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7/28/2019 Basic Geometry Review
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6/17/2013 Ventura College Mathematics Department 1
Basic Geometry Review
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Undefined Geometric Terms
PointA
Line
PlaneABC
AB
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This is rayAB
PointAis the vertex or
endpoint ofAB; always write
the name of the endpoint first
Definition:AB is the set of all
points ConAB such thatA is
not strictly between B and C
Half-lines (Rays)
AB
AB
AB
AB
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Line Segments
The green portion
is line segmentAB
PointsA & Bare endpoints;
distance between them isAB
Definition:AB is bounded by
endpointsA and B; it contains
every point onAB that is
between endpointsA and B
AB
AB
AB
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Circles (1 of 5)
Definition: A circle is
the set of all points
lying in a plane at a
fixed distance r(theradius) from a given point (the
center of the circle)
A diameter dis any line segment
whose endpoints lie on the circle,and which passes through
(contains) the center of the circle
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Circles (2 of 5)
A secant line is anyline that touches thecircle at exactly twopoints
A tangent line is anyline that touches the circle atexactly one point
A chord is any line segmentwhose endpoints lie on the circle,but which does not pass throughthe exact center of the circle
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Circles (3 of 5)
The circumference isthe full outer edge of thecircle, or the length of it
An arc is any continuousportion of the circumference
A sector is the wedge-like shapebounded by two radii and the arc
that lies between them A segment is the shape formed
by a chord and the arc thatextends between its endpoints
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Circles (4 of 5)
Formulas:
Diameter: d= 2r
Circumference: C= 2r= d
Area: A = r2
Pi: 3.14159265358979
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Circles (5 of 5)
Equations and unit circles
The equation of a circle whose
center is located at the origin of a
Cartesian coordinate system isx2 + y2 = r2
A unit circle is a circle that has a
radius of one unit (r= 1)
So the equation of a unitcirclewhose center is located at the origin
of a Cartesian coordinate system is
x2 + y2 = 1
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Angles
An angle BAC(orCABorA, if the shorter nameis clear) is the figure formed whentwo rays (the sides or legs ofthe angle) share a single endpoint
A(the vertex of the angle); thevertex is always the middle letter
Latin or Greek lowercase letters,such as a, b, , , , or, are alsoused to name angles intrigonometry and higher math
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Angle Measure (1 of 3)
Pac-Mans jaw forms
an angle (the black
wedge in the figure);
the measure of the angle is anumber that tells us about the
size of the wedge (how far open
Pac-Mans jaw has become)
The angles measure increasesas Pac-Man opens up wider
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Angle Measure (2 of 3)
One unit often used to measure
angles is the degree (symbol: )
Visit this web page* to learn about
different kinds of angles: Acute angles (measure m < 90)
Right angles (m = 90)
Obtuse angles (90 < m < 180)
Straight angles (m = 180)
Reflex angles (180 < m < 360)____________
* http://www.mathopenref.com/angle.html
http://www.mathopenref.com/angle.htmlhttp://www.mathopenref.com/angle.html7/28/2019 Basic Geometry Review
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Angle Measure (3 of 3)
If line segments, rays, orlines cross at a right angle(perpendicular), then asmall square is oftenadded to indicate this
Two angles whosemeasures add up to 90are complementary
Two angles whose measuresadd up to 180 aresupplementary
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Polygons (1 of 5)
The intuitive polygon: Draw a random assortment
of 3 or more points in a plane
Connect them so that each
point is the endpoint ofexactly two line segments,and no point lies on a givenline segment unless it is oneof that segments twoendpoints
The result is a polygon(some examples are shownin the figure at right)
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Polygons (2 of 5)
The strictly defined polygon(you wont be tested onthis): A polygon is aclosed path composed of afinite sequence of straightline segments
Other terms (you may betested on these): The line segments are called
sides of the polygon
Each corner is called avertex of the polygon
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Polygons (3 of 5)
Polygons are what anaverage person would callshapes but there aresome restrictions: Polygons have no curvy
parts; the definition (see theprevious slide) requires eachside to be straight
So, although circles, ellipses,parabolas, and other curvythings are called shapesalso, they are notpolygons
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Polygons (4 of 5)
Mathematicians classifypolygons by the number ofsides (or vertices) theyhave; the names usedhave mostly Greek roots: 3 sides = triangle or trigon
4 sides = quadrilateral ortetragon
5 sides = pentagon
6 sides = hexagon
8 sides = octagon, etc.
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Polygons (5 of 5)
Some polygons possesssymmetry; terms used forcertain types of symmetryinclude: Equiangular: All the vertex
angles have equal measures
Cyclic: All the vertices lieon a circle
Equilateral: All the sides, oredges, have the same length
Regular: The polygon isboth cyclic and equilateral
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Triangle Properties (1 of 2)
The measures of the three vertex
angles always add up to 180
An equilateral triangle is always
equiangular (and vice-versa); ifeither of these is true, then both
are true, and the measure of each
vertex angle is exactly 60
An equilateral triangle is the only
kind of triangle that is regular
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Triangle Properties (2 of 2)
If the lengths of at least two
sides of a triangle are equal,
then it is called an isosceles
triangle
If all three sides of a triangle
have different lengths, then it
is called a scalene triangle
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Right Triangles
If one vertex angle of a triangle isa right angle (has a measure of90), then the triangle is a righttriangle, having these properties: The two remaining vertex angles are
automatically complementary
It may be either scalene orisosceles; if it is isosceles, then the
two remaining vertex angles bothhave equal measures of exactly 45
The Pythagorean theorem (AppendixA) relates the lengths of the 3 sides
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Quadrilateral Properties (1 of 2)
The measures of the four vertexangles always add up to 360
An equilateral quadrilateral is
called a rhombus; it is notnecessarily equiangular or square
An equiangular quadrilateral iscalled a rectangle; it is not
necessarily equilateral All four vertices of a rectangle areright angles, and therefore havemeasures of 90
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Quadrilateral Properties (2 of 2)
A square is a quadrilateral
that is both equilateral and
equiangular
A square is the only kind of
quadrilateral that is regular
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Appendix A:
Pythagorean Theorem Ifcis the length of the
hypotenuse (longest side),
and a and b are the lengths of the
legs (shorter sides), thena 2 + b 2 = c2
The hypotenuse is always the side
that does nottouch the right angle
The figure depicts a scalene triangle;some right triangles might also be
isosceles, but they can never be
equilateral
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Appendix B:
The Greek Alphabet alpha ()
beta ()
gamma ()
delta ()
epsilon () zeta ()
eta ()
, theta ()
iota () kappa ()
lambda ()
mu ()
nu ()
xi ()
omicron ()
pi ()
rho () , sigma ()
tau ()
upsilon ()
, phi () chi ()
psi ()
, omega ()