CIRCLES

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CIRCLES

• BASIC TERMS AND FORMULAS
• Natalee Lloyd

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Basic Terms and Formulas

• Terms
• Center
• Radius
• Chord
• Diameter
• Circumference

• Formulas
• Circumference formula
• Area formula

3
Center The point which all points of the circle
are equidistant to.
4
Radius The distance from the center to a point
on the circle
5
Chord A segment connecting two points on the
circle.
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Diameter A chord that passes through the center
of the circle.
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Circumference The distance around a circle.
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Circumference Formula C 2?r or C ?dArea
Formula A ?r2
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Circumference Example

• C 2?r
• C 2?(5cm)
• C 10? cm

5 cm
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Area Example

• A ?r2Since d 14 cm then r 7cm
• A ?(7)2
• A 49? cm

14 cm
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Angles in Geometry
Fernando Gonzalez - North Shore High School
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Intersecting Lines

• Two lines that share
• one common point.
• Intersecting lines can
• form different types of
• angles.

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Complementary Angles

• Two angles that
• equal 90º

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Supplementary Angles

• Two angles that equal 180º

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Corresponding Angles

• Angles that are
• vertically identical
• they share a common vertex and have a line
  running through them

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Geometry

• Basic Shapes
• and examples in everyday life

Richard Briggs NSHS
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GEOMETRY

• Exterior Angle Sum Theorem

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What is the Exterior Angle Sum Theorem?

• The exterior angle is equal to the sum of the
  interior angles on the opposite of the triangle.

40
70
70
110
110 70 40
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Exterior Angle Sum Theorem

• There are 3 exterior angles in a triangle. The
  exterior angle sum theorem applies to all
  exterior angles.

128
52
64
64
116
116
128 64 64 and 116 52 64
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Linking to other angle concepts

• As you can see in the diagram, the sum of the
  angles in a triangle is still 180 and the sum of
  the exterior angles is 360.

160
20
80
80
100
100
80 80 20 180 and 100 100 160 360
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Geometry

• Basic Shapes
• and examples in everyday life

Barbara Stephens NSHS
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GEOMETRY

• Interior Angle Sum Theorem

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What is the Interior Angle Sum Theorem?

• The interior angle is equal to the sum of the
  interior angles of the triangle.

40
70
70
110
110 70 40
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Interior Angle Sum Theorem

• There are 3 interior angles in a triangle. The
  interior angle sum theorem applies to all
  interior angles.

128
52
64
64
116
116
128 64 64 and 116 52 64
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Linking to other angle concepts

• As you can see in the diagram, the sum of the
  angles in a triangle is still 180.

160
20
80
80
100
100
80 80 20 180
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Geometry

• Parallel Lines with a Transversal
• Interior and exterior Angles
• Vertical Angles
• By
• Sonya Ortiz
• NSHS

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Transversal

• Definition
• A transversal is a line that intersects a set of
  parallel lines.
• Line A is the transversal

A
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Interior and Exterior Angles

• Interior angels are angles 3,4,56.
• Interior angles are in the inside of the parallel
  lines
• Exterior angles are angles 1,2,78
• Exterior angles are on the outside of the
  parallel lines

1
2
3
4
5
6
7
8
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Vertical Angles

• Vertical angles are angles that are opposite of
  each other along the transversal line.
• Angles 14
• Angles 23
• Angles 58
• Angles 67
• These are vertical angles

1
2
3
4
5
6
7
8
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Summary

• Transversal line intersect parallel lines.
• Different types of angles are formed from the
  transversal line such as interior and exterior
  angles and vertical angles.

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Geometry

• Parallelograms

M. Bunquin NSHS
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Parallelograms

• A parallelogram is a a special quadrilateral
  whose opposite sides are congruent and parallel.

A
B
D
C

• Quadrilateral ABCD is a parallelogram if and only
  if
• AB and DC are both congruent and parallel
• AD and BC are both congruent and parallel

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Kinds of Parallelograms

• Rectangle
• Square
• Rhombus

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Rectangles

• Properties of Rectangles
• 1. All angles measure 90 degrees.
• 2. Opposite sides are parallel and congruent.
• 3. Diagonals are congruent and they bisect each
  other.
• 4. A pair of consecutive angles are
  supplementary.
• 5. Opposite angles are congruent.

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Squares

• Properties of Square
• 1. All sides are congruent.
• 2. All angles are right angles.
• 3. Opposite sides are parallel.
• 4. Diagonals bisect each other and they are
  congruent.
• 5. The intersection of the diagonals form 4 right
  angles.
• 6. Diagonals form similar right triangles.

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Rhombus

• Properties of Rhombus
• 1. All sides are congruent.
• 2. Opposite sides parallel and opposite angles
  are congruent.
• 3. Diagonals bisect each other.
• 4. The intersection of the diagonals form 4 right
  angles.
• 5. A pair of consecutive angles are supplementary.

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Geometry

• Pythagorean Theorem

Cleveland Broome NSHS
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Pythagorean Theorem

• The Pythagorean theorem
• This theorem reflects the sum of the
• squares of the sides of a right triangle
• that will equal the square of the hypotenuse.
• C2 A2 B2

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A right triangle has sides a, b and c.
c
b
a
If a 4 and b5 then what is c?
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Calculations
A2 B2 C2
16 25 41
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To further solve for the length of C
Take the square root of C
?41 6.4
This finds the length of the Hypotenuse
of the right triangle.
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The theorem will help calculate distance when
traveling
between two destinations.
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GEOMETRY

• Angle Sum Theorem
• By Marlon Trent
• NSHS

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Triangles

• Find the sum of the angles of a three sided
  figure.

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Quadrilaterals

• Find the sum of the angles of a four sided figure.

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Pentagons

• Find the sum of the angles of a five sided figure.

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Hexagon

• Find the sum of the angles of a six sided figure.

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Heptagon

• Find the sum of the angles of a seven sided
  figure.

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Octagon

• Find the sum of the angles of an eight sided
  figure.

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Complete The Chart
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What is the angle sum formula?

• Angle Sum(n-2)180
• Or
• Angle Sum180n-360

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• A presentation by

Mary McHaney
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A SQUARE IS RECTANGLE
QUADRILATERAL DILEMMA

• THE SQUARE IS A RECTANGLE
• OR
• THE RECTANGLE IS A SQUARE

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SQUARE Characteristics

• Four equal sides
• Four Right Angles

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RECTANGLE Characteristics

• Opposite sides are equal
• Four Right Angles

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Square and Rectangle share

• Four right angles
• Opposite sides are equal

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SQUARE AND RECTANGLE DO NOT SHARE

• All sides are equal

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SO

• A SQUARE IS RECTANGLE
• A RECTANGLE IS NOT A SQUARE

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Charles Upchurch
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Types of Triangles

• Triangles Are Classified Into 2 Main Categories.

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Triangles Classified by Sides
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Triangles Classified by Their SidesScalene
Triangles

• These triangles have all 3 sides of different
  lengths.

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Isosceles Triangles

• These triangles have at least 2 sides of the same
  length. The third side is not necessarily the
  same length as the other 2 sides.

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Equilateral Triangles

• These triangles have all 3 sides of the same
  length.

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Triangles Classified by their Angles

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Acute Triangles

• These Triangles Have All Three Angles That Each
  Measure Less Than 90 Degrees.

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Right Triangles

• These triangles have exactly one angle that
  measures 90 degrees. The other 2 angles will
  each be acute.

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ObtuseTriangles

• These triangles have exactly one obtuse angle,
  meaning an angle greater than 90 degrees, but
  less than 180 degrees. The other 2 angles will
  each be acute.

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Quadrilaterals

• A polygon that has four sides

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Quadrilateral Objectives

• Upon completion of this lesson, students will
• have been introduced to quadrilaterals and their
  properties.
• have learned the terminology used with
  quadrilaterals.
• have practiced creating particular quadrilaterals
  based on specific characteristics of the
  quadrilaterals.

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Parallelogram

• A quadrilateral that contains two pairs of
  parallel sides

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Rectangle

• A parallelogram with four right angles

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Square

• A parallelogram with four congruent sides and
  four right angles

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Group Activity

• Each group design a different quadrilateral
  and prove that its creation fits the desired
  characteristics of the specified quadrilateral.
  The groups could then show the class what they
  created and how they showed that the desired
  characteristics were present.

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Geometry

• Classifying Angles
• Dorothy J. Buchanan--NSHS

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Right angle 90
Straight Angle 180
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• Examples

Acute angle 35
Obtuse angle 135
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• If you look around you, youll see angles are
  everywhere. Angles are measured in degrees. A
  degree is a fraction of a circlethere are 360
  degrees in a circle, represented like this 360.
• You can think of a right angle as one-fourth of a
  circle, which is 360 divided by 4, or 90.
• An obtuse angle measures greater than 90 but
  less than 180.

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Complementary Supplementary Angles

• Olga Cazares
• North Shore High School

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Complementary Angles

• Complementary angles are two adjacent angles
  whose sum is 90

60
30
60 30 90
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Supplementary Angles

• Supplementary angles are two adjacent angles
  whose sum is 180

120
60
120 60 180
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Application

• First look at the picture. The angles are
  complementary angles.
• Set up the equation
• 12 x 180
• Solve for x
• x 168

12
x
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Right AnglesbySilvester Morris
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RIGHT ANGLES

• RIGHT ANGLES ARE 90 DEGREE
• ANGLES.
  

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STREET CORNERS HAVE RIGHT ANGLES
SILVESTER MORRIS NSHS
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Parallel and Perpendicular LinesbyMelissa
Arneaud
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Recall

• Equation of a straight line YmXC
• Slope of Line m
• Y-Intercept C

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Parallel Lines Symbol

• Two lines are parallel if they never meet or
  touch.
• Look at the lines below, do they meet?

Line AB is parallel to Line PQ or AB PQ
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Slopes of Parallel Lines

• If two lines are parallel then they have the same
  slope.
• Example
• Line 1 y 2x 1
• Line 2 y 2x 6
• THINK What is the slope of line 1?
• What is the slope of line 2?
• Are these two lines parallel?

90
Perpendicular Lines

• Two lines are perpendicular if they intersect
  each other at 90.
• Look at the two lines below

A
D
C
B
Is AB perpendicular to CD? If the answer is yes,
why?
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Slopes of Perpendicular Lines

• The slopes of perpendicular lines are negative
  reciprocals of each other.
• Example
• Line 3 y 2x 5
• Line 4 y -1/2 x 8
• THINK What is the slope of line 3?
• What is the slope of line 4?
• Are these two lines perpendicular. If so, why?
• Show your working.

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What do you need to know

• Parallel Lines
• Do not intersect.
• If two lines are parallel then their slopes are
  the same.

• Perpendicular Lines
• Intersect at 90(right angles).
• If two lines are perpendicular then their slopes
  are negative reciprocals of each other.

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Questions

• Write an equation of a straight line that is
  parallel to the line y -1/3 x 7
• State the reason why your line is parallel to
  that of the line given above.
• Write an equation of a straight line that is
  perpendicular to the line y 4/5 x 3.
• State the reason why the line you chose is
  perpendicular to the line given above.

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Basic ShapesbyWanda Lusk
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Basic Shapes

• Two Dimensional
• Length
• Width

• Three Dimensional
• Length
• Width
• Depth (height)

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Basic ShapesTwo Dimensions

• Circle
• Triangle
• Parallelogram
• Square
• Rectangle

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Basic ShapesTwo Dimensions

• Circle

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Basic ShapesTwo Dimensions

• Triangle

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Basic ShapesTwo Dimensions

• Square

100
Basic ShapesTwo Dimensions

• Square
• Rectangle

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Basic ShapesThree Dimensions

• Sphere
• Cone
• Cube
• Pyramid
• Rectangular Prism

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Basic ShapesThree Dimensions

• Sphere
• Cone
• Cube
• Pyramid
• Rectangular Prism