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Chapter 1- Study Guide
Geometry- Mr. Barnes
Vocabulary
1-1 Patterns and Inductive Reasoning
Inductive Reasoning-(p.4) is a type of reasoning in
which all possibilities are considered and then the
unwanted ones are proved false. The remaining
possibilities must be true.
Ex. Eduardo spent more than $60 on two books at the store.
Prove that at least one book costs more than $30
Proof: Suppose neither costs more than $30. Then he spent
no more than $60 at the store. Since this contradicts the
given information, at least one book costs more than $30.
In Math: Find a pattern (change) for each sequence. Use
the pattern to show the next two terms in the sequence.
4, 8, 16, 32
x2 x2 x2
Each term is twice the preceding term.
The next two terms are:
2x32=64 and 2x64=128
Hint: Find the “change” between terms. Sometimes it’s
addition, sometimes its multiplication, sometimes it is
something all together different
Coordinate (p.5) A coordinate is a point defined on a
line segment.
Conjecture (p.5) a conjecture is a conclusion reached
by using inductive reasoning.
Ex. As you walk down the street, you see many people
holding unopened umbrellas. You conjecture that the
weather forecast must call for rain.
In Math: Consider one more example. Suppose I start with the
number 60 and try to divide 60 by a few numbers smaller that 60.
ex. I try to divide 60 by 1,2,3,4,5 etc. I observe that 60 is divisible by
each of these numbers. So I might be tempted to make a conjecture
that every number less than 60 is a divisor of 60. To check the validity
of my conjecture I may try a few more numbers less than 60 and see
whether they divide 60. ex. I may try 6, 10, 12, 15, 20 etc. as divisors
and say that my conjecture seems to be true. But if you want to shoot
down my conjecture all you need to do is to point a single
exception to the rule I have guessed. For example you may point out
that 7 is less than 60 but 60 is not divisible by 7. Hence my conjecture
has been demonstrated to be wrong. The value of 7 is called
a 'counter example' in this context.
Thus we observe: Every conjecture may not be true. A conjecture
can be refuted by giving a counter example.
Almost all mathematical results, before they are proved; start their life
as conjectures. Some conjectures remain unproved for many many
years. So unless and until a conjecture is proved to be true or a counter
example is found it remains a conjecture. Only when a conjecture has
been proved to be true it becomes a mathematical result (theorem)
Counter Examples (p.17) a counter example to a
statement is a particular example or instance of the
statement that makes the statement false
Ex. 'The opposite of a number is always positive.'
Counterexample for the above statement: The opposite of 2
is - 2, a negative number.
1-2 Drawing, Nets and Other Models
Isometric Drawing (p.10) An isometric drawing of a
three dimensional object shows a corner view of a
figure. It is not drawn in perspective and distances
are not distorted.
Ex.
Make an isometric drawing of the cube structure at the left.
Orthographic Drawing (p.11) another way to show a
three dimensional figure. It shows the top view, front
view and right-side view.
Ex.
Make an orthographic drawing from the isometric drawing at the left.
Foundation Drawing (p.11) a foundation drawing shows
the base structure and the height of each part.
Ex.
Net (p.12) a net is a two dimensional pattern that you
can fold to form a three dimensional figure
Ex. Identifying a Net
The pattern is a net because you can fold it to form a cube. A
and C, B and D, and E and F are on opposite faces.
Ex. Drawing a Net
Packaging Draw a net for the graham cracker box. Label the
net with its dimensions.
1-3 Points, Lines and Planes
Point (p.17) A point is a location. A point has no size. It
is represented by a small dot and is named by a capital
letter.
Space (p.17) the set of all points
Line (p.17) A line is a series of points that extends into
two opposite directions without end. You can name a
line by any two points on the line, such as AB or line
AB. Another way to name a line is with a single
lowercase letter, such as line t.
Ex.
Collinear Points (p.17) Collinear points lie on the same
line
Ex.
Are points E, F, and C collinear?
If so, name the line on which they lie.
Points E, F, and C are collinear. They lie on line m.
Are points E, F, and D collinear?
If so, name the line on which they lie.
Points E, F, and D are not collinear.
Plane (p.17) A plane is undefined. You can think of a
plane as a flat surface that has no thickness. A plane
contains many lines and extends without end in the
direction of its lines
Ex.
Postulate (p.18) A postulate, or axiom, is an accepted
statement or fact.
Postulate 1-1
Through any two points there is exactly
one line.
Line t is the only line that passes
through points A and B.
Postulate 1-2
If two lines intersect, then
they intersect in exactly one point.
and
intersect at C.
Postulate 1-3 If two
planes intersect, then they intersect in
exactly one line.
Plane RST and plane STW intersect in
Postulate 1-4
Through any three noncollinear points
there is exactly one plane.
1-4 Segments, Rays, Parallel Lines and Planes
Segment (p.23) a segment is the part of the line
consisting of two points, called endpoints, and all
points between.
Ex.
Ray (p.23) A ray is the part of a line consisting of one
endpoint and all of the points of the line on one side of
the endpoint
Ex.
Opposite rays (p. 23) Opposite Rays are two collinear
rays with the same endpoint. Opposite rays always form
a line.
Ex.
Parallel Lines (p.24) Two lines are parallel if they lie in
the same plane and do not intersect. The symbol
means “is parallel to”
Ex.
Segments or rays are parallel if they lie in parallel lines.
They are skew if they lie in skew lines.
and
are skew
because
and
are skew.
Skew lines (p.24) Skew Lines are noncoplanar;
therefore, they are not parallel and do not intersect.
Parallel Planes (p. 24) are planes that do not intersect.
1-5 Measuring Segments and Angles
Coordinate of a point (p.25,43)The coordinate of a point
is its distance and direction from the origin of a number
line. The coordinates of a point on a coordinate plane
are in the form (x, y), where x is the x-coordinate and y
is the y-coordinate.
Congruent Segments (p. 31) Two segments with the
same length. The congruence symbol ( ) shows that two
figures are equal (=) in size and similar (~) in shape.
Ex. Two segments with the same length are congruent ( )
segments. In other words, if AB = CD, then
these statements interchangeably.
. You can use
As illustrated above, segments can be marked alike to
show they are congruent.
Midpoint (p.32) a midpoint of a segment is the point that
divides the segment into two congruent segments. A
midpoint, or any line, ray, or other segment through a
midpoint, is said to bisect the segment.
Ex.
1-6 Measuring Angles (p.36)
Angle (p.36) An angle( ) is formed by two rays with the
same endpoint. The rays are the sides of the angle and
the common endpoint is the vertex of the angle
One way to measure an angle is in degrees. To indicate
the size or degree measure of an angle, write a lowercase m in
front of the angle symbol. The degree measure of angle A is 80.
You show this by writing m A = 80.
Acute Angle (p.37) an acute angle is an angle whose
measure is between 0-90
Ex.
Congruent Angles (p.37) congruent angles are angles
that have the same measure
Obtruse Angle (p.37) an obtruse angle is an angle
whose measure is between 90 and 180
Ex.
Right Angle(p. 37) A right angle is an angle whose
measure is 90.
Ex.
Straight Angle (p. 37) A Straight Angle is an angle
whose measure is 180.
Ex.
Construction (p. 44) A construction is a geometric
figure made with only a straightedge and a compass
Perpendicular Lines (p.45) Perpendicular lines that
intersect and form right angles. The symbol means is
“perpendicular to”
Angle Bisector (p. 46) an angle bisector is a ray that
divides an angle into congruent angles
CHAPTER 1 FORMULAS
Find the Length of a Segment
Postulate 1-5- Ruler Postulate (p.31)
The points of a line can be put into one-to-one correspondence
with the real numbers so that the distance between any two
points is the absolute value of the difference of the
corresponding numbers.
Ex.
The distance between points C and D on the ruler
is 3. You can use the Ruler Postulate to find the distance
between points on a number line.
Comparing Segment Lengths (p. 32)
Find AB and BC.
AB = | −8 − (−5) | = | −3 | = 3
BC = | −5 − (−2) | = | −3 | = 3
AB = BC or
Segment Addition Postulate Postulate 1-6
If three points, A, B and C are collinear and B
is between A and C, then AB+BC+AC
Ex. Algebra If DT = 60, find the value of x.
Then find DS and ST.
Finding the Midpoint (p.33)
A midpoint of a segment is a point that divides a
segment into two congruent segments. A midpoint, or
any line, ray, or other segment through a midpoint, is
said to bisect the segment.
Ex.
Finding Lengths
Algebra C is the midpoint of
. Find AC, CB, and AB.
AC and CB are both 11, which is half of 22, the length of
.
Using the Angle Addition Postulate 1-8 (p.38)
What is m TSW if m RST = 50 and m RSW = 125?
Using the Distance Formula