminal ray is called the angle.
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Types of Angles
Measure

1. Right Angle ________________
2. Straight Angle ________________
3. Acute Angle ________________
4. Obtuse Angle _________________

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Types of Lines

• Lines in the same plane that do not intersect are
  called ____________________ lines.
• Lines that intersect at right angles are called
  _______________________ lines.

B
A
AB CD
D
C
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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.

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When two lines intersect, the angles that are
formed on opposite sides of the point of
intersection are called ___________________
angles. _______________ angles are equal in
measure (congruent).
2
3
1
4
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If a line intersects two or more lines in a
plane, it is called a __________________________
___.

• 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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• Pairs of corresponding angles ________________
  _ _________________ _________________
  _________________
• Pairs of alternate interior angles
• _________________ _________________
• Pairs of alternate exterior angles
• _________________ _________________

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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.

Ratios of corresponding segments of the
transversal are equal.
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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)

• Scalene
• Isosceles
• Equilateral

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Right Triangles
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).
Example If a 5 and c 11, find the exact
value of b.
c
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).
Area of a triangle
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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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Another Formula for the Area of a TriangleA
Super Hero!

• If we know the three lengths of the sides of a
  triangle, we can use Heros Formula (Alternate
  Herons Formula) to calculate the area.

Heros Formula
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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.

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Similar Triangles

• Similar triangles have congruent
  ________________ but do not necessarily have
  congruent _______________.

• Properties of Similar Triangles
• Corresponding angles of similar triangles are
  equal.
• Corresponding sides of similar triangles are
  proportional.

?ABC ? ?ADE
Corresponding angles Corresponding sides
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Example

• A woman is standing next to a building on level
  ground. The woman, who is 56 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
• Perimeter of a Rectangle P ________________
• Perimeter of a Square P ________________

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Area Formulas

• Area of a Parallelogram A ________________
• Area of a Rectangle A ________________
• Area of a Square A
  ________________
• 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

• ________________ The set of all points
  equidistant from a fixed point, called the
  __________.
• ________________ A line segment with endpoints
  on the circle that passes through its center.
• ________________ A line segment from the center
  to a point on the circle.
• ________________ A line segment with endpoints
  on the circle.

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More Circle Vocab

• ________________ A line that intersects
  (touches) the circle at EXACTLY ONE POINT.
• ________________ A line that passes through two
  points of the circle.

The radius is perpendicular to the tangent at the
point of tangency.
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An example
Let O be the center of the circle. MN is tangent
to the circle.

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.

Circumference of a Circle
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Area
Area of a Circle
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 25 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.

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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.)
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.
2. A ____________ is a region bounded by a chord and
   its arc.

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More Angles

• An ____________ ___________ is formed by two
  chords of the circle and has its vertex on the
  circle.

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 in cm (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
Cube
Right Circular Cylinder
Right Circular Cone
Regular Pyramid
Sphere
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