Geometry Unit 1 Practice

Transcription

1 Lesson Persevere in solving problems. Identify each figure. hen give all possible names for the figure. a. S Geometry Unit 1 Practice e. P S G Q. What is a correct name for this plane? W R Z X b.. plane XZR. plane WZ. plane WXZ D. plane X c. D F m 3. How many different rays are in the figure? Name them. W Z Y X d. H K L P 015 ollege oard. ll rights reserved. 1 Springoard Geometry, Unit 1 Practice

2 4. How many different angles are in the figure? Name them. X Lesson 1-6. Use this diagram. Y Z P D W 5. Use appropriate tools strategically. Draw each figure. a. RS a. How many radii are shown? Name them. b. V b. How many diameters are shown? Name them. c. WX c. How is a chord similar to a diameter? How is a chord different from a diameter? d. plane containing points D and E e. DE d. Make use of structure. Suppose you have two different diameters of a circle. What must be true about the point where the diameters intersect?

3 7. wo angles are complementary. hey also have the same measure. Which statement is correct?. he two angles are obtuse.. he two angles are acute.. he two angles must be adjacent. D. he two angles cannot be adjacent. 10. hink about a chord of a circle and a radius of the same circle. a. What two things do the chord and the radius have in common? b. How are the chord and the radius different? 8. Reason quantitatively. P and Q are supplementary. he measure of P is 30 degrees more than the measure of Q. What is the measure of each angle? Lesson Use inductive reasoning to determine the next two terms in each sequence. a. 1, 17,, 7,... b. 1, 17, 7, 4,... c. 1, 17, 3, 30,... d. 1, 60, 300, 1500, his diagram shows lines PQ, RS, and V. How many angles appear to be obtuse? Name them. e. 1, 4, 9, 16,... P R M N S V 1. he fourth term of a sequence is 40. Write the first five terms of two different sequences that satisfy that condition. Q 3

4 13. Persevere in solving problems. Use this picture pattern. 14. Which rule describes how to find the next term in the sequence? 0, 3, 9, 1, 45, 93,.... Multiply the previous term by 3.. dd 3 to the previous term, and then multiply the result by 3.. Multiply the previous term by, and then add 3. D. Divide the previous term by 3, and then add 3. a. Draw the next two shapes in the pattern. 15. onstruct viable arguments. Explain how you knew that the rules you did not choose in Item 14 were incorrect. b. What numbers represent the next three figures in the pattern? Lesson Use expressions for even integers to show that the product of two even integers is an even integer. c. Verbally describe the pattern of the sequence. d. How many dots are added from the first diagram to the second? From the second diagram to the third? From the third to the fourth? Explain how to find the nth term. 17. onsider these true statements. ll glass objects are breakable. ll windshields are made of glass. onette s car has a windshield. ased on deductive reasoning, which of the following statements is not necessarily true?. onette has a breakable object.. onette has a glass object.. ll windshields are breakable objects. D. ll breakable objects are glass. 4

5 18. Make use of structure. Use deductive reasoning to prove that x 5 5 is not in the solution set of the inequality x 1 1, 7. e sure to justify each step in your proof. Lesson Make use of structure. In each statement, tell whether each bold term is undefined or defined. a. n angle is formed by two rays that have a common endpoint. 19. During the first month of school, students recorded each day on which they had a quiz in math class. student stated that there is a math quiz every uesday morning. Is the student s statement a conjecture or a theorem? Explain. b. line segment consists of two points and all the points between them. 0. Reason abstractly. student knows that (1) any two diameters in a circle bisect each other and () RS and V are two different diameters in the same circle. he student concludes that RS and V bisect each other. a. Is this an example of inductive or deductive reasoning? Explain. c. triangle is the union of three segments that intersect at their endpoints. d. If two lines intersect, then there is exactly one plane that contains the two lines. b. Is the conclusion correct? Support your answer. 5

6 . Model with mathematics. omplete this twocolumn proof by providing the reasons for each statement. 3x 5 Given: 57 Prove: x 5 3 Statements Reasons 3x a.. 3x b. 3. 3x c. 4. x d. 3. Suppose you are given that p 5 q 1 1 and that p 5 8. Which of the following statements can you prove?. p q. p 1 q 5 9. q D. p q 4. Identify the property that justifies the statement: If 3x 5 1, then x ddition Property of Equality. Distributive Property. Division Property of Equality D. ransitive Property of Equality Lesson 3-6. Reason abstractly. Write each statement in if-then form. a. he only time I wake up early is when I set my alarm clock. b. I eat breakfast at a restaurant only if it is a weekend. c. n obtuse angle has a measure between 90 and State or describe a counterexample for each conditional statement. a. If x 5 5, then x 5 5. b. If three points,, and are collinear, then is between and. 5. a. Write a geometric statement that does not use any defined terms in geometry. c. If a triangle is obtuse, then it cannot be isosceles. b. Write a geometric statement that does not use any undefined terms. 6

7 8. Suppose that this statement is true: If I wear boots or a raincoat, then I carry an umbrella. lso suppose that the hypothesis of that statement is true. Which statement must also be true?. I am wearing boots.. I am not wearing a raincoat.. I am not carrying an umbrella. D. I am carrying an umbrella. 9. Write a true conditional statement that includes 3x 11 this hypothesis: Write the following biconditional statement as two conditional statements: People have the same ZIP code if and only if they live in the same neighborhood. 33. Make use of structure. Use this statement: If 3x 5 0, then x fi 0. a. Is the statement true? Explain. 30. Model with mathematics. Write a two-column proof to prove that your conditional statement in Item 9 is true. Statements Reasons 3x Given a.. b. 3. c. 3. d. 4. e. 4. Division (or Multiplication) Property of Equality b. Write the converse of the statement, and explain whether or not the converse is true. c. Write the inverse of the statement, and explain whether or not the inverse is true. Lesson Write the inverse and the contrapositive of each statement. a. If it is raining, then I stay indoors. d. Write the contrapositive of the statement, and explain whether or not the contrapositive is true. b. If I have a hammer, then I hammer in the morning. 7

8 34. Which two forms of a conditional statement always have the same truth value? a. statement and inverse b. inverse and contrapositive c. converse and contrapositive d. converse and inverse 38. ttend to precision. Use the centimeter ruler shown. R a. What is the length of RW? W 35. Reason abstractly. Use this statement: If two lines form equal adjacent angles, then the lines are perpendicular. hen tell whether each statement is the inverse, converse, or contrapositive of the original statement. a. If two lines are not perpendicular, then they do not form equal adjacent angles. b. What number on the ruler represents the midpoint of RW? c. Suppose Q is a point on RW. If QW 5 1, what are the possible coordinates of point Q? b. If two lines do not form equal adjacent angles, then they are not perpendicular. Lesson Suppose point is between points R and V on a line. If R units and RV units, then what is V? a..5 units b. 6.8 units c. 7.8 units d units d. Suppose point is between points R and W and R 1 5. What is the length of R? W 39. Reason quantitatively. On a number line, the coordinate of point is negative and the coordinate of point is positive. a. When will the midpoint of be positive? b. When will the midpoint of be negative? 37. Suppose P is between M and N. a. If MN 5 10, MP 5 x 1, and PN 5 x 1 1, what is the value of x? c. When will the midpoint of be zero? d. When will the distance from to be negative? b. If PM 5 x 5, PN 5 6x, and MN 5 5x 1 4, what is the value of x? 8

9 40. Points P, M, and are on a line and P PM 5 M. Which point is between the other two? Explain your answer. Lesson Which expression represents the distance between points (m, n) and (p, q)?. ( mn) 1( pq) Lesson Make sense of problems. Suppose that PQ bisects MPN. What conclusion can you make?. ( m1p) 1( n1q). ( mp) 1 ( nq) D. ( mp) 1( nq) 4. Suppose bisects R. If m 5 5x and m R 5 9x 1 7, what is m R? 47. segment has endpoints L(, 7) and K(5, 3). What are the coordinates of the midpoint of LK? 43. Suppose two angles are supplementary. Which of the following terms NNO describe both angles?. acute. adjacent. congruent D. vertical 48. ttend to precision. he coordinates of the vertices of a triangle are (4, 6), (4, ), and (6, 4). a. Find. b. Find. 44. D and E are complementary. If m D 5 5x 1 3 and m E 5 3x 1, what is x? c. Find. 45. a. ttend to precision. Draw a single diagram to represent the statements shown. Obtuse angle PQR is bisected by Q. PQ is bisected by Q. Q is bisected by Q. QR is bisected by QD. b. Suppose m PQR Find m QR. d. ased on the lengths of the sides, what kind of triangle is? 49. he coordinates of the vertices of a triangle are D(5, 6), E(7, 5), and F(4, 3). Find the perimeter of the triangle. 9

10 50. Model with mathematics. Every point on a circle is the same distance from the center of the circle. If (x, y) represents any point on a circle and (5, ) is the center of the circle, use the Distance Formula to represent the length of the radius r of the circle. Lesson Model with mathematics. has endpoints (3, 5) and (1, 1). Find the coordinates of the midpoint of. 5. In the diagram shown, points S and are the midpoints of PQ and PR, respectively. y Q(, 3) 4 S P(4, 7) R(10, 3) x 54. Which expression represents the midpoint of the line segment with endpoints (x, y) and (p, q)? a. x1y, p1q b. xp, yq. x1p, y1q d. xp yq, 55. For the coordinates (5, 8) and (9, 14), one is an endpoint of a line segment and the other is the midpoint. How many possibilities are there for the other endpoint? Find each one. Explain your method. Lesson onstruct viable arguments. Use the diagram shown. 6 1 a. Find the coordinates of points S and. b. Find the length of S. Write a statement that can be justified by each of the following: a. definition of angle bisector 53. onstruct viable arguments. Given: (, 5), (0, 0), and (4, ). a. Find the coordinate of M, the midpoint of. b. ngle ddition Postulate b. Which of the points,, M, or, is closest to? 10

11 57. Use the diagram shown. P Q R a. What is the justification for the statement that PQ 1 QR 5 PR? 59. What can you use to prove that is the midpoint of? y x b. Suppose Q is the midpoint of PR. What is the justification for the statement that PQ 5 QR? a. ruler b. protractor c. definition of midpoint d. folding on point 60. Reason abstractly. Which statement NNO be justified by the use of the Distance Formula? y 58. Use the diagram shown. (5, 5) 6 D(0, 5) (5, 5) D 4 x (0, 0) 6 x 4 6 Suppose m 5 90 and m D 5 x 1 5. a. Write a statement that can be used as a justification that m D 5 90 (x 1 5). a. 5 b. D is the midpoint of. c. D d. D 5 D b. Write a justification for the statement that. 11

12 Lesson onsider the diagram and the given statements for a proof. D Statements 1. D is the midpoint of.. D 5 D 3. D > D E Reasons Which of the following could be the correct Given and Prove statements? a. Given: D > D; Prove: D 5 D. b. Given: D > D; Prove: D is the midpoint of. c. Given: D is the midpoint of ; Prove: E is the midpoint of. d. Given: D is the midpoint of ; Prove: > D. 6. Make sense of problems. omplete the proof. Statements Reasons 1. D bisects E. 1. Given. > 3. a. 3. m 5 m 3 3. Definition of congruent angles 4. 1 is supplementary 4. b. to. 5. m 1 1 m Definition of supplementary angles 6. m 1 1 m c. 7. d. 7. e. 63. omplete the proof. Given: m ; m PR 5 53 Prove: m 5 16 Statements Reasons 1. a. 1. Given. m 1 1 m 5. b. m PR m Substitution 4. c. 4. d. 64. omplete the proof. 1 P Q R 1 3 D S R E Given: 1 is supplementary to ; D bisects E. 1 Prove: 1 is supplementary to 3. V 1 Given: m RV 5 (m 1) Prove: VS bisects RV.

13 0 Name class date Statements Reasons 1. m RV 5 (m 1) 1. Given. a.. ngle ddition Postulate 3. (m 1) 5 m b. m 4. m 1 5 m 4. Subtraction Property of Equality 5. 1 > 5. c. 6. d. 6. e. 65. ritique the reasoning of others. student says that the statement below can be justified by the definition of complementary angles. If and are both complementary to, then >. Is the student s reasoning correct? Explain. 67. Describe the relationship between each pair of angles a. 1 and 5 b. 4 and 5 c. 7 and Lesson Use appropriate tools strategically. Use the protractor shown. d. 4 and D In the diagram shown, lines, and m are parallel l m p a. Find m. b. Find m D. c. Find m D. d. Suppose P bisects angle D. t what degree measure will P lie on the protractor? Which pair of angles does NO represent corresponding angles? a. 1 and 4 b. 5 and 7 c. 6 and 8 d. and 4 13

14 69. ttend to precision. Suppose that 1 and are same-side interior angles formed by two parallel lines cut by a transversal, and that m 1 5 7x 4 and m 5 0x 5. a. What is the value of x? Lesson Use the diagram shown. t 1 m 3 4 b. What is m 1? n c. What is m? a. Suppose m What is m 3 so that m n? d. Explain how you found your answers. b. Suppose m What is m 4 so that m n? 70. omplete the proof that if parallel lines are cut by a transversal, then same-side exterior angles are supplementary. c. Suppose m What is m 6 so that m n? t m 1 n d. Suppose m What is m 1 so that m n? Given: m n Prove: m 7 1 m Statements 1. m n 1. a.. 3 > 7, 1 > 5. b. 3. m 3 5 m 7, m 1 5 m 5 Reasons 3. If two angles are congruent, then they have the same measure. 4. m 3 1 m c. 5. d. 5. Substitution Property of Equality 7. Use the diagram shown t 1 m 3 4 Suppose m 3 5 5x 1 11 and m x 1 1. What must the value of x be in order for line m to be parallel to line n? n 14

15 73. Make use of structure. Use the diagram below. Determine which pair of lines, if any, must be parallel for each statement to be true. m n p 74. onstruct viable arguments. omplete the proof of the onverse of the orresponding ngles heorem. F Q P E D q Given: EP > EQD Prove: D a. > 4 b. is supplementary to 13. c. > 13 d. 5 > 10 Statements Reasons 1. a. 1. Given. b.. Definition of congruent angles 3. m EP 1 m PQ 3. Linear Pair Postulate m EQD 1 m PQ 4. c EQD and PQ are supplementary. 5. Definition of supplementary angles. 6. d. 6. onverse of Same- Side Interior ngles Postulate 75. student found that m 5 89 in the diagram shown. Which angle must have a measure of 91 in order for m and n to be parallel? m n e. 5 > t f. > 4 a. 3 b. 5 c. 6 d. 7 15

16 Lesson In the diagram shown, PQ. If P 5 Q, which statement is true? 78. In the diagram shown, line m is parallel to line n, and point P is between lines m and n. n P P m Q. 5. PQ is the perpendicular bisector of.. is the perpendicular bisector of PQ. D. P is the perpendicular bisector of. a. Determine the number of rays with endpoint P that are perpendicular to line n. Explain your answer. b. hink about a ray with endpoint P that is perpendicular to line m. How is this ray related to the ray from Part a? 77. Reason quantitatively. Suppose p is the perpendicular bisector of RS in the diagram shown. R p 79. omplete the proof. m n P Q D S a. If R 5 5x 1 7 and RS 5 15x 1, what is the value of x? b. If R 5 7x 3 and S 5 8, what is RS? c. If RS 5 18 and S 5 3x 1, what is the value of x? d. Suppose Q forms a 35 angle with R. What is the measure of the angle formed by rays Q and S? Given: > m is the bisector of. n is the bisector of D. Prove: P > DQ Statements 1. >, m is the bisector of, and n is the bisector of D. Reasons 1. Given. 5 D. a Definition of 3. P 5 bisector 1 4. b. 4. DQ 5 D 5. P 5 DQ 5. Substitution 6. c. 6. d. 16

17 80. Express regularity in repeated reasoning. In this diagram, 5 x 1 3, 5 3x 1, 5 x 1 5, D 5 4x 1 1, and m D 5 41x 1 8. If x 5, which segment is the perpendicular bisector of the other? Explain your reasoning MN has a slope of and PQ 4 has a slope of. 5 5 re the lines parallel, perpendicular, or neither? Justify your answer. D 84. PQ contains the two points (0, 3) and (5, 7). he slope of RS is 1. re the two lines parallel, perpendicular, or neither? Justify your answer. Lesson ttend to precision. Use the ordered pairs (3, 7), (, 4), (0, 5), and D(10, 0). a. Find the slope of. 85. Make use of structure. For rectangle D, two vertices are (, 3) and (4, 6). Find the slopes of, D, and D. Explain your answer. b. Find the slope of D. c. Find the slope of any line parallel to. Lesson onsider the equation x 3y a. Write the equation in slope-intercept form. d. Find the slope of any line perpendicular to D. b. Identify the slope and y-intercept of the line. 8. Use the three ordered pairs X(1, 0), Y(10, 3), and Z(15, 4). Which of the following statements NNO be true? a. here is a line through Z that is parallel to XY. b. here is a line through Z that is perpendicular to XY. c. here is a line through Z that is the same line as XY. d. here is a line through Z that intersects XY. c. nother point on the line is (1, ). Use that ordered pair to write an equation for the line in point-slope form. 17

18 87. Use the two ordered pairs (, 9) and (0, 1). a. Suppose. Find the slope of. b. Write an equation in point-slope form for. 89. Reason abstractly. Suppose you are given two ordered pairs and. Explain how to write the equation of a line parallel to through a given point. c. Write an equation for in slope-intercept form. d. Write an equation for in point-slope form. 88. Which of the following is NO an equation for a line perpendicular to y 5 3 x 1? 3. y 5 x 16. 3x 1 y y 5 6x 3 D. y5 x Model with mathematics. segment has endpoints P(4, 5) and Q(, 1). Find the equation, in slope-intercept form, of the perpendicular bisector of PQ. Explain your solution. 18