Geometry – Unit 1 Targets & Info Name: This Unit’s theme – Tools for Geometry August - September Use this sheet as a guide throughout the chapter to see if you are getting the right information in reaching each target listed

By the end of Unit 1, you should know how to…

Target found in…

Target discussed on (date)

Did I reach the target?

DIAGRAMS & EXAMPLES!

Use inductive reasoning to make a conjecture and prove a conjecture false

Chapter 2, Section 1 page 82

Define, identify, and name a point, line, plane, collinear points, coplanar points, segments, and rays

Section 2, pages 11-19

Identify congruent segments and solve problems involving properties of line segments

Section 3 pages 20-26

Classify angles and angle pairs and solve problems using their properties

Sections 4 & 5 pages 27-40

Find the midpoint and distance between two points when given coordinates

Section 7 pages 50-56

Review perimeter, circumference, and area

Section 8 pages 57-67

All material covered on the test will be based on these targets. So keep track of your readiness for the test by updating the “Did I reach the target?” column.

ESSENTIAL VOCABULARY: inductive reasoning, conjecture, counterexample, Euclid’s undefined terms, collinear, coplanar, line segment, ray, between, intersect, postulate/axiom, theorem, segment addition postulate, midpoint, congruent, angle, angle addition postulate, adjacent angles, vertical angles, linear pair, complementary, supplementary

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Suggested Textbook Practice Problems

***DAILY QUIZ QUESTIONS WILL BE TAKEN FROM THESE TEXTBOOK PROBLEMS AS WELL AS PRACTICE FROM YOUR UNIT WORKBOOK***

Lesson 1: Patterns and Inductive Reasoning Page 86: 6 – 30, 33 – 43, 48 – 50, 53 Lesson 2: Points, Lines, and Planes Page 16: 8 – 46 Lesson 3: Postulates and Axioms, Midpoint, Distance Page 24: 8 – 30, 37, 42, 43 Page 54: 6 – 51, 58 Lesson 4: Angle Measures Page 31: 6 – 23, 29 – 39 Lesson 5: Angle Pair Relationships Page 38: 7 – 40 Lesson 6: Perimeter and Area Page 64: 7 – 50

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Lesson 1: Patterns and Inductive Reasoning What is Geometry? Who is Euclid? What other types of Geometry exist? Inductive Reasoning:

1. Look for a pattern Ex 1) 1, 2, 4, 16, 64, ____, ____, ____ Ex 2) -5, -2, 4, 13, ____, ____ Ex 3) , , , _______ 2. Make a conjecture 3. Verify the conjecture Conjecture: an unproven statement based on observations Is the conjecture TRUE or FALSE?

Ex 1) Every road in Illinois is currently under construction. Ex 2) All math teachers are attractive. Ex 3) The sum of the first n odd integers is n2. To prove a conjecture false, you need ONLY ONE example that shows the conjecture to be false. To prove a conjecture true, however, you must prove that the conjecture is true in ALL CASES.

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Counterexample: example showing a conjecture false Ex 4) For all x ∈, x

2 ≥ x . True or False? Ex 5) Every even number greater than 2 can be written as the sum of two prime numbers. True or False? Find a counterexample to show each conjecture false. (TEST QUESTION) a) The product of two numbers is always greater than either number. b) The difference of two integers is less than either integer. c) The sum of two prime numbers is an even number.

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Lesson 1 Practice: Patterns and Reasoning 1. Who is considered the Father of Geometry? _________________________ 2. What book did he write? ___________________________________ 3. How can you prove a conjecture false? _________________________________ 4. Sketch the next figure. 5. Sketch the next figure. 6. Find the next term in the pattern. a. 1, 5, 9, 13, _____ b. 1, 4, 9, 16, _____ c. 1, 3, 9, 27, _____ d. 1, 1, 2, 3, 5, 8, 13, _____ e. O, T, T, F, F, S, S, E, _____ f. 1, 2, 6, 24, 120, _____ g. Aquarius, Pisces, Aries, Taurus, _____ 7. What is the next row in the following pattern? 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 ____ ____ ____ ____ ____ ____ ____ Complete the conjecture based on the specific cases. 8. Conjecture: The sum of any two odd numbers is _________. 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506

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9. Conjecture: The product of a number (n – 1) and the number (n + 1) is always equal to __________. 10. Riley makes a conjecture about slicing pizza. She says that if you use only straight cuts, the number of pieces will be twice the number of cuts. Is this true? If false, provide a counterexample. Show each conjecture is false by providing a counterexample. 11. The square root of a number is always 12. The sum of two numbers is always greater than less than the number. the larger number.

13. All prime numbers are odd. 13. If m is a nonzero integer, then is always

greater than 1. 14. The product of two positive numbers is 15. Triangle ABC is a right triangle, so ∠A is a greater than either number. right angle. Make a conjecture for each scenario. Show your work. 16. the sum of the first 100 positive odd numbers 17. the sum of an even and odd number 18. the product of two odd numbers 19. Find the perimeter when 100 triangles are put together in the pattern shown. All triangles have side lengths that measure 1 cm. A. 100 cm B. 102 cm C. 202 cm D. 300 cm 20. Describe a real life situation in which you have used inductive reasoning.

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Lesson 2: Points, Lines, and Planes Definition: uses known words to describe a new word Euclid’s 3 Undefined Terms: 1) Point – has no dimension, represented by a small dot and labeled with a capital letter 2) Line – extends in one dimension, represented by a straight line with two arrows to show continuing forever

in both directions 3) Plane – extends in two dimensions, represented by a shape similar to a slanted tabletop extending forever in all directions Collinear Points: Coplanar Points: Ex 1) a. Name 3 collinear points b. Name 4 coplanar points c. Name 3 noncollinear points Between: Consider line AB. Line Segment AB is all points between A and B on AB Ray AB is the initial point A and all points on AB on the same side of A as point B Be very careful in labeling.

A

B

D

F

G H

A B

A B A B

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If C is between points A and B, then we call CA and CB Opposite Rays. Ex 1) Draw 4 noncollinear points M, N, O, P. Then draw MO, OP, MN, and NM Ex 2) Given EH, name two pairs of opposite rays.

Are GF and GE the same ray? Intersect: having one or more points in common Sketch the following: 1) line AB and a plane that do not intersect 2) two planes that do not intersect and a line that intersects each plane at exactly one point 3) a line BC intersecting a plane at one point B 4) a line intersecting a plane at infinite points 5) Three points that are coplanar but not collinear 6) Two lines that intersect in a point and all lie in the same plane 7) two planes that intersect at line AB

E F G H

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Lesson 2 Practice: Points, Lines, and Planes

1. Who is considered the Father of Geometry? _________________________ 2. What were his three undefined terms? a. ________________ b. ________________ c. ________________ 3. How can you prove a conjecture false? _________________________________ 4. How can you prove a conjecture true? _________________________________ Determine if each statement is true or false. T F 5. Points A and B are collinear. T F 6. Points A, B, and C are collinear. T F 7. Points D and E are collinear. T F 8. Points J and K are collinear. T F 9. Points J, K, and L are collinear. T F 10. Points J, K, and L are coplanar. T F 11. Points J, K, and M are coplanar. T F 12. Points L, M, and N are coplanar. T F 13. Points J, K, L, and M are coplanar. Name a point that is coplanar with the given points. 14. A, B, and C _______ 15. D, C, and H _______ 16. F, A, and E _______ 17. E, F, and G _______ 18. A, B, and H _______ 19. B, C, and F _______ 20. AB and BC intersect at ________. 21. AD and AE intersect at ________. 22. Plane ABC and plane DCG intersect at ________. 23. Plane EAD and plane BCD intersect at ________.

D

B C

A E

M

N

J K L

A

B C D

F

E

G

H

A

B

C

D

E

G

H

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Sketch the figure described. 24. Draw two points, X and Y. Then sketch XY. Add a point W between X and Y so that WX and WY are opposite rays. 25. Two lines that lie in a plane but do not intersect. 26. Two lines that intersect and another line that does not intersect either one. 27. Two rays that are coplanar but not collinear. 28. Two planes that intersect at line MN. 29. Three planes that intersect at line AB. 30. Points K, L, M, and N are not coplanar. What is the intersection of plane KLM and plane KLN? A. K and L B. M and N C. KL D. KL

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Lesson 3: Postulates and Axioms, Measuring Segments, Distance, Midpoint Postulate/Axioms: Ruler postulate: points on a line can be matched on the number line to real numbers so that the distance is equal to the absolute value of the difference between the coordinates (p 20)

Between: when 3 points lie on a line we say one is between the others Segment Addition Postulate: If B is between A and C, then Two friends leave their homes and walk in a straight line toward the other’s home. When they meet, one has walked 425 yds and the other has walked 267 yds. How far apart are their homes?

A C B

AB + BC = AC

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A

B

M

Finding the distance between two points using coordinates. Find AB. Distance Formula: Midpoint: the point between two endpoints of a segment that divides a segment into two congruent segments Midpoint Formula:

A

B

A

B

A B M

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Ex 1) Find the midpoint of AB. A(-2, 3) B(5, -2) Ex 2) Find the midpoint of CD. C(5, -8) D(9, -5) Ex 3) Find MN. M(8, -3) N(4, 2) Ex 4) The midpoint of RP is M(2, 4). One endpoint is R(-1, 7). Find the coordinates of the other endpoint P.

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Lesson 4: Angle Measures Vocabulary Review: Euclid’s Undefined Terms – Midpoint – Collinear – Coplanar – Line Segment – Ray – Congruent – Counterexample – Postulate/Axiom – Find MN and the midpoint of MN. Find RT if RS = 2x + 3, ST = 5x – 12, and RT = 4x + 15 Angle: two rays with the same initial point Name all the angles in the diagram. If point F is placed between points A and C, have more angles been created?

A

E D

C

B

A

B C

M(-3, 4)

N (5, -11)

R T S

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Angle Addition Postulate: If point P is in the interior of ∠RST, then m∠RSP + m∠PST = m∠RST. Ex 1) If m∠PQR = 30o and m∠RQS = 40o, then m∠ PQS = _____ Ex 2) Given m∠ PQR = 45o, m∠RQS = 20o,

m∠ SQT = 50o, and m∠TQV = 65o m∠ PQT = ______ m∠ SQV = ______

m∠VQP = ______ m∠RQT = ______ Ex 3) NO bisects ∠MNP Bisect: cut in half, make angles congruent (≅ )

P

Q R

V S

T

M

N

O

P

(3x + 8)o

(4x – 9)o

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Ex 4) x = _______ 4 Classifications of Angles:

1. Acute – 2. Right – 3. Obtuse – 4. Straight –

Adjacent Angles: two angles sharing a common vertex and a common side Name a pair of adjacent angles in the diagram.

(2x – 10)o (x – 20)o

(2x + 5)o

M

N

O

P

P

Q R

V S

T

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Lesson 4 Practice: Angle Measures 1. m∠QRY = __________ 2. m∠XRZ = __________ 3. m∠XRS = __________ 4. m∠QRS = __________ 5. Classify each angle as acute, obtuse, right, or straight. a) ∠QRY b) ∠XRZ c) ∠XRS d) ∠QRS 6. m∠LMN + m∠NMO = m∠ _______________ Why? _____________________________________ 7. ∠LMN ≅ _______ 8. m∠LMN = _______ 9. If m∠LMN = 24°, then m∠NMO = _________ m∠LMO = _________ 10. If m∠LMO = 64°, then m∠LMN = _________ m∠NMO = _________ 11. If m∠LMN = (3x + 13)° and m∠NMO = (5x – 7)°, then x = __________ m∠LMN = __________ m∠NMO = __________ 12. If m∠LMN = (5x + 6)° and m∠LMO = 62°, then x = __________ m∠LMN = __________ m∠NMO = __________ 13. m∠SXT = (4x + 1)°, m∠QXS = (2x – 2)°, m∠QXT = 125 x = __________ m∠QXS = __________

MN bisects ∠LMO

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14. m∠PXR = (3x) °, m∠RXT = (5x + 20)° x = __________ m∠RXT = __________ 15. Find the distance between (1, 2) and (-5, 4). 16. Find the distance between (2, 9) and (-3, -3). 17. GH + HI = _______ Why? ____________________________________________ 18. Determine if each statement is true or false. T F a. All the points shown are coplanar. T F b. P and S are collinear points. T F c. R, S, and T are collinear points. T F d. Q, R, and S are collinear points. T F e. P, R, and S are coplanar points. T F f. Q, R, and S are coplanar points. T F g. RS and PT intersect. T F h. US and PT intersect. T F i. RT and TR are opposite rays. T F j. RS and RT are the same. Use the diagram to complete each statement. 19. ∠CBJ ≅ _______ 20. ∠FJH ≅ _______ 21. If m∠EFD = 75, then m∠JAB = _______

P

Q

R S

T

U

19

22. If m∠GHF = 130, then m∠JBC = _______ 23. Points E and G are collinear, and point F is between points E and G. If EG = 49, EF = 2x + 3, and FG =

4x – 2, find x, EF and FG. 24. Find the midpoint of ST given endpoints S(-5, 8) and T(-2, 4). 25. The midpoint of ST is M(-6, 4). Find the coordinates of point T given endpoint S(8, 2). 26. Make a conjecture for the following scenario. Show support for your conjecture. The sum of the first 100 positive even numbers is ________. Provide a counterexample to show that each conjecture is false. 27. ∠1 and ∠2 are supplementary, so one of the angles is acute. 28. If two angles are adjacent, then they are also congruent.

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Lesson 5: Angle Pair Relationships Vertical Angles: two angles whose sides form two pairs of opposite rays Linear Pairs: two adjacent angles whose non-common sides form opposite rays Ex 1) a) ∠1 and ∠3 are vertical? b) ∠2 and ∠4 are vertical? c) ∠3 and ∠4 are a linear pair? Ex 2) Find x, y, and z. Make a conjecture about vertical angles: Make a conjecture about linear pairs:

1 3

2 4

x

y

z 40

o

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Ex 3) Use your conjectures from example 2 to find the values of x and y. Is your conjecture supported? Is your conjecture true or false? Complementary Angles: two angles whose sum = ________ Supplementary Angles: two angles whose sum = ________

(y + 20)o (3x + 5)o

(4y – 15)o (x + 15)o

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Lesson 6: Perimeter and Area Rectangle: P = A = Triangle: P = A = Circle: C = A = Ex 1) The radius of a circle is 6 cm. Find the circumference and area in exact terms. Ex 2) Find the area of a triangle defined by the points D(1, 3), E(8, 3), and F(4, 7). Ex 3) Find the area and perimeter of the triangle defined by H(-2, 2), J(3, -1), and K(-2, -4).

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Lesson 5 & 6 Practice: Angle Pair Relationships, Perimeter, Area Write the Distance Formula Write the Midpoint Formula Find the coordinates of the midpoint of each segment with the given endpoints. 1) A(0, 0) B(-8, 6) 2) J(-1, 7) K(3, -3) 3) C(10, 8) D(-2, 5) Find the coordinates of the other endpoint of a segment with the given endpoint and midpoint M. 4) R(2, 6) M(-1, 1) 5) T(-8, -1) M(0, 3) 6) W(-3, 12) M (2, -1) Use the marks on the diagram to name the congruent segments and congruent angles. 7) 8) 9)

A

B

C

D

E

F G

W

X

Y

Z

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QS is the angle bisector of PQR. Find the two angle measures not given in the diagram. 10) 11) 12) BD bisects ABC. Find the value of x. 13) 14) Find the distance between the two points. 15) A(3, 12) B(-5, -1) 16) C(-6, 9) D(-2, -7) 17) Who is considered to be the Father of Geometry? _________________________

18) What were his three undefined terms? _____________, _____________, and _____________.

19) A statement that we accept without proof is called a(n) ___________________ or a(n)_____________.

22o

P

R

S

Q 80o P

R

S

Q

91o

P

R

S

Q

(5x – 22)o

A

C

D

B

(2x + 35)o ( x + 20)o

A

C

D

B (3x – 85)o

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T F S, T, V, and W are coplanar. T F S, R, and X are collinear. T F Q, P, and R are collinear T F P, R, Q, and W are coplanar. T F TS and TU are opposite rays. T F PQ and QP are opposite rays. T F T and Q are collinear. T F S, X, and Q are coplanar. T F R is the midpoint of PQ. T F ST and SU are opposite rays. T F If two planes intersect, then they intersect at a line. Use the figure at the right. Answer with yes or no. 20) Are 5 and 6 a linear pair? __________ 21) Are 5 and 9 a linear pair? __________ 22) Are 5 and 8 a linear pair? __________ 23) Are 5 and 8 vertical angles? __________ 24) Are 5 and 7 vertical angles? __________ 25) Are 9 and 6 vertical angles? __________

Decide whether the statement is always, sometimes, or never true. 26) If m 1 = 40°, then m 2 = 140° __________ 27) If m 4 = 130°, then m 2 = 50° __________

28) 1 and 4 are congruent __________

29) m 2 + m 3 = m 1 + m 4 __________ 30) m 2 ≅ 1 __________ 31) m 2 = 90° – m 3 __________

P Q R

S

T U

V W

X

5 6 7 8 9

4 1

2 3

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Use the figure at the right to answer questions #33 – 40. 32) If m 6 = 72°, then m 7 = _______ 33) If m 8 = 80°, then m 6 = _______ 34) If m 9 = 110°, then m 8 = _______ 35) If m 9 = 123°, then m 7 = _______ 36) If m 7 = 142°, then m 8 = _______ 37) If m 6 = 13°, them m 9 = _______ 38) If m 9 = 170°, then m 6 = _______ 39) If m 8 = 26°, then m 7 = _______ Find the value(s) of the variables(s). 41) 42) 43) 44) 45) 46)

9 6

7 8

105° (2x – 11)°

(6x + 19)°

x° (2x – 20)° (3y – 8)°

(6x – 32)° (y – 12)°

(5x - 50)°

y° (3x + 20)° 6x° (4x + 16)°

11y° 56° y°

7x° 2x°

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47) A and B are complementary. The measure of B is three times the measure of A. Find m A and m B. 48) C and D are supplementary. The measure of D is eight times the measure of C. Find m C and m D.

A and B are complementary. Find m A and m B. 49) 50)

m A = 5x + 8 m A = 3x - 7 m B = x + 4 m B = 11x – 1

A and B are supplementary. Find m A and m B. 51) 52) m A = 2x m A = 6x – 1 m B = x + 8 m B = 5x – 17 Given points A(-3, 6), B(4, 8), and C(4, -4). Find the following. 53) BC 54) midpoint of AB 55) AC

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56) FG + GH = ________ Why? 57) Find the coordinates of endpoint F of segment EF with the midpoint M. Find the perimeter and area of each figure. 58) 59) 60) 61) Find the perimeter of the figure with vertices at A(-5, 3), B(7, -2), C(7, -6), D(-5, -6). Find the area of the shaded region. All angles are right angles. 62) 63) 64)

F H G

E(-2, 8) F(x, y) M(4, -1)

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Review for Unit 1 Test 1. Find the area and perimeter (or circumference) of each figure. Give exact answers. a) b) square rectangle area = ___________ area = ___________ perimeter = ___________ perimeter = ___________ c) d) triangle circle area = ___________ area = _______________ perimeter = ___________ perimeter = ___________ 2. Find the perimeter of a square if its area is 100 square meters. 3. The length of a rectangle is 4 more than two times its width. If the area of the rectangle is 48 meters, find

its length and its width. 4. Find the area of a triangle if its vertices are at (-2, 2), (6, 2), and (0, 10) 5. Who is considered to be the Father of Geometry? 6. What were his 3 undefined terms? 7. What do we call a statement that we accept without proof? 8. What do we call a statement that we accept because we have proven it?

12 in

5 cm

8 m

16 cm

8 cm

15 cm 17 cm

20 cm

25 cm

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T F 10. Points A and B are collinear. T F 11. Points D and E are collinear. T F 12. Points A, B, and C are collinear. T F 13. Points D, E, and F are collinear. T F 14. Points A, B, and C are coplanar. T F 15. Points F, G, and H are coplanar. T F 16. Points A, B, D, and E are coplanar. T F 17. Points F, G, H, and J are coplanar. T F 18. All points shown are coplanar. T F 19. AB and FH intersect. T F 20. FH and GJ intersect. T F 21. AB and AC are the same. T F 22. AB and BA are the same. T F 23. AB and AC are the same. T F 24. AB and BA are the same. T F 25. AB and BA are opposite rays. T F 26. BC and BA are opposite rays. T F 27. AB and BA are the same. T F 28. AB and AC are the same. T F 29. ∠1 and ∠2 form a linear pair. T F 30. ∠4 and ∠5 form a linear pair. T F 31. ∠1 and ∠4 are vertical angles. T F 32. ∠2 and ∠5 are vertical angles. T F 33. m∠1 + m∠2 + m ∠3 = 180 T F 34. m∠4 + m ∠5 = 180 T F 35. ∠1 ≅ ∠ 4

D A

B

C

F G

H

J

E

1 2 3

4 5

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36. If m∠1 = 40° and m∠2 = 90°, then m∠3 = __________ m∠4 = __________ m∠5 = __________ 37. CG + GF = _________ Why? 38. If G is the midpoint of AD, then ___________ Why? 39. If AD bisects BE, then _________________________________ Why? 40. m∠AGB + m∠BGC = m∠ ______________ Why? 41. If GF bisects ∠AGE, then _______________________ Why? 42. One angle is 10 degrees more than 3 times its complement. Find the measure of the angle. 43. An angle is 15 degrees less than half its supplement. Find the measure of the angle. 44. x = _____________ 45. x = __________ m∠HLJ = __________ y = __________ m∠HLK = ___________

(2x + 30)° (4x + 20)°

H K

J

L

I

x (3x – 8)° (2y – 17)°

A

B C

D

E F

G

1 2 3

4 5

32

46. Point B is between A and C. If AB = 10 and BC = 6, then AC = ___________. 47. Point B is between A and C. If AB = 5 and AC = 12, find BC. 48. Find the midpoint of AB for A(-2, 8) and B(4, -3). 49. If M(-3, 6) is the midpoint of XY, and the coordinates of X are (7, 1), find the coordinates of point Y. 50. For A(5, -2) and B(8, 4), find AB. 51. Use inductive reasoning to predict the next two terms in the sequence. 2, 6, 7, 21, 22, 66, 67, ___, ____ 52. Find a counterexample to show that the conjecture is false. a. The sum of two integers is always positive. b. All four sided figures are rectangles. 54. Answer each question and provide a reason that supports your answer. a. SQ + QR = _______________ Why? __________________________ b. If Q is the midpoint of SR, then ______________. Why? _______________________________ c. m∠XYW + m∠WYZ = m∠_______________ Why? ______________________________ d. If YW bisects ∠XYZ, then ___________________ Why? ______________________________

X

Y Z

W

Q

R

S