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Why is the use of inductive
reasoning important to
understanding
mathematics?
Make a conclusion about each of the following:
Scenario #1:
• You see a dark cloud in the sky…
Scenario #2:
• You hear a fire truck coming down the road…
Scenario #3:
• You hear fans cheering at a football game…
When you make a conclusion
based on a pattern of
examples or past events.
The conclusion made based on
inductive reasoning.
CONJECTURES ARE NOT
ALWAYS TRUE!!!
Find the next three terms of the sequence:
33, 39, 45, ____, ____, ____
Find the next three terms of the sequences:
1.25, 1.45, 1.65, ____, ____, ____
13, 8, 3, ____, ____, ____
1, 3, 9, ____, ____, ____
32, 16, 8, ____, ____, ____
Find the next three terms of the sequence:
1, 3, 7, 13, 21, ____, ____, ____
Find the next three terms of the sequences:
10, 12, 15, 19, ____, ____, ____
1, 2, 6, 24, ____, ____, ____
Sometimes there may be more than one pattern to
study. What is the next figure?
Find the next figure in the sequence:
If six points on a circle are joined by as many
segments as possible, how many non-overlapping
pieces are created?
2 Points
3 Points
4 Points
If six points on a circle are joined by as many
segments as possible, how many non-overlapping
pieces are created?
6 Points
Find the sum of the first 20 odd numbers:
1=
1+3=
1+3+5=
1+3+5+7=
Assignment on Moodle
Homework: Lesson #1 – Inductive
Reasoning
Compare and contrast a
line, line segment, and a
ray.
• The basic unit of Geometry
•A point has NO size
• Points are named using capital letters
• We call them “point A”
A
B
• A series of points that extends without end in two directions
•A line is made up of an infinite number of points
• The arrows show that the line extends in both directions
forever.
• A line is named with a single lower case letter or by two
points on the line.
A
AB
B
l
Name two points on line m.
Give any three names for the line.
P
Q
R
m
Hint: The order of the letters DOES NOT
matter when naming a line.
Three or more points that lie on the same line.
S
P
Q
R
T
S, Q, T are collinear points.
Points that are NOT on the same lines
are said to be NONCOLLINEAR.
S, Q, P are noncollinear points.
Name three OTHER points that are collinear.
S
P
Q
R
T
Name three OTHER points that are noncollinear.
• A flat surface that extends without end in all directions
• For any three NONCOLLINEAR points, there is only one
plane.
• Named by a single uppercase script letter or by three or
more NONCOLLINEAR points.
Points and lines that are in the same plane.
A, B, C are coplanar.
Points and lines that are
NOT on the same plane
are said to be
NONCOPLANAR.
A, B, F are noncoplanar
points.
A
B
C
D
E
F
Not all planes CREATE the figure, some planes pass
through the figure.
How many planes are in the figure?
H
E
G
F
Name the “front” plane in 3
different ways?
Name a point that is coplanar with
E, F, and G?
C
D
A
B
Name a point that is coplanar with
D, C, and F?
Name two lines that are coplanar
with AB and DC?
Assignment on Moodle
Homework: Lesson #2a – Points, Lines,
Planes, Segments, and Rays
• Has a definite starting point and extends without
end in one direction.
• The starting point is called the ENDPOINT.
• A ray is named using the endpoint first then
another point in the ray.
A
AB
B
• Part of a line that has a definite beginning and
ending
• Named using the endpoints.
A
AB
B
Name two segments and two rays.
S
Q
R
P
Hint: Be sure to use the correct symbols.
Parallel Lines
Coplanar lines that
do not intersect.
Skew Lines
Noncoplanar lines that
are neither parallel or
intersecting.
The foundations of Geometry.
Facts about Geometry that we
accept are true.
1-1: Two points determine a unique line.
1-2: If two lines intersect, they intersect at
exactly one point.
1-3: Three noncollinear points determine a
plane.
1-4: Two planes intersect at exactly one line.
Points D, E, and F are noncollinear. Name all of the
different lines that can be drawn through these
points.
D
E
F
What postulate did you use?
Points Q, R, S, and T are noncollinear. Name all of
the different lines that can be drawn through these
points.
Q
R
T
S
Name the intersection of plane CGA and plane HCA.
G
H
C
A
What postulate did you use?
Name the intersection of plane ABC and plane DFC.
A
B
C
D
E
F
Page 15 #22-26 and 34-39
Page 20 #1-8, 13-30
Make a distinction between
lines or angles being congruent
and lines or angles being equal.
You and your family are driving on I-80 through Nebraska.
You entered the interstate at mile marker 126. You decide
to drive as far as you can before stopping for breakfast
within 1.5 hours. Assume that on the highway you drive an
average speed of 60 mph.
How far will you travel in 1.5 hours?
At what mileage marker will you exit to get to breakfast?
Does the direction you travel affect the distance you travel?
A
C
B
D
Two segments with the same length are
CONGRUENT (≌).
If AB ≌ CD, then AB = CD
The segment AB
The length of the segment AB
2 in
A
3 in
B
B
C
What is AC?
If three points A, B, and C are collinear and B is
between A and C, then AB + BC = AC
If DT = 60 find the value of x, then find DS
and ST.
2x - 8
D
3x - 12
S
T
If F is between E and G, and EG = 94,
EF = 4x – 20, and FG = 2x + 30 find the
value of x, EF, and FG.
3 in
A
3 in
B
B
C
What can you say about point B?
A point that divides a segment in half is called
a MIDPOINT.
3 in
A
3 in
B
C
A segment bisector, bisects a segment at the
midpoint, therefore, cutting it into two equal
parts.
If TQ bisects AB at M and MB = 8x + 7 and
AB = 126, find x.
T
B
M
A
Q
If M is the midpoint of TQ and TM = 3x + 5
and MQ = x + 17, find x.
T
B
M
A
Q
Page 28 #13, 14, 27, 28
Page 36 #35-40
Defined by two noncollinear rays
Side
Vertex
Side
Angles could be named in three ways:
1.
2.
3.
By the vertex angle
By the number inside the angle
By three points on the angle (vertex point must
be in the middle)
P
C
Q
1
R
B
Refer to the figure to answer each question
What other names could
be used to identify ∠BCD?
Name the vertex of ∠ 1.
What are the sides of ∠ 2?
If m∠PQR = 30 m∠RQS = 25 what is the
m∠PQS?
S
Q
If R is in the interior of ∠PQS, then
m∠PQR + m∠RQS = m∠PQS
S
Q
If m∠AEG = 75, m∠1 = 25 – x, and m∠4 =
5x + 20, find the measure of x.
If m∠3 = 32, find the m∠CED.
If m∠2 = 6x - 20, m∠4 = 3x + 18, and
m∠CED = 151, find the value of x.
If m∠1 = 49 – 2x, m∠4 = 4x + 12, and m∠2 =
15x, find x.
Acute Angle
Angles whose
measure is < 90°
Obtuse Angle
Angles whose
measure is > 90°
Right Angle
Angles whose
measure is = 90°
Straight Angle
Angles whose
measure is = 180°
Divides an angle into two
congruent angles.
P
S
Q
R
AB bisects ∠ CAD. Solve for x and
find m∠CAD.
C
B
(7x + 4)°
A
(10x - 20)°
D
BX bisects ∠ABC. If the m∠ABC = 5x + 18,
m∠CBX = 2x + 12, find m∠ABC.
C
X
B
A
A
B
C
Two lines that intersect to form a right angle.
3 in
A
3 in
B
C
A segment, line, or ray that is perpendicular to
a line at its midpoint.
If AC = 7, then find AB.
D
E
_____ is the angle
bisector of ______.
F
A
C
G
B
m∠ACG = ______
CG = 2x + 2, DC = 5x – 1,
Find x and CG.
Page 30 #32-35
Page 37 #48-51
Explain in words what
vertical angles are and
WHY they are congruent.
What conclusion can you make about
∠1 and ∠2?
T
B
120°
1
2
120°
A
Q
Angles across from one another at an
T intersection.
B
120°
1
2
120°
A
Q
VERTICAL ANGLES ARE CONGRUENT
Find the value of the variables
T
B
(7x + 3)°
(4x + 1)° 65°
A
Q
B
Two coplanar
angles with a
common side and
vertex, but no
common interior
points.
T
Q
A
B
Two angles whose
sum is 90°
T
A
Q
Find the value of z:
(4z - 10)°
z°
Two angles whose
sum is 180°
T
A
Q
B
Find the value of y:
(6y – 10)°
(6y + 10)°
Page 50 #13-18 and 21