Download Topic 1 Activities Packet

Geometry
Topic 1
The Language of Geometry
ACTIVITIES PACKET
Name
___________________________________
1.1 Mathematical Word Roots
Name: _________________
Directions: Many words in geometry have meanings that we can figure out by breaking down the
words into their “roots.” These roots are often based in Latin or Greek. In the chart below, fill in
English words that appear to go with the given root and meaning of the root.
Word
Root/Prefix
L/G Meaning in
English
Angulus
L
Angle, corner
Bi
G
Two
Centrum
L
Center
Circa
L
Around
Cum (changed to Co
L
With
Congruere
L
Agree
Deka
G
Ten
Dia
G
Through
Ex
L
Out of
Gonia
G
Angle
Hemi
G
Half
Hexa
G
Six
Inter
L
Between
Isos
G
Equal
Latus
L
Side
Linea
L
Line
Metron
G
Measure
Medius
L
Middle
in English variations)
Example(s) of English Words
Novem
L
Nine
Okta
G
Eight
Orthos
G
Straight
Para
G
Side by side
Penta
G
Five
Planus
L
Flat, level
Poly
G
Many
Rectus
L
Right
Quadrum
L
Square
Secare
L
Cut
Semi
L
Half
Trans
L
Across
Tri
G
Three
Uni
G
One
Vertere
L
Turn, change
1.2 Points, Lines, and Planes

Geometry:

Undefined terms:
Terms
Point
Characteristics
Line
Plane

Intersection:

Collinear:

Coplanar:
Name______________________________
Dimensions
Intersection
Notation
Diagram
Term
Segment
Definition
Endpoint
Ray
Opposite
rays
~Through any two points there is
~Through any three noncollinear points
~If two points lie in a plane,
~If two lines intersect,
~If two planes intersect,
Notation
Diagram
1.3 Naming Angles
Name_____________________________________
Angle:
Sides of an angle:
Vertex:
Naming Angles
Graphic Organizer: Classifying Angles
In each oval, name, define, and draw a diagram for each type of angle.
ONE
ANGLE
1.3 Naming Angles (cont.)
In each rectangle, name, define, and draw a diagram for each pair of angles.
PAIRS OF
ANGLES
A
3 4
5
E
6
2
L
1
9 8
7
T
S
State another name for each
angle.
State whether the angle appears
to be acute, right, obtuse, or
straight.
1.  1
6.  2
2.  3
7.  LAS
3.  5
8.  ATL
4.  AST
9.  1
5.  LES
10.  EDT
1.4 Midpoints & Bisectors
Name________________________________
Example 1: Find the distance from A to B on the number line.
B
A
-3
-2
-1
0
1
2
3
4
5
Distance =
Example 2: Find the measure of MN if M is between K and N, KM = 4 cm, MN = x, and KN =
25.
Example 3: Find the measure of AB if A is between B and T, AB = 2x, AT = 3x, and BT = 35
cm.
Segment Addition Postulate: If B is between A and C, then
1.4 Midpoints & Bisectors (cont.)
Congruent:
Symbol for congruent:
Midpoint:
Segment Bisector:
Example 1. Suppose M is the midpoint of AB . If AM  6.5 , find BM .
Example 2. Suppose WR bisects CP at Z. If CP  8 cm, find CZ .
Example 3. Given: CX bisects AB at X and Y bisects both CD and XB
C
If AX = 3 cm, and CD = 12, find:
a) XB
B
b) AB
Y
c) CY
d) YB
X
A
D
1.5 Measuring Angles and Segments
I
Name_______________________________
H
L
J
K
G
Find the measure of each of the following angles on the protractor.
1. m  LFG
6. m  HFL
2. m  HFG
7. m  IFH
3. m  IFG
8. m  JFL
4. m  JFG
9. m  KFH
5. m  KFG
10. m  KFI
11. What is the point F called in all of these angles?
1.5 Measuring Angles and Segments (cont.)
Directions: Use a protractor to measure the following angles in degrees. Round your answer to
the nearest degree.
1.
2.
3.
4.
5.
6.
1.5 Measuring Angles and Segments (cont.)
When measuring segments in inches, each tick mark is worth ______________
When measuring segments in cm, each tick mark is worth _________________
A ruler in inches
A ruler in cm
Directions: Use a ruler to measure the following segments in both inches and cm. Round your
answer to the nearest 10th of a unit.
1.
3.
2.
4.
1.5 Measuring Angles and Segments (cont.)
Each problem below gives you information about a particular diagram. For each, you must draw
one diagram that meets all of the requirements.
1.
XY is horizontal
XY = 1 in.
VW is the segment bisector of XY .
Z is the midpoint of XY .
m YZW = 90
2.
AB is vertical
m AB = 2 in.
C is the midpoint of AB
m ACE = 90
E is the midpoint of DF
DF is congruent to AB
DF is parallel to AB
What is m
3.
m PQR = 40
m PQ = 2 in.
PQ is congruent to QR
ST bisects PQ and QR
QX bisects PQR
?
4.
AB and CD bisect each other.
AB = CD = 2 in.
H is the midpoint of AB and CD
m BHC  60
EF bisects AHC and BHD
What is m AHF and m BHE ?
What is m PQX and m XQR ?
1.6 Inductive Reasoning
Name__________________________________
Warm-Up: Find the next three numbers in the sequence and explain the pattern you found: 1, 2, …
Inductive Reasoning is __________________________________________________________
_____________________________________________________________________________
Directions: Use inductive reasoning to solve the following problems:
1. Draw the next shape in the pattern:
2. Draw the next shape in the pattern:
3. Find the next number in the pattern:
1 2 3 4
, , , ,...
2 3 4 5
4. Find the next number in the pattern: 1, 4, 9,16, 25,...
1.6 Inductive Reasoning (cont.)
Activity #1
1. Draw the next 3 towers in this sequence of towers:
2. Without drawing more towers, figure out how many
squares would in the 10th tower. Explain your reasoning.
3. How many squares would be in the “zero”th tower?
4. If you start with the “zero”th tower, how many MORE squares
would you need to build the third tower? Why?
5. How many squares are in the 20th tower? How did you figure this out?
6. Express the number of squares that would exist in the nth tower. Your answer will have an “n”
in it.
1.6 Inductive Reasoning (cont.)
Activity 2: The table below shows the number of squares in a sequence of towers:
Tower
1 2 3 4 5
# of Squares 7 12 17 22 27
1. How many squares are added from one to tower to the next?
2. How many squares would there be in a “zero”th tower?
3. How many squares would there be in the 10th tower? How do you know?
Activity 3: The table below is called an “In/Out” Table. The “rule” for the table can be
calculated much like the previous tower problems. The IN would
represent the tower number and the OUT would be the number of
squares in each tower.
IN
1 2 3 4 5
OUT 5 9 13 17 21
1. Explain what you do to the “IN” to get the “OUT”
2. How can you express the answer to the previous question symbolically? That is, if the IN were
“n”, what would the OUT be?
1.6 Inductive Reasoning (cont.)
The Handshake Problem
You’re at a party. As each new person arrives, they shake the hand of everyone who is already at
the party. This has been going on since the very beginning of the party.
Later on in the evening, you notice that there are 25 people at the party (including yourself) and no
one has left yet. How many handshakes have taken place?
(a) Often times in math, it’s much easier to look at a simpler problem to solve a harder one.
Lets try this strategy here: how many handshakes would have taken place if there were 3 people at
the party? How about 4? 5?
(b) Make a chart until you notice a pattern
# of People
# of Handshakes
(c) Explain the pattern you found.
(d) So…., how many handshakes for 25 people?
(e) Find a shortcut that requires less than 5 calculations to find the # of handshakes if there are “n”
people at the party.
1.6 Inductive Reasoning (cont.)
Question: How many different angles are formed when 10 rays are drawn in the interior of
an angle?
For example, when one ray is drawn, 3 angles are formed:
(a) When two rays are drawn, how many are formed? _____________
(b) Complete this table and predict the number angles formed when ten rays are drawn. (use scrap
paper for your drawings)
# of Rays
1
# of Angles Formed
3
(c) Describe the pattern you found in the above table.
(d) How many different angles are formed when 10 rays are drawn in the interior of an
angle?
1.7 Deductive Reasoning
Statements:
(1) If it is 2011, then
Obama is president.
(2) It is 2011.
Conclusion:
Obama is president
Warm-Up: Consider the two arguments
Statements:
One of these arguments is considered
(1) If it is 2011, then
valid and the other is considered
Obama is president.
invalid. Which one is valid and why?
(2) Obama is president.
Which one is invalid and why?
Conclusion: It is 2011
Deductive Reasoning is __________________________________________________________
_____________________________________________________________________________
A valid argument is _____________________________________________________________
A counterexample is _____________________________________________________________
For each of the following examples, decide if the argument is valid or invalid. If the
argument is invalid, explain why and give a counterexample.
1. If someone buys a new Lamborghini, she or he will pay over $200,000.
Marie does not buy a new Lamborghini.
Therefore Marie does not pay over $200,000 for her new car.
2. If people listen to Rush Limbaugh, they will hear a lot of gossip about Washington.
Mr. Jones hears gossip about Washington all the time.
Mr. Jones must listen to Rush Limbaugh.
3. If you do all of your Geometry homework, you will do well on the quizzes.
Sevara does all of her Geometry homework.
Sevara must do well on the quizzes.
1.7 Deductive Reasoning (cont.)
4. In China, job applicants do not ask how much they will be paid when they are hired.
When Jin Tai was hired, he asked his employer how much he would be paid.
Jin Tai must have been hired outside of China.
5. If Alec washes the school’s windows, he will be paid $5.00 an hour.
Alec washes the school’s windows for four hours and so gets paid $20.00.
6. If Maria leaves work at five o’clock, she will run into rush-hour traffic.
If Maria runs into rush-hour traffic, she will arrive home in a bad mood.
Therefore if Maria leaves work at five o’clock, she will arrive home in a bad mood.
7. Some solutions to the equations are integers.
Some integers are less than zero.
We can conclude that some solutions to the equation are less than zero.
8. If a politician decides to run for president, then he or she will make many visits to New
Hampshire.
Senator Dole has decided to run for president.
Senator Dole will make many visits to New Hampshire.
From the Mathematics Teacher, September 1995
PRACTICE:
1:
Given: If two numbers are negative, then their product is positive.
Given: -3 and -4 are negative numbers.
What can we conclude?
2:
Given: If you want to pick apples, go to Bolton.
Given: I went to Bolton.
What can we conclude?
3:
Given: If I go to the store, then I will stop at the post office.
Given: If I stop at the post office, then I will buy stamps.
What can we conclude?