Geometry

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Geometry

• deals with the measurement, properties, and
  relationships of points, lines, angles, surfaces,
  and solids

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Lines, Segments, Rays
Area Perimeter
Angles, Vertex, Midpoint
Area of a Parallelogram
Area of a Triangle
Parallel, Perpendicular
Area of a Trapezoid
Angles
Circles again
Triangles
Surface Area
More Angles
Volume
Congruent
Area Formulas
Squares roots
Squares roots again
Polygons
Coordinate graphing
Special Quads
Polyhedron

• Back to the WEB

Circles
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Lines, Segments, Rays
Lines straight paths in a plane. They have no
end and no beginning. They can be named with two
points. Segments pieces of a line with a
beginning and ending Rays pieces of a line with
a beginning but NO ending Points identify a
location on an object in space.
A B
A
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Angle ( ) two rays that share an
endpoint Vertex a point at which two line
segments, lines, or rays meet to form an
angle Midpoint The point on a line segment
that divides it into two congruent (equal)
parts Plane a flat surface that extends
infinitely in all directions
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Intersecting lines are two lines that cross

with a transversal a line intersecting two or
more lines.
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Angles

• Acute - less than 90 degrees
• Right - exactly 90 degrees
• Obtuse - more than 90 degrees but less than 180.
• Straight line - exactly 180 degrees
• acute right obtuse straight

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All triangles

• sum of the angles equals 180 degrees
• Angle 1 Angle 2 Angle 3
• equals 180 degrees
• for every triangle

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Classifying Triangles by their Sides

• three sides congruent (match) - equilateral
• at least two sides match - isosceles
• no congruent sides - scalene
• Equilateral Isosceles
  Scalene

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Classifying Triangles by their Angles

• If the largest angle is
• Acute (all three are smaller than 90 degrees)
• Right (two smaller ones total 90 degrees)
• Obtuse (two smaller ones total less than 90
  degrees)
• Acute Right
  Obtuse

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Adjacent angles

• are next door neighbors
• they share a side (ray)
• 1 2 3
• Angle 1 is adjacent to 2 2 is adjacent to 1 and
  3 3 is adjacent to only 2.

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Vertical angles

• Two angles that are straight across or up and
  down from each.
• They only share a vertex.
• The blues ones are vertical to one another as are
  the black ones vertical to one another.

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Complementary angles

• two angles that together total 90 degrees
• they do not have to be next to each other
• touching not next to each other
• (Adjacent) (Not adjacent)

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Supplementary angles

• two angles that together total 180 degrees
• They do not have to be next to each other.
• Next door Not touching

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Alternate interior/exterior angles when a line
(called a transversal intersects two other lines,
it forms eight angles. The alternate interior
and exterior pairs will flash. Corresponding
(matching) angles will occur when a transversal
intersects a set of parallel lines. Corresponding
(matching) sides occur on congruent shapes.
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Congruent

• means matching
• it can be sides, angles, entire shapes etc.
• The symbol for congruent is as below

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Check

• Classify the angles
• obtuse right
  acute
• Classify the triangles
• scalene equilateral
  isosceles
• obtuse acute acute

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Define

• Supplemental
• two angles totaling 180 degrees
• Complementary
• two angles totaling 90 degrees
• Adjacent
• next door to each other
• Vertical
• across from each other and measuring the same

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16 4 Squares Square Roots radical
sign indicates a root to be taken 16 Square
the product of a number multiplied by itself 4
Root of 16 the number multiplied by itself 1, 4,
9, 16, 25, 36, 49Perfect squares the product
of a whole number and itself.
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Polygons

• are lines closed with no overlaps to form two
  dimensional shapes
• three sided is a
• triangle
• four sided is a
• quadrilateral
• five sided is a
• pentagon

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More Polygons

• six sided is a
• hexagon
• seven sided is a...
• Heptagon
• eight sided is an
• octagon
• nine sided is a
• nonagon

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More Polygons

• ten sided is a decagon
• You can go on and on however,
• just call anything over 10 a n-agon
• where n is your numbers of sides
• Regular means equal sides and angles such as a
  square.

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Special Quads

• Remembera quadrilateral has four sides
• Special ones are as follows
• One set of parallel lines - trapezoid
• Two sets of parallel lines - parallelogram
• Special parallelograms are
• four congruent angles - rectangle
• four congruent sides - rhombus
• four congruent sides angles - square

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Polyhedron Vertex the point at which two or
more line segments meet to form an angle Edge a
line segment where two faces intersect Face is
a flat surface of a solid figure Prisms
polyhedron with at least two faces that are
congruent and parallel. They are named by their
congruent bases. Volume area of base X
height Pyramids are polyhedrons with only one
base that is a polygon and all other faces are
triangles. V1/3bh Other examples octahedrons,
dodecahedrons, etc.
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Circle is a set of points, all of which are the
same distance from a given point Center of the
circle Circumference distance around a
circle Radius from the center point to any
point on the circle Diameter is a chord (line
segment w/both end points on the circle) that
passes through the center of the circle
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Circle is a set of points, all of which are the
same distance from a given point Central angle
angle with the center of the circle as its
vertex Semi-circle an arc that is exactly half
of a circle Arc part of the curve between any
two points Chord line segment with both end
points on the circle
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A set of points, all of which are the same
distance from a given pointParts of a circle are

• a radius.
• a segment that has one endpoint at the center and
  the other on the circle
• a diameter
• a segment that passes through the center w/both
  endpoints on the circle
• a central angle
• is an angle with its vertex at the center of a
  circle

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Circle parts cont.

• a chord
• is a segment that has both endpoints on the
  circle
• an arc AB
• is part of a circle name with two points
• a semicircle ABC
• is a half a circle and is named with three
  letters with an arc symbol

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Circle parts cont.

• inscribed
• is a polygon inside a circle whose sides form
  chords into the circle
• a circle
• is a set of points on a plane that are all the
  same distance from a given point

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Perimeter is the distance around the outside of a
polygon. So the perimeter of this shape is 14 cm.
3 4 3 4 14 cm.
4
3
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Area is covering the inside. There are 12 square
centimeters inside. To find it for a rectangle
multiply base times height.
4
So 3 x 4 12 cm squared
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A parallelogram can become a rectangle if the
side is sliced at a 90 degree angle and moved to
the other side. So we can multiply a base of 5
times a height of 4. Remember Base and height
must meet at 90 degrees. Note Still use all
outsides for perimeter.
Height4 cm.
Base 5 cm.
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Any parallelogram can be divided into two
triangles with one-half the original area. So to
find the area of a triangle multiply base times
height and divide by 2.
BH 44 2
2
4 cm
8 cm2.
4 cm.
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5 cm.
2 cm. Height
4.5 cm. X 2 cm.
4 cm.
Trapezoids have one set of parallel lines which
forms a set of uneven bases.
One forms an area too large when multiplied by
the height.
The another an area too small when multiplied by
the height.
Therefore, you must average the bases before
multiplying by height.
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So if the radius is 4 cm, the diameter is 8
cm. If the diameter is 2 inches, the radius is 1
inch.
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Find the diameter and radius. Diameter 8 cm.
Radius 4
8 cm.
Circumference is going around the outside of a
circle. It is a little over 3 times the diameter
so we multiply 3.14 times diameter. For this one
it is about 3.14 times 8 or about 25.12 cm.
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Find the diameter and radius. Diameter 8 cm.
Radius 4
8 cm.
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Surface area is the area of each surface. A
right prism has 6 surfaces. 2 ((LXW)(LXH)(WXH))
2(1286) 52 cm squared
4 cm
2 cm
3 cm
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6 cm
Volume is the amount something will hold. To
find it find the area of the base times
height. For a cylinder it is about 3.14 r r
H 3.14 6 6 H For a prism it is LXWXH.
14 cm
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Area formulas
Triangle ½BH or (BH)/2
Rectangle L times W
r
Parallelogram B times H
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100 is the square of 10 as 10 x 10 100. 10
is the square root of 100. 100 is read as the
square root of 100 and equals 10 because 10 is
the square root of 100
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Ordered pairs are

• (x, y) where the first number controls the
  side-to-side movement and the second number
  controls the up-and-down movement
• Example (3, 6) three right and six up
• (-4, 4) four left and four up

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Coordinate graphing

• (x, y)
• x controls sideways
• y controls up-and-down (2,
  1)
• Remember You must roll over (sideways movement
  first) before you can get up out of bed or fall
  down out of bed.

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Transformations in Geometry

• rotational symmetry is
• turning an image to fit on itself
• translation is .
• Sliding an image up, down, sideways, or
  diagonally
• reflection is.
• Where one half is a mirror image of the other

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Transformations

• translation - slide
• dilation - make larger or smaller
• rotation - turn
• reflection - flip

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Symmetry
A line of symmetry divides a figure so that two
parts of the figure are congruent.
Yes
Yes
No
Yes