Geometry

Topic 1The Language of Geometry

PROBLEM PACKET*Do Not Write In This Packet*

Packet #

Topic 1 Outline

1.1 Mathematical Word Roots1.2 Points, Lines, and Planes 1.3 Naming Angles1.4 Midpoints and Bisectors1.5 Measuring Angles and Segments1.6 Inductive Reasoning1.7 Deductive Reasoning

1.1 Mathematical Word Roots

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

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 angleDefinition 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

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

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.

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 K2. A line containing H and I: HI

s ru

3. A plane: plane GHI4. A segment with K and G as endpoints: KG5. A ray starting at A and passing through J: AJ

u ru

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 STu ru

10. Draw AB 11. Draw J 12. Draw NBs ruu

Directions: Problems 13 – 22 refer to the picture to the right

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 CDs ruu

, BGs ruu

, and EFs ruu

, with C on all three lines.

13. Name four points.14. Name three lines.15. Name three rays.16. Name two segments that contain I17. Name two lines that contain J18. Name a plane.19. Name the point that is between J and I20. Name that point that is common to three lines.21. Name three non-collinear points22. Name three collinear points

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

Part I: Place a word from the list below in each of the blank spaces in the paragraph. (Each word is used only once)

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.

Angles have different parts and we name them in a certain way:

The sides of an angle are the two rays that intersect: DG

u ruu and DM

u ruuu

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 .

Vertex Three AngleSides Rays Vertex Vertex

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

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. HKu ruu

bisects ∠PHR 6. M is the midpoint of TV

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 M is the midpoint of AB . If AB = 6 cm, then AM

= MB = 3cm

Ex: Suppose ABu ruu

bisects ∠CAD . If m∠CAD =60o , then, m∠CAB=m∠BAD =30o

7. GHs ruu

bisects BX at H

Directions: For #s 8 – 14, answer the following questions based on this diagram.

1.4 Midpoints and Bisectors (cont.)

Directions: For #s 15 – 18, answer the following questions based on this diagram.

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.

8. Name a ray9. Name a line10. Name a segment11. Name the segment bisector12. Name a midpoint13. Name a point that is not a midpoint14. Name an angle

15. Name the angle bisector16. Name the angle that is being bisected17. 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.

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

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.

1.

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:

This angle is roughly 75 degrees

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.

2. 3.

4.

1.5 Measuring Angles and Segments (cont.)

Directions: For #s 7 – 11, use the diagram below. Note that m∠HAF=90o

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 QSu ru

13. Segment named VW , bisected by JKs ruu

14. Acute angle named ∠DEF , bisected by EG

Directions: For #s 15 – 19, use the diagram below along with the following information: XVu ruu

bisects ∠TXW . XUu ruu

bisects ∠TXV . The measure of ∠TXU = 20o .

5. 6.

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

15. m∠UXV =

16. m∠VXW =

17. m∠TXW =

18. m∠TXV =

19. m∠UXW =

1.6 Inductive Reasoning

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, …

Inductive reasoning is the use of patterns to make conjectures about a problem.

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?

20. How many total squares in the second tower?

21. How many squares do you add to the “zero”th tower to get to the third 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.

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?

25. How many squares do you add to the “zero”th tower to get to the nth tower?

26. How many total squares in the nth

tower?

26)

In Out2 43 611 2227 ?? 18

27)

In Out2 74 137 2210 3112 ?? 76

28)

In OutHouse 4Cup 2

Writer 5Elephant 7

Spin ?Mathematics ?

? 3? 8? 0

29)

I n Out1 32 73 114 ?5 19

1.7 Deductive Reasoning

Deductive ReasoningUse 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:

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

Conclusion: I am not in school

Example of a Invalid Argument(1) If it is July, I am not in school

(2) I am not in school

Conclusion: It is July(Counterexample: It is August)

5. All Greeks are good mathematicians. Pythagoras was Greek.Conclusion:

6. All squares are rectangles. The diagonals of a rectangle bisect each other.Conclusion:

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. Carrie is a junior. 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:

1.7 Deductive Reasoning (cont.)