Additional Practice
Additional Practice 5. Name the radii of circle A.
ACTIVITY 1.1 1. Identify the type of angle formed by the hour hand and a minute hand on the face of a clock at each of the times listed below. a. b. c. d.
UNIT 1
2:00 5:00 9:00 6:00
6. Name the diameter(s) of circle A. 7. Name the chord(s) of circle A.
ACTIVITY 1.2 8. Use inductive reasoning to determine the next two terms in the sequence. 1 , __ 1 __ 1 __ 1 a. __ 3 5, 7, 9, … b. 2, B, 4, D, 8, F, 16, …
2. Use this diagram for parts a–c below. A
B
D
E
9. Write the first five terms of two different sequences that have 6 as the second term.
C
10. Generate a sequence using this description: the first term of the sequence is 1, the second term is 2 and each term after the second is the sum of the two previous terms.
F ‹___›
a. Give another correct name for AB. b. Name the lines that are parallel. c. Name the lines that are perpendicular.
11. Use this picture pattern for parts a–c below.
3. The measure of ∠P is 37°. a. What is the measure of an angle that is complementary to ∠P? b. What is the measure of an angle that is supplementary to ∠P?
© 2010 College Board. All rights reserved.
4. In the diagram below, lines WT and XC intersect at point A, and m∠XAW = 23°. W
S
12. Use expressions for odd integers to confirm that the sum of three odd numbers is odd.
C
A
X
ACTIVITY 1.3
T
13. Identify the property that justifies this statement:
a. Name another angle that has measure of 23°. b. Name a pair of supplementary angles.
D
14. Write a two-column proof:
15. Write the following statement in if-then form.
B
A
If 2x = 6, then x = 3. Given: 4(x - 1) = 2x + 10 Prove: x = 7
Use circle A for Items 5–7. C
a. Draw the next shape in the pattern. b. Write a sequence of numbers that could be used to express the pattern. c. Verbally describe the pattern of the sequence.
An angle whose measure is 37° is acute. F
E
Geometry, Unit 1 • Proof, Parallel and Perpendicular Lines
1
Additional Practice
UNIT 1
Use the following statement for Items 16–19. If two lines intersect, then they are perpendicular. 16. State the hypothesis.
In Items 31–33, use the figure and given information to determine the answers. Show all work. 31. Given: m∠1 = (6x + 40)° and m∠2 = (9x + 4)°
17. Write the inverse. 18. Write the converse. 19. Write the contrapositive.
1
3
2
20. Write a counterexample for this false conditional statement. If two lines intersect, then they are perpendicular. 21. Give an example of a true statement that has a false converse.
m∠2 =
; m∠3 =
.
32. Given: EA &&' ⊥ ED &&', EB &' bisects ∠AEC, m∠CEB = (4x + 3)°, and m∠DEC = (3x - 4)°
ACTIVITY 1.4
A
Construct a truth table for each compound statement.
B
22. p∨ ∼ q
C
23. (p ∧ q) % (∼q) E
24. (p ∨ q) % (p ∧ q)
D
25. (p % q) ∧ (p ∨ q)
ACTIVITY 1.5 26. Given: point E is between points G and T, GE = 2x, ET = 3x - 1, and GT = 14. Find the value of x.
; m∠AEB =
33. Given: m∠2 = (3x)°, m∠3 = (2x + 13)°, and m∠4 = (5x + 7)°
___
27. If M is the midpoint of SB and BM = x + 5 and ? . SM = 4x - 1, then SB = 28. Given point R is in the interior of ∠PAT, m∠PAR = (5x + 12)°, m∠TAR = (3x + 10)°, ? and m∠PAT = 78°. m∠PAR = . 29. ∠A and ∠B are complementary. If m∠A = (4x + 6)°, and m∠B = (5x + 3)°, then x = ? . 30. EG &&' bisects ∠DEF. If m∠DEG = (3x + 7)° and m∠DEF = (4x + 34)°, then m∠GEF = ? .
2
SpringBoard® Mathematics with Meaning™ Geometry
1 2
3
4
a. x = b. m∠1 = c. Is ∠2 complementary to ∠3? Explain.
© 2010 College Board. All rights reserved.
x=
Additional Practice ACTIVITY 1.6 Use the figure for Items 34–47. In the diagram, t ! r and line m is not parallel to line n.
UNIT 1
Determine whether each statement is true or false. 43. ∠5 is supplementary to ∠9. 44. m∠3 = m∠7
t
r
45. ∠16 ) ∠10 46. ∠4 ) ∠3
8
1 2
47. m∠9 + m∠10 = 180°
7 6 4 5
ACTIVITY 1.7
3
n m
14 15 13 12
Use this figure for Items 1–8. Lines AM, RE, and NG intersect at point T. Write the definition, postulate, property, or theorem that justifies the statement.
16 9 11 10
R
34. List all pairs of alternate interior angles.
1 2
35. List all pairs of same side interior angles. 36. If m∠7 = 110°, then m∠8 =
?
T 4
.
?
© 2010 College Board. All rights reserved.
40. If m∠3 = 150°, then m∠5 =
.
41. If m∠10 = (8x + 7)° and m∠12 = (5x + 55)°, then x = and m∠16 = . 42. If m∠15 = 85° and m∠2 = 45°, determine the measure of each angle. Justify each answer. a. b. c. d. e.
m∠4 = m∠13 = m∠16 = m∠7 = m∠11 =
M E
.
39. If m∠8 = (5x - 2)° and m∠4 = (3x + 24)°, then x = and m∠7 = . ?
U
3
N
37. If m∠16 = (5x - 8)° and m∠15 = (2x + 13)°, then x = and m∠13 = . 38. If m∠15 = 84°, then m∠11 =
G
A
48. If TG &&' bisects ∠RTU, then ∠2 ) ∠3. 49. AT + TM = AM 50. ∠2 ) ∠4 ___
___
____
51. If NT ) GT, then T is the midpoint of NG. ‹___›
‹___›
52. If m∠ATN = 90°, then AT ⊥ TN. 53. m∠2 + m∠GTE = 180° 54. If m∠1 + m∠2 = m∠2 + m∠3, then m∠1 = m∠3. 55. If m∠2 + m∠3 = 90°, then ∠2 is complementary to ∠3.
Geometry, Unit 1 • Proof, Parallel and Perpendicular Lines
3
Additional Practice
UNIT 1
56. Supply the missing statements and reasons. Given: a * b, c * d Prove: m∠1 = m∠4 a 1
ACTIVITY 1.9 61. Calculate the distance between points A(-7, 2) and B(5, -3).
b
62. Calculate the distance between points M(-3, -8) and N(4, 13).
2 3
4
c
d
63. Find and explain the error(s) made in the following calculation of distance. Then, fix the error(s) and determine the correct answer. Find the length of the segment with endpoints (4, 1) and (7, 2). ________________
√(4 + 7)2 - (1 + 2)2 _______
Statements
Reasons
= √112 - 32
1. a * b
1.
= √121 - 9
2. ∠1 )
2.
= √112
3.
3. Vertical angles are congruent.
4. c * d
4.
5. ∠3 ) ∠4
5.
6. ∠1 )
6.
7.
7.
57. Determine the slope of a line that contains the points (-3, -2) and (3, 2). 58. One line passes through the points (-1, -2) and (1, 2); another line passes through the points (-2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither? Justify your answer. 59. One line passes through the points (0, -4) and (-1, -7); another line passes through the points (3, 0) and (-3, 2). Are these lines parallel, perpendicular, or neither? Justify your answer. 60. Write the equation of a line that has y-intercept 1 and is parallel to the line y = 4x - 3.
4
SpringBoard® Mathematics with Meaning™ Geometry
____
64. Determine the coordinates of the midpoint of the segment with endpoints (-7, 2) and (5, -3). 65. Determine the coordinates of the midpoint of the segment with endpoints (7, -2) and (-5, 5). 66. Find and explain the error(s) made in the following calculation of midpoint. Then, fix the error(s) and determine the correct answer. Find the midpoint of the segment with endpoints (1, 5) and (6, -4). 1 + 5 _____ 6-4 (_____ 2 , 2 ) 6 , __ 2 = __ 2 2 = (3, 1)
( )
© 2010 College Board. All rights reserved.
ACTIVITY 1.8
_______
ACTIVITY 1.1 1. Identify the type of angle formed by the hour hand and a minute hand on the face of a clock at each of the times listed below. a. b. c. d.
UNIT 1
2:00 5:00 9:00 6:00
6. Name the diameter(s) of circle A. 7. Name the chord(s) of circle A.
ACTIVITY 1.2 8. Use inductive reasoning to determine the next two terms in the sequence. 1 , __ 1 __ 1 __ 1 a. __ 3 5, 7, 9, … b. 2, B, 4, D, 8, F, 16, …
2. Use this diagram for parts a–c below. A
B
D
E
9. Write the first five terms of two different sequences that have 6 as the second term.
C
10. Generate a sequence using this description: the first term of the sequence is 1, the second term is 2 and each term after the second is the sum of the two previous terms.
F ‹___›
a. Give another correct name for AB. b. Name the lines that are parallel. c. Name the lines that are perpendicular.
11. Use this picture pattern for parts a–c below.
3. The measure of ∠P is 37°. a. What is the measure of an angle that is complementary to ∠P? b. What is the measure of an angle that is supplementary to ∠P?
© 2010 College Board. All rights reserved.
4. In the diagram below, lines WT and XC intersect at point A, and m∠XAW = 23°. W
S
12. Use expressions for odd integers to confirm that the sum of three odd numbers is odd.
C
A
X
ACTIVITY 1.3
T
13. Identify the property that justifies this statement:
a. Name another angle that has measure of 23°. b. Name a pair of supplementary angles.
D
14. Write a two-column proof:
15. Write the following statement in if-then form.
B
A
If 2x = 6, then x = 3. Given: 4(x - 1) = 2x + 10 Prove: x = 7
Use circle A for Items 5–7. C
a. Draw the next shape in the pattern. b. Write a sequence of numbers that could be used to express the pattern. c. Verbally describe the pattern of the sequence.
An angle whose measure is 37° is acute. F
E
Geometry, Unit 1 • Proof, Parallel and Perpendicular Lines
1
Additional Practice
UNIT 1
Use the following statement for Items 16–19. If two lines intersect, then they are perpendicular. 16. State the hypothesis.
In Items 31–33, use the figure and given information to determine the answers. Show all work. 31. Given: m∠1 = (6x + 40)° and m∠2 = (9x + 4)°
17. Write the inverse. 18. Write the converse. 19. Write the contrapositive.
1
3
2
20. Write a counterexample for this false conditional statement. If two lines intersect, then they are perpendicular. 21. Give an example of a true statement that has a false converse.
m∠2 =
; m∠3 =
.
32. Given: EA &&' ⊥ ED &&', EB &' bisects ∠AEC, m∠CEB = (4x + 3)°, and m∠DEC = (3x - 4)°
ACTIVITY 1.4
A
Construct a truth table for each compound statement.
B
22. p∨ ∼ q
C
23. (p ∧ q) % (∼q) E
24. (p ∨ q) % (p ∧ q)
D
25. (p % q) ∧ (p ∨ q)
ACTIVITY 1.5 26. Given: point E is between points G and T, GE = 2x, ET = 3x - 1, and GT = 14. Find the value of x.
; m∠AEB =
33. Given: m∠2 = (3x)°, m∠3 = (2x + 13)°, and m∠4 = (5x + 7)°
___
27. If M is the midpoint of SB and BM = x + 5 and ? . SM = 4x - 1, then SB = 28. Given point R is in the interior of ∠PAT, m∠PAR = (5x + 12)°, m∠TAR = (3x + 10)°, ? and m∠PAT = 78°. m∠PAR = . 29. ∠A and ∠B are complementary. If m∠A = (4x + 6)°, and m∠B = (5x + 3)°, then x = ? . 30. EG &&' bisects ∠DEF. If m∠DEG = (3x + 7)° and m∠DEF = (4x + 34)°, then m∠GEF = ? .
2
SpringBoard® Mathematics with Meaning™ Geometry
1 2
3
4
a. x = b. m∠1 = c. Is ∠2 complementary to ∠3? Explain.
© 2010 College Board. All rights reserved.
x=
Additional Practice ACTIVITY 1.6 Use the figure for Items 34–47. In the diagram, t ! r and line m is not parallel to line n.
UNIT 1
Determine whether each statement is true or false. 43. ∠5 is supplementary to ∠9. 44. m∠3 = m∠7
t
r
45. ∠16 ) ∠10 46. ∠4 ) ∠3
8
1 2
47. m∠9 + m∠10 = 180°
7 6 4 5
ACTIVITY 1.7
3
n m
14 15 13 12
Use this figure for Items 1–8. Lines AM, RE, and NG intersect at point T. Write the definition, postulate, property, or theorem that justifies the statement.
16 9 11 10
R
34. List all pairs of alternate interior angles.
1 2
35. List all pairs of same side interior angles. 36. If m∠7 = 110°, then m∠8 =
?
T 4
.
?
© 2010 College Board. All rights reserved.
40. If m∠3 = 150°, then m∠5 =
.
41. If m∠10 = (8x + 7)° and m∠12 = (5x + 55)°, then x = and m∠16 = . 42. If m∠15 = 85° and m∠2 = 45°, determine the measure of each angle. Justify each answer. a. b. c. d. e.
m∠4 = m∠13 = m∠16 = m∠7 = m∠11 =
M E
.
39. If m∠8 = (5x - 2)° and m∠4 = (3x + 24)°, then x = and m∠7 = . ?
U
3
N
37. If m∠16 = (5x - 8)° and m∠15 = (2x + 13)°, then x = and m∠13 = . 38. If m∠15 = 84°, then m∠11 =
G
A
48. If TG &&' bisects ∠RTU, then ∠2 ) ∠3. 49. AT + TM = AM 50. ∠2 ) ∠4 ___
___
____
51. If NT ) GT, then T is the midpoint of NG. ‹___›
‹___›
52. If m∠ATN = 90°, then AT ⊥ TN. 53. m∠2 + m∠GTE = 180° 54. If m∠1 + m∠2 = m∠2 + m∠3, then m∠1 = m∠3. 55. If m∠2 + m∠3 = 90°, then ∠2 is complementary to ∠3.
Geometry, Unit 1 • Proof, Parallel and Perpendicular Lines
3
Additional Practice
UNIT 1
56. Supply the missing statements and reasons. Given: a * b, c * d Prove: m∠1 = m∠4 a 1
ACTIVITY 1.9 61. Calculate the distance between points A(-7, 2) and B(5, -3).
b
62. Calculate the distance between points M(-3, -8) and N(4, 13).
2 3
4
c
d
63. Find and explain the error(s) made in the following calculation of distance. Then, fix the error(s) and determine the correct answer. Find the length of the segment with endpoints (4, 1) and (7, 2). ________________
√(4 + 7)2 - (1 + 2)2 _______
Statements
Reasons
= √112 - 32
1. a * b
1.
= √121 - 9
2. ∠1 )
2.
= √112
3.
3. Vertical angles are congruent.
4. c * d
4.
5. ∠3 ) ∠4
5.
6. ∠1 )
6.
7.
7.
57. Determine the slope of a line that contains the points (-3, -2) and (3, 2). 58. One line passes through the points (-1, -2) and (1, 2); another line passes through the points (-2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither? Justify your answer. 59. One line passes through the points (0, -4) and (-1, -7); another line passes through the points (3, 0) and (-3, 2). Are these lines parallel, perpendicular, or neither? Justify your answer. 60. Write the equation of a line that has y-intercept 1 and is parallel to the line y = 4x - 3.
4
SpringBoard® Mathematics with Meaning™ Geometry
____
64. Determine the coordinates of the midpoint of the segment with endpoints (-7, 2) and (5, -3). 65. Determine the coordinates of the midpoint of the segment with endpoints (7, -2) and (-5, 5). 66. Find and explain the error(s) made in the following calculation of midpoint. Then, fix the error(s) and determine the correct answer. Find the midpoint of the segment with endpoints (1, 5) and (6, -4). 1 + 5 _____ 6-4 (_____ 2 , 2 ) 6 , __ 2 = __ 2 2 = (3, 1)
( )
© 2010 College Board. All rights reserved.
ACTIVITY 1.8
_______
