Download What is Geometry? - University of Arizona Math

What is Geometry?
Make 2 lists with your table:
What geometry content are you
confident about?
What geometry content are you
nervous about?
Geometry
•
•
•
•
Points, lines, planes, angles
Curves, Polygons, circles, polyhedra, solids
Congruence, similarity
Reflections, rotations, translations,
tessellations
• Distance, Perimenter, Area, Surface Area,
Volume, Temperature, Time, Mass, Liquid
vs. Solid Capacity
• Above, below, beside, left, right, upsidedown, perception, perspective
Geometry
• Notice that nowhere on the previous list is
the word “proof.”
• An example shows that something is true at
least one time.
• A counter-example shows that something is
not true at least one time.
• A proof shows that something is true (or not
true) all of the time.
• This is what all of mathematics is based
upon, not just geometry.
Geometry
• If we believe something to be true…
– Assumption/Axiom/Postulate
– Conjecture/Hypothesis
– Definition
• If we can prove something to be true…
– Theorem
– Property
These words are not interchangeable!!
Some words are hard to
define
• Describe the color red to someone.
• Can you define the color red?
• Can you define or describe the color
red to someone who is blind?
Some words are hard to
define
• Point: a dot, a location on the number line or
coordinate plane or in space or time, a pixel
• Line: straight, never ends, made up of
infinite points, has at least 2 points
• Plane: a flat surface that has no depth that is
made up of at least 3 non-collinear points.
• We say these terms are undefined.
With undefined terms, we
can describe and define…
•
•
•
•
•
•
•
•
Segment
Ray
Angle
Collinear points
Coplanar lines
Intersecting lines
Skew lines
Concurrent lines
D•
•
A
•
B
C
p
Symbols
• We use some common notation.
• Line, line segment, ray: 2 capital letters
AB
AB
AB
BA or t
• Point: 1 capital letter D
• Plane: 1 upper or lower case letter Pp
• Angle: 3 capital letters with the vertex in the
center, or the vertex letter or number  ACD
Try these
Name 3 rays.
Name 4 different
angles.
D•
F•
Name 2 supplementary
angles.
Name a pair of vertical
angles.
Name a pair of adjacent angles.
Name 3 collinear points.
B
A•
C•
E
G
•
Try these
Name 2 right angles.
•
• H F
E
Name 2 complementary
•
D
angles.
G•
Name 2 supplementary
•
angles.
C
B
Name 2 vertical angles.
A •
True or false: AD = DA.
If m  EDH = 48˚, find m  GDC.
Euclid’s Postulates
• 1. A straight line segment can be drawn
joining any two points.
• 2. Any straight line segment can be
extended indefinitely in a straight line.
• 3. Given any straight line segment, a
circle can be drawn having the segment as
radius and one endpoint as center.
• 4. All right angles are congruent.
Euclid’s Fifth Postulate
• 5. If two lines are drawn which intersect a
third in such a way that the sum of the inner
angles on one side is less than two right
angles, then the two lines inevitably must
intersect each other on that side if extended
far enough. This postulate is equivalent to
what is known as the parallel postulate.
A
C
Try these
• Assume lines l, m, n
are parallel.
• Copy this
diagram.
• Find the value of
each angle.
l
Exploration 8.1
• Exploration 8.1
• Part 4 #1a - e--copy or cut and tape the
figures so that the groups are easy to
distinguish.
How did you group the
polygons?
• Compare your answers. Were they all
the same or different? Write a few
sentences to describe your group’s
findings.
Use Geoboards
• On your geoboard, copy the given segment.
• Then, create a parallel line and a
perpendicular line if possible. Describe how
you know your answer is correct.
Exploration 8.6
• Do part 1 using the pattern blocks--make
sure your justifications make sense.
• You may not use a protractor for part 1.
• Once your group agrees on the angle
measures for each polygon, trace each onto
your paper, and measure the angles with a
protractor.
• List 5 or more reasons for your protractor
measures to be slightly “off”.
More practice problems
• Given m // n.
7 6
• T or F:  7 and  4
3
5
4
are vertical.
2
1
n
• T or F:  1   4
m
• T or F:  2   3
• T or F: m  7 + m  6 = m  1
• T or F: m  7 = m  6 + m  5
• If m  5 = 35˚, find all the angles you can.
More practice problems
• Think of an analog clock.
• A. How many times a day will the minute
hand be directly on top of the hour hand?
• B. What times could it be when the two
hands make a 90˚ angle?
• C. What angle do the hands make at 7:00?
3:30? 2:06?
More practice problems
• Sketch four lines such that three are
concurrent with each other and two are
parallel to each other.
True or False
• If 2 distinct lines do not intersect, then they
are parallel.
• If 2 lines are parallel, then a single plane
contains them.
• If 2 lines intersect, then a single plane
contains them.
• If a line is perpendicular to a plane, then it is
perpendicular to all lines in that plane.
• If 3 lines are concurrent, then they are also
coplanar.