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Geometry
Topic 1
The Language of Geometry
PROBLEM PACKET
*Do Not Write In This Packet*
Packet #
Topic 1 Outline
1.1
Mathematical Word Roots
1.2
Points, Lines, and Planes
1.3
Naming Angles
1.4
Midpoints and Bisectors
1.5
Measuring Angles and Segments
1.6
Inductive Reasoning
1.7
Deductive Reasoning
1.1 Mathematical Word Roots
The definitions of many words in geometry are directly related to their word roots.
1. Pentagon – Word Root 1: PENTA meaning 5, Word Root 2: GONIA meaning angle
Definition of Pentagon: a five sided figure (has five angles too)
2. Concentric – Word Root 1: CON meaning with or same, Word Root 2: CENTRE
meaning center.
Definition of Concentric : Concentric circles have the same center
Directions: Complete the table by finding the one or two latin/greek roots, their meaning,
and the definition of each geometry term in the table. Use your notes!
Geometry Term
Root
1
Meaning
Root
2
Meaning
Definition of Geometry Term
Polygon
Trisect
Quadrilateral
Triangle
Isosceles
Circumference
Hemisphere
Intersect
Rectangle
Bisect
Colinear
Sesquicentennial*
Democracy*
Calorie*
*These words aren’t directly related to geometry. See if you can break them down into their
roots. Use the internet if you need help.
1.2 Points, Lines, and Planes
Geometry is based on the fact that points, lines, and planes have no definitions
It is upon these words that the basic structures of geometry are built. Each of these
structures has a unique way in which we write them. Consider the following example:
Some examples from the diagram
to the right:
1. 3 points: J, A, and K
2. A line containing H and I: HI
3. A plane: plane GHI
4. A segment with K and G as
endpoints: KG
5. A ray starting at A and passing
through J: AJ
Directions: For #s 1 – 6, write each of the following using the appropriate symbols:
1. The segment with endpoints T and S
2. The ray that begins at R and passes through P
3.
4.
5.
6.
7. Draw a line that passes through A, B, and C. Name it using appropriate symbols.
8. Draw a line segment and name it two different ways.
1.2 Points, Lines, and Planes (cont.)
Directions: For #s 9 – 12, sketch the following:
9. Draw ST
10. Draw AB
11. Draw J
12. Draw NB
Directions: Problems 13 – 22 refer to the picture to the right
13. Name four points.
14. Name three lines.
15. Name three rays.
16. Name two segments that contain I
17. Name two lines that contain J
18. Name a plane.
19. Name the point that is between J and I
20. Name that point that is common to three
lines.
21. Name three non-collinear points
22. Name three collinear points
23. Use the idea of a line segment to describe a triangle.
24. Use the idea of a line segment to describe a six-sided figure
25. Draw a figure with points B, C, D, E, F, and G that shows lines CD , BG , and EF , with
C on all three lines.
Directions: For #s 26 – 28, how many different lines can you create that contain at least 2
points?
26.
27.
28.
1.3 Naming Angles
Angles have different parts and we name them in a certain way:
The sides of an angle are the two
rays that intersect: DG and DM
are the sides of this angle
The vertex of an angle is the
point where the two sides meet:
D is the vertex of this angle
You name an angle using
3 points where the vertex
of the angle must go in
the middle: this angle is
called GDM or
MDG .
Sometimes angles are
numbered and it’s OK to
use that number. We can
also call this angle 1 .
Part I: Place a word from the list below in each of the blank spaces in the paragraph.
(Each word is used only once)
Vertex
Sides
Three
Rays
Vertex
Angle
Vertex
An _______________________ is formed when two rays meet at their endpoints. The two
_________________ intersect to form an angle called the _________________ of the angle.
The endpoint where the two rays intersect is called the ______________________ of the
angle. An angle is named using ____________________ points: one point on one of its
sides, then the ______________________, and a point on the other side. The
____________________ is always the second point used to name the angle.
Part 2: Draw the following angles:
1. TVE
2. TEV
3. ETV
4. For the angle in question 3, how many different ways could you correctly name it?
1.3 Naming Angles (cont.)
Part III. Name the numbered angles in each drawing.
5.
6.
7.
8.
9.
1.4 Midpoints and Bisectors
A midpoint is a point that cuts a line segment into 2 congruent parts.
A bisector is a more general term for a ray, line, or line segment that divides something
else into 2 congruent parts.
Ex: Suppose AB bisects
Ex: Suppose M is the midpoint
CAD . If mCAD  60 ,
then, mCAB  mBAD  30
of AB . If AB = 6 cm, then AM
= MB = 3cm
1. Explain what a midpoint is. Draw a diagram that contains a midpoint and mark it
appropriately to show that the point you drew is actually a midpoint.
2. Explain what a segment bisector is.
3. Explain what an angle bisector is.
Directions: For #s 4 – 7, sketch the following pictures and be sure to mark them
appropriately.
4. AB bisects CD
5. HK bisects PHR
6. M is the midpoint of TV
7. GH bisects BX at H
Directions: For #s 8 – 14, answer the following questions based on this diagram.
8. Name a ray
9. Name a line
10. Name a segment
11. Name the segment bisector
12. Name a midpoint
13. Name a point that is not a
midpoint
14. Name an angle
1.4 Midpoints and Bisectors (cont.)
Directions: For #s 15 – 18, answer the following questions based on this diagram.
15. Name the angle bisector
16. Name the angle that is
being bisected
17. What is mTPK ?
18. What is P called?
Often times, we want to describe points that aren’t necessarily midpoints.
The segment addition postulate says that if C is a point on AB , then AC  CB  AB
Ex: Since AC+ CB = AB, we know that
AB = 7 cm.
19. If C is a point on AB and AC = 3 in and CB = 1.5 cm, find AB.
20. If C is a point on AB and AB = 12 mm, and AC = 8 mm, find CB.
21. If E is a point on DF and DE = x + 2, EF = 8, and DF = 15, find the value of x.
22. If E is a point on DF and DE = 3x and, EF = 6, and DF = 24, find DE.
1.5 Measuring Angles and Segments
To use a protractor to measure an angle, place
the protractor over the angle as shown. You’ll
usually have to choose between 2 angle
measures, one that is acute and one that is
obtuse. Choose the one that makes sense with
the angle you’re measuring:
To use a ruler to measure a segment in inches,
remember that each tick mark is 1/16 of an
inch. Notice that the segment below is 2
inches long and then an extra 8 tick marks.
Therefore, the line segment is 2 and 8/16
inches long or 2.5 inches stated more simply.
This angle is roughly 75 degrees
Directions: Measure the following angles and segments (in inches). If you don’t have a
protractor and ruler, use the ones above to estimate your answers.
3.
2.
1.
5.
4.
6.
1.5 Measuring Angles and Segments (cont.)
Directions: For #s 7 – 11, use the diagram below. Note that mHAF  90
7. How many angles are
there in this diagram?
8. List each angle and name
it in 2 different ways.
9. Name all the acute
angles.
10. Name all the right
angles.
11. Name all the obtuse
angles
Directions: For #s 12 – 14, draw a diagram based on the information given. Be sure to
include any necessary markings.
12. Obtuse angle named PQR , bisected by QS
13. Segment named VW , bisected by JK
14. Acute angle named DEF , bisected by EG
Directions: For #s 15 – 19, use the diagram below along with the following information:
XV bisects TXW . XU bisects TXV . The measure of TXU  20 .
15. mUXV =
16. mVXW 
17. mTXW 
18. mTXV 
19. mUXW 
1.6 Inductive Reasoning
Inductive reasoning is the use of patterns to make conjectures about a problem.
For #s 1 - 13 , find the next three terms in the sequence and explain the pattern in a a few
words.
1. 3, 6, 9, 12, …
2. 7, 10, 13, 16, 19, …
3. 5, 10, 15, 20, 25, …
4. 2, 7, 12, 17, 22, …
5. 3/2, 2, 5/2, 3, 7/2, …
6. 17, 25, 33, 41, 49, …
7. 4, 7, 12, 19, 28, …
8. 3, 7, 13, 21, 31, …
9. 2, 7, 16, 29, 46, …
10. 2, 5, 9, 14, 20, …
11. 1, 8, 27, 64, 125, …
12. 2, 4, 8, 16, 32, …
13. O, T, T, F, F, S, S, E, N, …
14. There are ten teams in a professional soccer league. Each team plays every team once in
a season. How many games are played in total? (Hint: start with an easier example and find
a pattern)
1.6 Inductive Reasoning (cont.)
Directions: For #s 15 – 26, use the diagram to the right:
15. How many squares are added to get from one
tower to the next?
16. How many squares would be in the “zero”th
tower?
17. How many squares do you add to the “zero”th
tower to get to the first tower?
18. How many total squares in the first tower?
19. How many squares do you add to the “zero”th
tower to get to the second tower?
23. How many squares do you add to
the “zero”th tower to get to the eighth
tower?
24. How many total squares in the
eighth tower?
20. How many total squares in the second tower?
25. How many squares do you add to
the “zero”th tower to get to the nth
tower?
21. How many squares do you add to the “zero”th
tower to get to the third tower?
26. How many total squares in the nth
tower?
22. How many total squares in the third tower?
Directions: For #s 26 – 29, fill in the missing terms in the IN and OUT tables.
26)
27)
In
2
3
11
27
?
Out
4
6
22
?
18
28)
In
2
4
7
10
12
?
Out
7
13
22
31
?
76
In
Out
House
4
Cup
2
Writer
5
Elephant
7
Spin
?
Mathematics ?
?
3
?
8
?
0
29)
In
1
2
3
4
5
Out
3
7
11
?
19
1.7 Deductive Reasoning
Deductive Reasoning is the process by which you come to a conclusion using facts,
logic, and rules.
A valid argument is one when the conclusion reached follows from the givens.
An invalid argument is one when the conclusion reached does not follow from the
givens and can be disproved using a counterexample.
Example of a Valid Argument
(1) If it is July, I am not in school
(2) It is July
Example of a Invalid Argument
(1) If it is July, I am not in school
(2) I am not in school
Conclusion: I am not in school
Conclusion: It is July
(Counterexample: It is August)
Deductive Reasoning
Use the following statements to reach a valid conclusion.
1.
All elephants are mammals. All mammals have hair.
Conclusion:
2.
All jumbos are mumbos. All mumbos are gumbos.
Conclusion:
3.
If it rains, then the grass will grow. It rains.
Conclusion:
4.
If you study hard, then you will succeed. Susan studies hard.
Conclusion:
5.
All Greeks are good mathematicians.
Conclusion:
6.
All squares are rectangles. The diagonals of a rectangle bisect each other.
Conclusion:
Pythagoras was Greek.
1.7 Deductive Reasoning (cont.)
7.
Jody lives a mile and a half farther from school than Jing does.
Jing lives 5 miles from school.
Conclusion:
8.
Paul is definitely taller than Karl is. Karl is 5 ft. 9 in. tall.
Conclusion:
9.
No juniors ordered a yearbook.
Conclusion:
10.
If you are an athlete, then you probably score high on spatial visualization tests.
People who score high on spatial visualization tests do well in geometry.
Conclusion:
11.
If Professor Moriarty wrote a paper about the binomial theorem, then he is familiar
with Pascal’s triangle.
Professor Moriarty is not familiar with Pascal’s triangle.
Conclusion:
12.
Coach said “If any of you don’t come to practice on Thursday, then you won’t play in
the game on Friday.”
Dylan did not play in the game on Friday.
Conclusion:
Carrie is a junior.
1.7 Deductive Reasoning (cont.)