Geometry Quarter 1 Test - Study Guide.

Transcription

1 Name: Geometry Quarter 1 Test - Study Guide. 1. Find the distance between the points ( 3, 3) and ( 15, 8). 2. Point S is between points R and T. P is the midpoint of. RT = 20 and PS = 4. Draw a sketch to show the relationship between the specified segments. Find ST. 3. Find AB and BC if BC = 7x 13, AB = 4x + 26, B is the midpoint of. 4. Find the coordinates of the midpoint of the segment with the given pair of endpoints: J(6, 6); K(2, 4) 5. Find the measures of and if bisects. The measure of is. Draw a sketch that shows the given information. 6. In the figure shown, m AED = 117. True or False: and are adjacent angles and. 7. a. Name a pair of vertical angles in the figure: b. Name an angle supplementary to in the figure. 8. Complete the table. n nth number 1 3 5??? 9. Identify the hypothesis and conclusion of the statement. If yesterday was Saturday, then tomorrow is Monday. 10. "If, then I will go to the game." What is the underlined portion called in this conditional statement? 11. Is the statement true or false? Explain your reasoning. Perpendicular lines always intersect at right angles. 12. Refer to the following statement: Two lines are perpendicular if and only if they intersect to form a right angle. A. Is this a biconditional statement? B. Is the statement true? 13. Write the converse of the true statement and decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample: If four points are not coplanar, then they are not collinear.

2 14. True or False: True biconditional statements make good definitions Use the given endpoint R (4, 6) and midpoint M ( 7,8) of RS to find the coordinates of the other endpoint S. State the postulate indicated by the diagram Identify the property that makes the statement true. If m 1 + m 2 = 25 and m 1 = 9, then. 19. True or False: If two angles are complements of the same angle, then they must be equal in measure and 2 are complementary, and 2 and 3 form a linear pair. If m 1 =, what is m 3? Explain your reasoning. 21. and. If =, what is? Justify your answer. 22. What is the distance from point A (1, 2) to line d with equation y = x You and your friend each drew 3 points. Your 3 points lie in only 1 plane, but your friend's 3 points lie in more than one plane. Explain how this is possible. 24. R, S, and T are collinear. S is between R and T. RS = 2w + 1, ST = w 1, and RT = 18. Use the Segment Addition Postulate to solve for w. Then determine the length of 25. If AB = 17 and AC = 32, find the length of.

3 26. In a class there are: (Hint: Draw a Venn diagram.) 8 students who play football and hockey 7 students who do not play football or hockey 13 students who play hockey 19 students who play football How many students are there in the class? 27. Find the length of 28. a. What are the approximate lengths of and? b. Find the midpoint of each segment. What is true about the midpoints? 29. m SQR = ( ) and m PQR = ( ) and m SQP = 70. Find m SQR and m PQR. 30. Open-ended: Write three facts you observe in the figure below. 31. In the figure (not drawn to scale), bisects, and. Solve for x and find 32. Which is not a possible value for y in the figure below? a) 96 c) 141 b) 76 d) 89

4 33. Give a counterexample to the following conjecture. All mammals cannot fly. 34. Use inductive reasoning to find the next two numbers in each pattern. 2, 3, 5, 8,, 35. Decide whether inductive or deductive reasoning is used to reach the conclusion. If you live in Nevada and are between the ages of 16 and 18, then you must take driver s education to get your license. Anthony lives in Nevada, is 16 years old, and has his driver s license. Therefore, Anthony took driver s education. 36. Given that: i. Tawana bought a new computer. ii. All computers depreciate in value. What conclusion can be logically deduced? 37. Use deductive reasoning to show that the product of two even numbers is even. 38. For each set of true statements, make a valid conclusion, if possible. If none can be made, write "no valid conclusion." a. If a triangle has two acute angles, then the third angle may be obtuse. Triangle ABC has two acute angles. b. If I go to school, I'll ride the school bus. If I ride the school bus, I'll sit next to Kerry on the bus. I went to school. c. If a polygon is a regular, then it is convex. Polygon RSTU is convex. 39. Use symbolic notation to write the converse, inverse and contrapositive of the statement: p q Write a two-column proof. 40. Given: bisects Prove: 41. Given: are vertical angles; form a linear pair Prove: are supplementary angles 42. Sketch an example of lines intersected by a transversal. Identify a pair of : alternate interior angles, corresponding angles, consecutive interior angles and alternate exterior angles.

5 43. Is the statement true or false? Explain your reasoning. Intersecting lines are always perpendicular. 44. Write the converse of the true statement and decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample: If two lines are parallel, then they never intersect. 45. Use the figure to name a pair of: parallel lines, skew lines, perpendicular lines, parallel planes and perpendicular planes. 46. Find m 1 in the figure below. PQ and RS are parallel. 47. Explain the difference between the Alternate Interior angles theorem and Alternate Interior Angles Converse. 48. Given: Lines q and r are parallel. Transversal t intersects lines q and r. Prove: are supplementary. 49. In the figure below, if l and k are parallel lines, what is the value of x and y? 50. Use the given angle measures to decide whether lines a and b are parallel.,

6 51. Calculate the slope of the line. Does it matter which points are used? Why? 52. Writing: Explain the difference between a horizontal line and a vertical line in terms of slope. Give an example of an equation for each type of line. 53. A line L1 has slope -1. State whether the line that passes through ( 3, 2) and (5, 10) is parallel or perpendicular to line L Decide whether Line 1 and Line 2 are parallel, perpendicular, or neither. Line 1 passes through (10, 7) and (13, 9) Line 2 passes through ( 4, 3) and ( 1, 5) 55. Find the slope of the line through the points ( 1, 3) and ( 1, 7). 56. Write the slope-intercept form of the equation of the line passing through the point ( 2, 5) and parallel to the line 57. Write the slope-intercept form of the equation of the line passing through the point (5, 4) and perpendicular to the line 58. Write an equation in slope-intercept form 59. Line l passes through (1, 1) and ( 2, 8). of the line shown. Graph the line perpendicular to l that passes through ( 2, 2). 60. Find the shortest distance between y = x 1 and y = x + 3.

7 Geometry Quarter 1 Test - Study Guide. Answer Section units AB = 78, BC = (4, 1) 5., 6. True 7. a. b. 8. 7, 9, hypothesis: yesterday was Saturday, conclusion: tomorrow is Monday 10. The hypothesis 11. True; definition of perpendicular lines. 12. A. yes B. yes 13. If four points are not collinear, then they are not coplanar. - False Counterexample: Four points may not be in the same line, but they may be in the same plane. 14. True 15. (-18, 22) 16. Through any three noncollinear points there exists exactly one plane. 17. If two points lie in a plane, then the line containing them lies in the plane. 18. Substitution Property of Equality 19. True 20.. Since 1 and 2 are complementary, the sum of their measures is. So, m 2 =. Since 2 and 3 form a linear pair, they are supplementary and the sum of their measures is. So, m 3 = =. 21. m = and =. and are supplementary so =. by the Angle Addition Postulate. by the Transitive Property. and by the Definition of Congruence. Using the Substitution Property,. Since, by the Division Property of Equality

8 23. An infinite number of planes intersect any one line so my friend's points are collinear. There is only 1 plane through any 3 points not on the same line so my 3 points are not collinear students a. = = = = = = = = b. The midpoint of = =. The midpoint of = =. The midpoints are the same. 29. m SQR = 20 and m PQR = Sample answer: 1) m WVX = m YVX; 2) m XVY m WVY; 3) bisects WVY , b) Answers will vary. For example, bats are mammals that can fly , Deductive it is based on logic and order. 36. Tawana's computer will depreciate in value. 37. Let 2x and 2y represent any two even numbers. Their product is (2x)(2y), or 4xy. Since 2 is a factor of 4 and thus of 4xy, (2x)(2y) is even. 38. a. The third angle of may be obtuse. b. I sat next to Kerry on the bus. c. no valid conclusion 39. Converse: q p; Inverse: p q; Contrapositive: q p. 40.

9 Sample Answer: Alternate interior angles: 4 and 6; Corresponding angles : 1 and 5; Consecutive interior angles: 3 and 6; Alternate exterior angles: 1 and False: Lines can intersect at any angle. 44. If two lines never intersect, then they are parallel. False. Counterexample: Two skew lines never intersect, but they are not in the same plane and therefore are not parallel. 45. Sample Answer: Parallel lines: AB and DC ; Skew lines: BC and GH ; Perpendicular lines: AD and CD ; Parallel planes: AED and BFC ; Perpendicular planes: CDH and BCG. 47. Alt. Int. Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alt. Int. Angles Converse: If two lines are cut by a transversal, so the alternate interior angles are congruent, then the lines are parallel x y No ; No - the slope ratio is the same for any two points on a line.

10 52. Sample answer: A line having a slope of 0 is a horizontal line, that is, a line that crosses the y-axis and is parallel to the x-axis; example: y = 3. A line with an undefined slope is a vertical line, that is, a line that crosses the x-axis and is parallel to the y-axis; example: x = perpendicular 54. parallel 55. undefined 56. y = 57. y = 58. y = Since y = x 1 and y = x + 3 have the same slope, they are parallel. The shortest distance between parallel lines can be found using a perpendicular line. Find the equation of a line perpendicular to y = x 1. Find the points where the perpendicular line intersects y = x 1 and y = x + 3. The distance between these two points is the shortest distance between the lines.