Chapter 1& 2: Introduction to Geometry

Transcription

1 ALL DATES ARE SUBJECT TO CHANGE. Name : Date Day Topic Block # Chapter 1& 2: Introduction to Geometry Assignment Points, Lines and Planes pgs 6-7{1 23 odd, odd} Segments, Rays, Parallel Lines and Planes 1.2 Measuring Segments pgs {3 21 every 3 rd } 1.3- Midpoint and Distance pg {7-35 odd} Formulas Measuring Angles pgs {15-42 every 3rd} 1.5 Angle Pair Relationships pgs {9 33 every 3rd,} 1.6 Classify Polygons pg {3-19 odd} 1.7 Perimeter, Circumference pgs {4 36 every 4 th problem} and Area 7.1 Pythagorean Theorem pg {3-13 odd} 3 Test Chapter 1 Read section 2.1 and complete pg 75 {1 11 all} Inductive Reasoning pgs {12-28 even} 2.2 Analyze Conditional pgs 90 {1 17 odd} Statements Logic Puzzles Deductive Reasoning pgs {2-14 even} 2.4- Use postulates and Diagrams (This is review) pg 100{14-23 all} Quiz 6 Quiz Reasoning in Algebra pgs {6-9 all, 28, 31, 32} What do you see? Worksheet Proving Angles and segments pg {4,17, 21, 23} 2.7 Prove Angle Pair relationships pg {12, , 39} Quiz 8 Quiz Proof day TBA 9 Review Finish Review Sheet & Study 10 Test Portfolio Worksheet

2 Name: Date Block # Day : Points, Lines, and Planes 1. The most basic figure in geometry: It is know as a. a. It is represented by a dot, but it really has no or. b. Points are named with letters! Example: c. Every geometric figure is made up of points! d. Two different types of arrangements of points (on a piece of paper). e. A group of points that line up are called points. 2. The second basic figure in geometry is a. a. Explanation: A series of points that extends in 2 opposite directions. b. We use at the end of the line to save time (and space!) c. Naming lines (two options) i. Option 1: List any two points with a line (with arrows) over it: ii. Option 2: With an italicized (scripted) lowercase letter: 3. The third basic figure in geometry is called a. a. Explanation: a flat surface with no thickness that extends forever in all directions. E D F R Plane E D F R Plane How we a plane. How planes work (extend forever). b. Naming Option 1: The word Plane followed by any points in the plane. Option 2: The word Plane followed by a italic letter. 1

3 4. The 3 basic shapes of geometry (,, and ) are the undefined terms of geometry because they are so basic, we can t define them. 5. At your seat: Describe the two different sets of points, name them if possible. Set #1: Set #2: 6. Set #1: points because all points lie in the same. 7. Set #2: points, not all points lie in the same. 8. Question: What is your name? How do you know? a. A is an accepted statement or fact. b. A synonym for the word is the word. 9. Fact: Through any two points there is exactly line. 10. Fact: If two lines intersect, then they intersect in exactly one. 11. Fact: If two planes intersect, then they intersect in exactly one. 2

4 Day 1: Segments, Rays, Parallel Lines & Planes 1. Lines that do not intersect are called lines a. Only when the lines are! lines (same ) 2. When lines are and they don t intersect, they are called. lines (different ) 3. Look at our line 4. Each of these new figures are called. A B Written, spoken. Called the, when written it MUST come * What do two rays that face opposite directions and have the same endpoint create? Definition (Opposite Rays): two rays with the same endpoint. Always, sometimes, never Two opposite rays form a line. 3

5 5. What do we get if we cut the line twice? This is called a. X Y Written, spoken. Called. Now order does matter. Why? 6. Quick Vocabulary Review. Fill in the Missing Information. b. Line: Extends forever in direction(s). c. Ray: Extends forever in direction(s). d. Segment: Extends forever in direction(s). 7. What s the difference between the two pairs of planes shown below? 8. Two planes that do not are said to be planes. 9. Look at the figure below and describe the connection between line t and plane ABC. t G H J More Questions E F a. Name a pair of parallel planes. b. Name a pair of skew lines. B C c. Name a pair of parallel lines. d. Name a ray. A D 4

6 Name: Date Block # 1.2: Measuring Segments 1. What s the distance between your house and Red Bank Regional? 2. Is your house and Red bank Regional in a straight line? e. Answer: establish a (straight) line. f. The Postulate: ( Postulate is like a definition) i. Any two points can be put onto a number line and measured. 3. A number line is like an endless ruler 4. Find the length of each segment listed below: A B C D E F G AB = BE = CF = DG = BG = BA = DF = 5. How do you find the distance between two points on a number when the units are variables? C D x y *** Distance formula on a number line: 1st point minus 2nd point 5

7 6. Two segments that have the same length are said to be. g. The symbol means. Definition: Congruent figures have the same &. Why do we only need to check one of these for segments? 3 ft. 3 ft. 3 ft. 5 ft. segments segments * Tick Marks are used to indicate segments. 7. Is there a place between your house and RBR where you are equally far from both? 10 ft. The halfway point is called the. 8. Length of our stick: Note: The stick can be broken any way you want, but the two pieces must add up to. Does this seem obvious? What do we call something that we accept as obvious? This illustrates the. Solve for x. (2x + 4) ft. 6 ft. 14 ft. 6

8 Name: Date Block # 1.3: Midpoint and Distance Formulas 1. The coordinates of point A is (6, 6) and point B is (3, 2). Plot the points below. 2. What is the length of segment AB (round to nearest tenth if necessary)? 3. What is the midpoint of segment AB? y x Commit these formulas to Memory it will help you later on Level 1 Question: The endpoint of a segment is (2, 3) and the midpoint is (3, -4). What is the other endpoint? 7

9 Name: Date Block # 1.3: Midpoint and Distance Formulas Given that B is between A and C, find each missing measure. 1. AB = 5.3, BC= AC = AB= 21, BC=4.3 AC= 3 3. AB=, BC= 18.9 AC= AB= 6 4, BC= AC= 10 If B is between A and C, find the value of x and the measure of BC. 5. AB = 3x, BC= 5x, AC= 8 6. AB= 3(x+7), BC= 2(x-3) AC =50 W, R, and S are points on a number line, and W is the midpoint of RS. For each pair of coordinates given find the coordinate of the third point. 7. R= 4 S= R= 12 W= W= -4, S= 2 Y is the midpoint of XZ. For each pair of points given find the coordinates of the third point. 10. X( -4, 3) Y(-1,5) 11. Z(2,8) Y(-2,2) 12. Z( -3,6), Y(0, 5.5) Use the diagram to the right to Find the Midpoint and Distance of the Following Segments 13. CD 14. FD 15. HE 16. AJ 8

10 Name: Date Block # 1.4: Angles Vocabulary Fill in the boxes below use all resources available (friends, books, etc.) Angle Type Draw an Example Describe or Define Acute Angle Obtuse Angle Right Angle Straight Angle Congruent Angles Complimentary Angles 9

11 Supplementary Angles Angle Addition Postulate Vertical Angles Adjacent Angles Angle Bisector Linear Pair 10

12 Name: Date Block # 1.5: Problem Set for Vocabulary Directions: Use the figure below to answer questions 1-5. A 1. Name an angle complimentary to AGB. F G 70 B 2. Name an angle supplementary to AGB. C E 3. What type of angle is AGD? D 4. What angle is vertical to BGC? What is its measure? 5. Name an angle that is congruent to AGB. 6. In the figure below SX bisects RST. Find the measure of RST. S (2x+10)º (5x-20)º T R X 11

13 7. Use the figure below to find m WYZ. X Y (6x-30)º W (2x+10)º Z 8. In the figure below, m ABC = 43+ x. Find x. A D 3x + 1 6x - 6 C B 9. What postulate did you need to use to solve the problem in #8? is twice the size of its compliment. What are the degree measures of both angles? 12

14 Name: Date Block # 1.6 Classify Polygons In Geometry a figure that lies in a plane is called a. A polygon is a closed plane figure with the following properties: What did the pirate say when his Parrott flew away???? Each endpoint of a side is a of the polygon. The plural of vertex is. A polygon can be named by all its vertices in consecutive order. Ex. the polygon to the right can be named Polygon ABCDE or DEABC Vertex Convex Polygon Concave polygon Definition: Definition: Picture: Picture: Equilateral: Equiangular: **** A is a convex polygon that is both equilateral and equiangular. 13

15 Polygons have names according to the number of sides. Fill in the chart below, you will be responsible to know the names of polygons below Number of sides Type of polygon Number of sides Type of polygon n Examples: Tell whether the figure is a polygon. If it is not, explain why. If it is a polygon, tell whether it is convex or concave The lengths (in meters) of two sides of a regular heptagon are represented by the expressions 1lx 32 and 6x 7. Find the length of a side of the heptagon. 5. The expressions 6x and 13x 54.5 represent the lengths (in feet) of two sides of a regular pentagon. Find the length of a side of the pentagon. 6. The vertices of a figure are given below. Plot and connect the points so that they form a convex polygon. Classify the figure. Then show that the figure is equilateral using algebra. A( 2, 1), B( 1, 2), C(2, 3), D(5, 2), E(4, 1),F(1, 2) 14

16 Name: Date Block # 1.7 Perimeter, Circumference and Area Area Perimeter Area Perimeter Height base Area Circumference Examples: 1. What is the perimeter and area of a rectangle with a height of 6 and base of 14? 2. What is the area of a circle with a circumference of 14π? 3. What is the perimeter of the figure created on the coordinate plane with the points A(-4, -1), B(4, 5), C(4, -2) 15

17 Name: Date Block # 7.1 Pythagorean Theorem Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs a 2 +b 2 =c 2 a and b are the C is always the How do we know which one is the hypotenuse????? Practice: Find the missing sides, x and y, in the triangles below leave answers in simplest radical form x 15 y What is the length of the hypotenuse of a right triangle with leg lengths of 21 inches and 28 inches? 5. A flagpole has cracked 9 feet from the ground and fallen as if hinged. The top of the flagpole hit the ground 12 feet from the base. How tall was the flagpole before it fell? 6. A rope 17 m long is attached to the top of a flagpole. The rope is able to reach a point on the ground 8 m from the base of the pole. Find the height of the flagpole. 16

18 Name: Date: Block: # 1. Name 3 non-collinear points. 2. Name the vertex of DGF. 3. Name a pair of adjacent angles. 4. Name a straight angle. 5. Give 2 other names for 2. Complete the following sentences: 6. points create 1 unique line. 7. A plane and a line intersection is at a. 8. When two planes intersected it at exactly. 9. Two segments that have the same length are said to be. Fill in the blanks below with always, sometimes, or never. 10. AC is in Plane Q, so point B is in Plane Q. 11. Two planes that do not intersect are parallel. 12. JK are JM the same ray. 13. Three points are coplanar. 14. Intersecting lines are parallel. 15. A and B are complimentary. A is twice as big as B, what are the measure of the two angles? 16. BD bisects ABC ; ABD = 6x + 2 and DBC = 3x Draw the figure and find m ABC. 17. Point X is the midpoint of W and Y. XW = 3x + 8 and XY = 9x 10. What is the length of WY? 18. The area of a circle is 36π, what is the circumference of that circle? y 19. A rectangle has coordinates A(5, 3), B(5, -2), C(1, -2), D(1, 3). What is the area of rectangle ABCD? x 17

19 20. M is the midpoint of segment AB. A(5, 9) and M(11, 19). What are the coordinates of B? 21. Line AB is in Plane Q. Line t intersects plane Q at point A. Draw and label the figure. 22. Solve for x and y in the figure below. 4x + 10 y 3x Draw skew lines. 24. Solve for x, then find CBD given that ABD = (43+x)º. C D A (6x-6)º B (3x+1)º 25. If 1 is supplement to 2, m 1= (2x+1) and m 2=(4x -7), Find the measure of both angles. 26. If bisects ABC, m ABE=(2x+11), and m CBE =(12x -19). Solve for x. 27. If J is between H and K, HJ= (2x+4), JK=(3x +3) and KH = 22. Solve for x. 28. Let H be the midpoint of, CH=(8x () and SH=(5x +18). Solve for x. Wrap Up: Confidence Meter for next class Test?

20 Name: Date Block # 2.1 Using Inductive Reasoning Warm Up: From the previous homework answer the following: Describe a pattern in the numbers. Write the next number in the pattern. 1. 5, 7, 9, 11, 13, 2. 22, 21, 19, 16, 12, Show the conjecture is false by finding a counterexample. 3. The sum of the squares of any two consecutive squared natural numbers is an even number. 4. The sum of the squares of any two squared natural numbers is an odd number. For the given ordered pairs, write a function rule relating x and y. 5. (1, 3), (2, 4), (3, 5), (4, 6) 6. (1, 4), (2, 9), (3, 16), (4, 25) Vocabulary * A conjecture is *When you find a pattern you are using reasoning. *A counterexample is. How many counterexamples do you need in order to prove a statement false? 19

21 Name: Date Block # 2.2 Conditional Statements, Biconditionals & Definitions A conditional is a logical statement that has two parts a hypothesis and a conclusion. A conditional statement can be written as an if-then statement. a. Example: If it is raining, then I will bring an umbrella. b. Each conditional is made up of two parts: i. The hypothesis the clause after the word if. ii. The conclusion the clause after the word then. If an angle is between 0-90, then it is a right angle. hypothesis conclusion Conditionals can be represented visually using a diagram. Conditional: If you live in Berwyn, then you live in Pennsylvania. Pennsylvania conclusion Berwyn hypothesis Example #1: If an animal is a reptile, then it is cold-blooded. Create a Euler Diagram for the statement. Example #2: Rewrite the conditional statement into if-then form. a. All birds have feathers. b. Two angles are supplementary if they are a linear pair. Example #3: Circle the hypothesis and underline conclusion of each conditional below: a. If 3x-7 = 32, then x= 13 b. I can t sleep if I am not tired. 20

22 The converse of a conditional is created by switching the hypothesis and conclusion. Conditional: If an angle s measure is between 0-90, then it is a right angle. Converse: If it is a right angle, then its measure is between (just switch the two to create the converse) Example #4 State the converse of each statement below. a. If today is Friday, then tomorrow is Saturday. b. If x>0, then x 2 >0. c. If a number is divisible by 6 then it is divisible by 3. d. If 6x =18, then x=3. When both the conditional and the converse are true, you can combine them into one statement. known as a. You combine them using the statement if and only if. If B is between A and C, then AB + BC = AC Conditional Statement ( True) If AB + BC =AC, then B is between A and C. Converse Statement ( True) B is between A and C if and only if AB + BC = AC Biconditional Example 5: In Example #4 if the converse and the conditional statement are true write the biconditional below in the space provided. If they are both not true give a counterexample for the converse. a. b. c. d. 21

23 The hypothesis and conclusion are both made up of statements. The negation of a statement has the opposite truth value. (Add the word not. Or take the word not out) Statement: x equals 8 Negation: x does not equal 8. * The symbolic form of a negation is ~p, which is read not p. (p is the statement) Two new words: inverse and contrapositive. Inverse: take the negation of both the hypothesis and conclusion for the conditional. Contrapositive: take the negation of both for the converse. Conditional statements can be represented symbolically p: hypothesis and q: conclusion ~: negation If then statement is presented as p q and is read as p implies q Statement Symbolic form Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p Example 6: Conditional: If it is raining, then I bring an umbrella Inverse: If it is not raining, then I do not bring my umbrella. Converse: If I bring my umbrella, then it is raining. Contrapositive: If I do not bring my umbrella, then it is not raining. Let s do one together: Statement: Rain will cancel the soccer game. Conditional: Converse: Inverse: Contrapositive: 22

24 Name: Block: # Partner: Pair work: Work with your partner, to solve the following problems, in Rally/Coach form. Student A does problem 1 with Student B watching and coaching when necessary. Student B completes problem 2 with Student A coaching where needed. After 2 problems are done review the work and then check with your other pair. Continue this until the entire worksheet is completed Directions: Write the negation of each statement below 1) You are not sixteen years old. 2) The soccer game is on Friday Directions: use the conditional below to write the inverse, converse and the contrapositive. 3. Conditional: If you like the Yankees, then you live in New York. Inverse: Converse: Contrapositive: If you can write the biconditional: 4. Conditional: If two lines are perpendicular, then they form right angles. Inverse: Converse: Contrapositive: If you can write the biconditional: 23

25 Rewrite the conditional statement in if-then form and Draw a Euler diagram representing the statement. 5. A car with leaking antifreeze has a problem. 6. Don t say anything at all when you don t have something nice to say. Decide whether the statement is a valid definition. 7. If a figure is an n-gon, then the figure is a polygon with n sides. 8. If a polygon is convex, has five sides, and is both equilateral and equiangular, then the polygon is a regular polygon. In a plane, point F lies between points C and D and EF intersects CD so that CFE DFE. Decide whether the given statement is true. Explain your answer using definitions and properties that you have learned. 9. FC and FE and are opposite rays. 10. CFE and DFE are adjacent angles. 11. CFD and EFD are a linear pair. 12. CFE and DFE are a linear pair. 13. CFE is an obtuse angle. 14. CD EF Little Review: 15. Define and draw vertical angles: 16. Explain the Angle Addition Postulate 17. Explain the segment addition postulate 18. Define and Draw Vertical Angles 24

26 Name: Date Block # 2.3 Deductive Reasoning & 2.5 Reasoning in Algebra Warm Up Directions: Make a valid conclusion from each set of statements below. 1. If a student wants to go to college, then a student must study hard. Rashid wants to go to the University of North Carolina. Conclusion: 2. If an animal is a red wolf, then its scientific name is Canis rufus. If an animal is named Canis rufus, then it is endangered. Conclusion: 3. If you read a good book, then you enjoy yourself. If you enjoy yourself, then your time is well spent. Conclusion: 4. If there is lightning, then it is not safe to be out in the open. Maria sees lightning from the soccer field. Conclusion: The Law of Detachment: The Law of Syllogism: These laws are forms of reasoning, which is strong reasoning based on facts. Inductive reasoning is more of a guess, and is based on continuing a. 25

27 1. We can use deductive reasoning to perform proofs. a. Remember, a theorem must be proven! 2. Example: An Algebraic Proof. Solve x + = Statements (Steps to Solve) Reasons (What you did) x + = x = x = Properties you are probably familiar with a. Addition Property of Equality b. Subtraction Property of Equality c. Multiplication Property of Equality d. Division Property of Equality e. The Distributive Property f. Simplifying 4. For the two examples below, describe the difference in the operations. Which reason would you use for each? a) If 8+ 2 x + 3= 23, then 2x + 11= 23 b) If 3x 5= 13, then 3x = Properties you probably are not familiar with a. The Reflexive Property b. The Symmetric Property c. The Transitive Property d. Substitution Property 26

28 a) The Reflexive Property: Think of when you look in a mirror and you see your reflection. Any time you have a number (angle, side length, etc.), you can always write that it is equal (or congruent) to itself. Examples: 10 = 10, x = x, AB AB, m ABC = m ABC When will this be used? Whenever two figures share something. C 1 Both the little triangle and the big triangle share angle 1, so 1 1. D Both the white triangle and the grey triangle share side CD, so CD CD b) The Symmetric Property: Think of when you solve equations and the x is on the right. You might like to always have your x on the left hand side, and you probably learned that you are allowed to switch sides this is the symmetric property. When solving an equation, if you end with this: 6 = x You can switch it to this: x = 6 Other examples: CD FG switches to FG CD 1 2 switches to 2 1 c) The Transitive Property (Substitution): Think of your two closest friends is friends with AND is friends with Kelly Jane Jane Beth is friends with Kelly (Jane is the link between Kelly and Beth) Beth 27

29 Example #1: Fill in each reason on the right that matches with the statement on the left 1. 5( 3) 5 x + = x + 15= x + 15= x = x = Example #2: Geometric Proof. Given: XY = 42 3( n + 4) 3n X Z Y 1. XZ + ZY = XY 1. ( ) n+ 4 + n= n n= n + 12= n = n = 5 6. Other Reasons you might use a. Definition of a midpoint b. Definition of an angle bisector c. Vertical angles property Reasons are comprised of properties, postulates, theorems, definitions, and sometimes a few other things Wrap Up: Partner Quizzo Partner A: use your notes and quiz your partner B about reasons we learned today. Once you have successfully described 3 reasons, switch places 28

30 Name: Date Block # 2.6 & 2.7 Proving Angles and Segment Congruence 1. Vocabulary Review: h. An accepted fact is known as a. i. An educated guess is known as a. j. A proven fact is known as a. 2. Let s prove a theorem: k. The Vertical Angles Theorem: Vertical angles are congruent i. Given: Intersecting lines that form angles 1 4. Sometimes this is stated for you, sometimes you must get it from a picture. 3. ii. Prove: 1 2 Usually it is just what the theorem says Sometimes interpreted to your picture. Statements Reasons (Postulates, Theorems, Definitions)

31 The type of proofs we ve done so far are called proofs. You can also do proofs by writing your steps out as sentences (this is usually more difficult especially for beginners). 4. Congruent Supplements Theorem: If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent Given: 1 and 2 are supplementary & 3 and 2 are supplementary Prove: Statements Reasons (Postulates, Theorems, Definitions) (The proof above can also be done as a paragraph proof) Wr ap up: List all Reasons on board (x + y +5) y 2x 30

32 Name: Date Block # 1. In the conditional below, underline the hypothesis, and circle the conclusion. If two segments have equal measures, then they are congruent. 2. Write the converse, inverse and contrapositive of the condition from #1. 3. Define/Describe a biconditional. Give a mathematical example 4. Write a valid conclusion or write no valid conclusion for the statements below. a. Conditional: If M is the midpoint of AB, then AM = BM. i.given: M is the midpoint of AB ii.conclusion: b. Conditional: Given 3 random points, they are always coplanar. i.given: Points X, Y, Z ii.conclusion: 31

33 5. Find the measures of the indicated angles below. B C D a. BFD = b. CFE = c. m AFB = 55; m BFC = A F E d. m CFD = 42; m DFE = Note: AE FC, BF FD 6. Complete the proof below Given: MI = LD Prove: ML = ID M I L D Statements 1. MI = LD IL = IL MI + IL = LD + IL MI + IL = ML and 4. LD + IL = ID 5. ML = ID 5. Reasons Write a statement to show each property of equality below. 7. Division Property 8. Reflexive Property 9. Subtraction property. 10. Transitive Property 11. Substitution Property 32

34 12. Given: MN= PN, NL= NO Prove: ML = PO 13. Given AB BC, m 1= m 3 Prove: m 1 + m 3 = Given : ABD and CBD form a linear pair. YXZ and WXZ form a linear pair. ABD YXZ Prove: CBD WXZ 33

35 Name: Date Block # - POW Homework Assignment (After Test) Math Forum - Problem of the Week Points, Lines, and Planes The picture on the left below shows three non-collinear points and all of the lines that are determined by these points. The picture on the right shows four non-collinear points and all of the resulting lines. 1. Name the lines formed by five non-collinear points A, B, C, D, and E. 2. How many lines would be formed by 6 non-collinear points? 3. If the points A, B, C and D in the second picture are non-coplanar, name all of the different planes that they could define. 4. How many planes could five non-coplanar points define? Extra: How many lines would be formed by N non-collinear and non-coplanar points? How many planes? Drexel University 34