First Semester (August - December) Final Review

Transcription

1 Name: lass: ate: I: First Semester (ugust - ecember) Final Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name three points that are collinear. a., G, F c. J, G, F b.,, H d. J,, G 2. re points,, and F coplanar? Explain. a. Yes; they all lie on plane P. b. No; they are not on the same line. c. Yes; they all lie on the same face of the pyramid. d. No; three lie on the same face of the pyramid and the fourth does not. 1

2 Name: I: Find the measurement of the segment. 4. QT = 0.4 in., QV = 1.9 in. 3. PR = 18.8 mm, RS = 13.7 mm PS =? a mm b. 5.1 mm c mm d mm TV =? a. 1.4 in. b. 2.3 in. c. 1.7 in. d. 1.5 in. 5. Find the value of the variable and GH if H is between G and I. GI = 5b + 1,HI = 4b 5,HI = 7 a. b = 1.2, GH = 6.8 c. b = 3, GH = 9 b. b = 1.22, GH = 7.11 d. b = 3, GH = 16 Use the number line to find the measure. 6. PH a. 4.5 b. 8 c. 9 d RK a. 2 b. 5 c. 7 d Use the istance Formula to find the distance between each pair of points. a. 50 b. 34 c. 6 d. 4 2

3 Name: I: 9. In the figure, GK bisects FGH. a. 50 b. 4 c. 34 d. 6 Find the coordinates of the midpoint of a segment having the given endpoints. 11. If m FGK = 3v 4 and m KGH = 2v + 7, find x. a. 33 b. 58 c. 11 d. 29 In the figure, KJ and KL are opposite rays. 1 2 and KM bisects NKL. 10. QÊ Ë 1, 3 Ê 11, 5 ˆ a. Ê 1, 8 ˆ b. Ê 10, 8 ˆ c. Ê 6, 1 ˆ d. Ê 5, 4 ˆ 12. What bisects JKN? a. P b. KP c. PK d Which is NOT true about KM? a. MKJ is acute. b. 3 MKL c. Point M lies in the interior of LKN. d. It is an angle bisector. 3

4 Name: I: 14. If m JKM = 5x + 18 and m 4 = x, what is m 4? a. 153 b. 33 c. 27 d. 12 Use the figure to find the angles. 15. Name two acute vertical angles. a. KQL, KQM b. KQL, IQH c. GQI, IQM d. HQL, IQK 16. Name a pair of obtuse adjacent angles. a. KQG, HQM b. GQL, LQJ c. GQI, IQM d. HQG, IQH 17. Name a linear pair. a. KQG, HQM b. GQL, LQJ c. GQI, IQM d. LQG, KQM 18. Name an angle supplementary to MQI. a. IQG b. GQL c. MQK d. IQH 19. The measures of two complementary angles are 12q 9 and 8q Find the measures of the angles. a. 42, 48 b c d. 96, Two angles are supplementary. One angle measures 26 o more than the other one. Find the measure of the two angles. a. 77, 103 b. 32, 58 c. 167, 193 d. 76, Find m Y if m Y is six more than three times its complement. a b c. 21 d The measure of an angle s supplement is 24 less than twice the measure of the angle. Find the measure of the angle and its supplement. a. 38, 52 b. 52, 38 c. 68, 112 d. 112, Rays and are perpendicular. Point lies in the interior of. If m = 4r 7 and m = 8r + 1, find m and m. a. 8, 90 b. 55, 125 c. 21, 69 d. 25, 65 4

5 Name: I: Find the length of each side of the polygon for the given perimeter. 26. P = 49 units. Find the length of each side. 24. P = 48 cm a. 8 cm b. 6 cm c. 4 cm d. 3 cm 25. P = 60 in. Find the length of each side. a. 11 in., 20 in., 35 in. b. 10 in., 18.5 in., 31.5 in. c. 12 in., 21.5 in., 38.5 in. d. 10 in., 15 in., 35 in. a. 18 units, 18 units, 13 units b units, units, units c. 27 units, 27 units, 22 units d. 26 units, 26 units, 21 units Make a conjecture about the next item in the sequence , 4, 16, 64, 256 a b c d , 8, 32, 30, 120 a. 122 b. 480 c. 488 d. 116 etermine whether the conjecture is true or false. Give a counterexample for any false conjecture. 29. Given: points,,, and onjecture:,,, and are coplanar. a. False; the four points do not have to be in a straight line. b. True c. False; two points are always coplanar but four are not. d. False; three points are always coplanar but four are not. 5

6 Name: I: 30. Given: point is in the interior of. onjecture: a. False; m may be obtuse. b. True c. False; just because it is in the interior does not mean it is on the bisecting line. d. False; m + m = Given: points R, S, and T onjecture: R, S, and T are coplanar. a. False; the points do not have to be in a straight line. b. True c. False; the points to not have to form right angles. d. False; one point may not be between the other two. 32. Given:, E are coplanar. onjecture: They are vertical angles. a. False; the angles may be supplementary. b. True c. False; one angle may be in the interior of the other. d. False; the angles may be adjacent. 33. Given: F is supplementary to G and G is supplementary to H. onjecture: F is supplementary to H. a. False; they could be right angles. b. False; they could be congruent angles. c. True d. False; they could be vertical angles. Write the converse of the conditional statement. etermine whether the converse is true or false. If it is false, find a counterexample. 34. If you have a dog, then you are a pet owner. a. If you are a pet owner, then you have a dog. True b. dog owner owns a pet. True c. If you are a pet owner, then you have a dog. False; you could own a hamster. d. If you have a dog, then you are a pet owner. True In the figure below, points,,, and F lie on plane P. State the postulate that can be used to show each statement is true. 35. and are collinear. a. If two points lie in a plane then the entire line containing those points lies in that plane. b. Through any two points there is exactly one line. c. If two lines intersect then their intersection is exactly one point. d. line contains at least two points. 6

7 Name: I: 36. Line contains points and. a. If two lines intersect then their intersection is exactly one point. b. If two points lie in a plane then the entire line containing those points lies in that plane. c. line contains at least two points. d. Through any two points there is exactly one line. Refer to the figure below. 40. In the figure, m RPZ = 95 and TU Find the measure of angle WSP. Ä RQ Ä VW. a. 85 b. 75 c. 95 d Name all planes intersecting plane I. a., G, I, FGH b., F, HGF c., GFI, G, GF d., G, F 41. In the figure, Ä. Find x and y. 38. Name all segments parallel to GF. a.,, HI b.,, HI c., HI d., 39. Name all segments skew to. a. FI,, F, I b. FG, GH, HI, FI c.,, G, H d. GF, HI, I, F a. x = 32, y = 140 b. x = 140, y = 52 c. x = 52, y = 140 d. x = 38, y = 154 7

8 Name: I: 42. In the figure, p Ä q. Find m 1. etermine the slope of the line that contains the given points. 43. T Ê Ë 6, 3 Ê 8, 8 ˆ a. 5/2 b. -2/5 c. 2/5 d. 0 a. m 1 = 61 b. m 1 = 35 c. m 1 = 55 d. m 1 = 64 etermine whether WX and YZ 44. WÊ Ë 0, 3 Ê Ë 5 Ê Ë 5 Ê 1, 2 ˆ a. parallel b. perpendicular c. neither are parallel, perpendicular, or neither. 8

9 Name: I: Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer a. c Ä d; congruent corresponding angles b. a Ä b; congruent corresponding angles c. c Ä d; congruent alternate interior angles d. a Ä b; congruent alternate interior angles Find the measures of the sides of Δ and classify the triangle by its sides. 46. Ê Ë 5, 2 Ê Ë 2 Ê 3, 5 ˆ a. equilateral c. scalene b. isosceles d. obtuse Find each measure. 47. m 1, m 2, m 3 a. m 1 = 64, m 2 = 74, m 3 = 52 c. m 1 = 47, m 2 = 74, m 3 = 69 b. m 1 = 64, m 2 = 47, m 3 = 52 d. m 1 = 47, m 2 = 59, m 3 = 64 9

10 Name: I: 48. m 1, m 2, m 3 a. m 1 = 77, m 2 = 41, m 3 = 37 c. m 1 = 82, m 2 = 41, m 3 = 37 b. m 1 = 77, m 2 = 36, m 3 = 30 d. m 1 = 82, m 2 = 92, m 3 = m 1, m 2, m 3 a. m 1 = 74, m 2 = 129, m 3 = 101 c. m 1 = 51, m 2 = 101, m 3 = 101 b. m 1 = 46, m 2 = 129, m 3 = 129 d. m 1 = 74, m 2 = 152, m 3 = 74 Refer to the figure. ΔRM, ΔMX, and ΔXFM are all isosceles triangles. 50. What is m RM? a. 23 b. 38 c. 42 d What is m MX? a. 80 b. 38 c. 64 d What is m MX? a. 16 b. 38 c. 36 d What is m RX? a. 74 b. 68 c. 64 d

11 Name: I: 54. Triangle FJH is an equilateral triangle. Find x and y. 56. Triangle RSU is an equilateral triangle. RT bisects US. Find x and y. a. x = 7 5, y = 16 b. x = 7, y = 16 c. x = 7 5, y = 14 d. x = 7, y = Triangles and F are vertical congruent equilateral triangles. Find x and y. a. x = 1, y = 32 3 b. x = 1, y = 62 2 c. x = 1, y = 62 3 d. x = 1, y = Triangle RSU is an equilateral triangle. RT bisects US. Find x and y. a. x = 7, y = 27 b. x = 7 3, y = 27 c. x = 7 3, y = 28 d. x = 7, y = 33 a. x = 4 5, y = 9 b. x = 4 3, y = 21 c. x = 4 5, y = 21 d. x = 4 3, y = 9 etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain , 9, 10 a. Yes; the third side is the longest. b. No; the sum of the lengths of two sides is not greater than the third. c. No; the first side is not long enough. d. Yes; the sum of the lengths of any two sides is greater than the third. 11

12 Name: I: 59. n isosceles triangle has a base 9.6 units long. If the congruent side lengths have measures to the first decimal place, what is the shortest possible length of the sides? a. 4.9 b c. 4.7 d. 9.7 Solve each proportion. 61. Identify the similar triangles. Find x. 60. x + 1 x 1 = a. -3/17 b. 10/7 c. 7/10 d. -17/3 a. Δ ΔEF; x = 12 b. Δ ΔEF; x = 3 c. Δ ΔEF; x = 4 d. Δ ΔEF; x = 3 12

13 Name: I: etermine whether each pair of triangles is similar. Justify your answer a. yes; ΔEF Δ by Similarity b. yes; ΔEF Δ by Similarity c. yes; ΔEF Δ by S Similarity d. No; there is not enough information to determine similarity. Find x and the measures of the indicated parts a. No; sides are not proportional. b. yes; ΔEF Δ by SSS Similarity c. yes; ΔEF Δ by SSS Similarity d. yes; ΔEF Δ by SSS Similarity and a. x = 5, = 14, = 8 b. x = 1.6, = 0.8, = 1.4 c. x = 5, = 6, = 2 d. x = 1.6, = 7.2, = a. No; the sides are not congruent. b. yes; ΔEF Δ by SSS Similarity c. yes; ΔEF Δ by S Similarity d. yes; ΔEF Δ by SS Similarity a. x = 7, = 20 c. x = 7, = 36 b. x = 7, = 16 d. x = 7, = 4 13

14 Name: I: a. x = , = 6 c. x =, = b. x = 3 2, = 3 d. x = 14 3, = 28 = 4x E = 2x + 2 and E a. x = 5, = 20, E = 12 c. x = , =, E = b. x = 5, = 20, E = 8 d. x = 3, = 12, E = ount the number of dots in each arrangement. How many dots will be in the sixth triangular number? a. 6 c. 21 b. 15 d

15 Name: I: Short nswer etermine whether each pair of triangles is similar. Justify your answer

16 I: First Semester (ugust - ecember) Final Review nswer Section MULTIPLE HOIE 1. NS: ollinear points are points on the same line. re those points on the same line? What does collinear mean? re those points on the same line? PTS: 1 IF: verage OJ: Identify collinear points. NT: NTM GM.2 ST: 1.0 TOP: Identify collinear points. KEY: ollinear Points 2. NS: Points that lie on the same plane are said to be coplanar. Three points are always coplanar but if the fourth point is not on the same plane with the first three, they are not all coplanar. o all four points lie on the same plane? Which plane? o all four points lie on the same plane? Which plane? What does coplanar mean? PTS: 1 IF: verage OJ: Identify coplanar points and intersecting lines in space. NT: NTM GM.2 ST: 1.0 TOP: Identify coplanar points and intersecting lines in space. KEY: oplanar Points Intersecting Lines Lines in Space 3. NS: PS has the same length as PR and RS combined. id you add correctly? PS contains both PR and RS. Try adding that again. PTS: 1 IF: asic OJ: Measure segments. NT: NTM ME.2 NTM ME.2a TOP: Measure segments. KEY: Measurement Line Segments 1

17 I: 4. NS: TV is the length of QV minus the length of QT. Try subtracting that again. You need to subtract, not add. Try subtracting that again. PTS: 1 IF: asic OJ: Measure segments. NT: NTM ME.2 NTM ME.2a TOP: Measure segments. KEY: Measurement Line Segments 5. NS: Solve for b first using HI s two values. GI = GH + HI. Solve for GH. Which two segments in the question are the same? Which two segments in the question are the same? Which segment are you solving for? PTS: 1 IF: verage OJ: ompute with measures. NT: NTM NO.1 TOP: ompute with measures. KEY: Measurement ompute Measures 6. NS: The distance between two points a and b is b a or a b. You are looking for the measure, not the half measure. dd those numbers again. You are looking for the measure, not the midpoint. PTS: 1 IF: verage OJ: Find the distance between two points on a number line. NT: NTM GM.2 NTM GM.2a TOP: Find the distance between two points on a number line. KEY: istance Number Lines istance etween Two Points 2

18 I: 7. NS: The distance between two points a and b is b a or a b. You are looking for the measure, not the midpoint. You are looking for the measure, not the half measure. dd those numbers again. PTS: 1 IF: verage OJ: Find the distance between two points on a number line. NT: NTM GM.2 NTM GM.2a TOP: Find the distance between two points on a number line. KEY: istance Number Lines istance etween Two Points 8. NS: The istance Formula is d = Ê x 2 x ˆ Ê y 2 y ˆ 1 2. With distance you subtract the coordinates. e a little more precise. id you use the distance formula correctly? PTS: 1 IF: asic OJ: Find the distance between two points on a coordinate plane. NT: NTM GM.2 NTM GM.2a TOP: Find the distance between two points on a coordinate plane. KEY: istance oordinate Plane istance etween Two Points 9. NS: The istance Formula is d = Ê x 2 x ˆ Ê y 2 y ˆ 1 2. With distance you subtract the coordinates. id you use the distance formula correctly? e a little more precise. PTS: 1 IF: asic OJ: Find the distance between two points on a coordinate plane. NT: NTM GM.2 NTM GM.2a TOP: Find the distance between two points on a coordinate plane. KEY: istance oordinate Plane istance etween Two Points 3

19 I: 10. NS: Ê Ê x The formula for the midpoint between two points Ê x 1, y ˆ 1, Ê x, y ˆ 1 + x ˆ is, 2 Ê y 1 + y ˆ ˆ 2. 2 id you use the midpoint formula? id you use the midpoint formula correctly? o you subtract then divide by two? PTS: 1 IF: verage OJ: Find the midpoint of a segment. NT: NTM ME.1 TOP: Find the midpoint of a segment. KEY: Midpoint Line Segment 11. NS: Since GK bisects FGH, x = y and 3v 4 = 2v + 7. Solve for v, then substitute into either side of the equation to find x. on t forget to subtract. You are not finding the measure of FGH. You are finding x. You are not finding v. You are finding x. PTS: 1 IF: asic OJ: Identify and use congruent angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use congruent angles. KEY: ngles ongruent ngles ongruency 12. NS: ray bisects an angle. K is the endpoint of that ray, not P. The answer is KP. Is a bisector a point? Is P the endpoint of that ray? Is a bisector an angle? PTS: 1 IF: asic OJ: Identify and use the bisector of an angle. NT: NTM GM.1 NTM GM.1a TOP: Identify and use the bisector of an angle. KEY: ngle isectors 4

20 I: 13. NS: MKH > 90 so it is obtuse. If answer d is true, then this must be true. eing in the interior means being between the two end rays of an angle. If answer b is true, then this must be true. PTS: 1 IF: asic OJ: Identify and use the bisector of an angle. NT: NTM GM.1 NTM GM.1a TOP: Identify and use the bisector of an angle. KEY: ngle isectors 14. NS: m JKM + m 4 = 180 5x x = 180 6x = x = 27 That is the measure of angle JKM. You forgot to add in x. Opposite rays add up to 180. PTS: 1 IF: verage OJ: Identify and use the bisector of an angle. NT: NTM GM.1 NTM GM.1a TOP: Identify and use the bisector of an angle. KEY: ngle isectors 15. NS: Vertical angles are two nonadjacent angles formed by two intersecting lines. cute angles measure less than 90 degrees. You are looking for vertical angles, not adjacent angles. You are looking for vertical angles, not a linear pair. What is the definition of acute? PTS: 1 IF: asic OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 5

21 I: 16. NS: djacent angles are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points. Obtuse angles measure greater than 90 degrees. You are looking for adjacent angles, not vertical angles. You are looking for adjacent angles, not a linear pair. What is the definition of obtuse? PTS: 1 IF: asic OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 17. NS: linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. You are looking for a linear pair, not vertical angles. You are looking for a linear pair, not just adjacent angles. You are looking for a linear pair which, by definition, must be adjacent. PTS: 1 IF: verage OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 18. NS: Supplementary angles are two angles whose measures have a sum of 180. What is the definition of supplementary? o the measures have a sum of 180 degrees? What is the definition of supplementary? PTS: 1 IF: asic OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 6

22 I: 19. NS: omplementary angles are two angles whose measures have a sum of 90. Is that the value of q, or the measure of the angles? What is the definition of complementary? Is the sum of those angles 90 degrees? PTS: 1 IF: verage OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 20. NS: Supplementary angles are two angles whose measures have a sum of 180. What is the definition of supplementary? What is the the sum of those two measures? Is the measure of one angle 26 more than the other? PTS: 1 IF: verage OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 21. NS: omplementary angles are two angles whose measures have a sum of 90. id you write and solve an equation? What is the definition of complementary? You are looking for its complement. PTS: 1 IF: verage OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 7

23 I: 22. NS: Supplementary angles are two angles whose measures have a sum of 180. What is the definition of supplementary? What is the definition of complementary? You want the angle first, then its supplement. PTS: 1 IF: verage OJ: Identify and use special pairs of angles. NT: NTM GM.1 NTM GM.1a TOP: Identify and use special pairs of angles. KEY: djacent ngles Vertical ngles Linear Pair omplementary ngles Supplementary ngles 23. NS: Lines that form right angles are perpendicular. right angle measures 90. Substitute back into the angles. What is the measure of the angle created by perpendicular rays? What is the value of r? PTS: 1 IF: asic OJ: Identify perpendicular lines. NT: NTM GM.1 NTM GM.1a TOP: Identify perpendicular lines. KEY: Perpendicular Lines 24. NS: In the case of a regular figure, the length of each side is the perimeter divided by the number of sides. Is the figure a hexagon? Is the figure a dodecagon? How many sides are there? PTS: 1 IF: asic OJ: Find the perimeters of polygons. NT: NTM PS.1 NTM PS.2 NTM PS.3 TOP: Find the perimeters of polygons. KEY: Perimeter Polygons 8

24 I: 25. NS: Perimeter is the sum of the sides. What is the sum of the sides? id you find the value of y? What is the value of y? PTS: 1 IF: verage OJ: Find the perimeters of polygons. NT: NTM PS.1 NTM PS.2 NTM PS.3 TOP: Find the perimeters of polygons. KEY: Perimeter Polygons 26. NS: Perimeter is the sum of the sides. heck your math. id you add all three sides? What is the sum of the three sides? PTS: 1 IF: verage OJ: Find the perimeters of polygons. NT: NTM PS.1 NTM PS.2 NTM PS.3 TOP: Find the perimeters of polygons. KEY: Perimeter Polygons 27. NS: Start with 1. dd, subtract, or multiply the same number to each number to get the next one. What operations are involved? idn t you carry the conjecture too far? heck your math. PTS: 1 IF: asic OJ: Make conjectures based on inductive reasoning. NT: NTM RP.2 ST: 1.0 TOP: Make conjectures based on inductive reasoning. KEY: Inductive Reasoning onjectures 9

25 I: 28. NS: Start with 6. o one operation to get the next number. o a different operation to get the next number. Repeat. What operation should come next? What operations are involved? heck your math. PTS: 1 IF: verage OJ: Make conjectures based on inductive reasoning. NT: NTM RP.2 ST: 1.0 TOP: Make conjectures based on inductive reasoning. KEY: Inductive Reasoning onjectures 29. NS: oplanar points always lie in the same plane. Three points are always coplanar but four are not. What does coplanar mean? What does coplanar mean? re more than two points always coplanar? PTS: 1 IF: asic OJ: Find counterexamples. NT: NTM RP.3 ST: TOP: Find counterexamples. KEY: ounterexamples 30. NS: ngles are congruent only if their measures are equal. Point may be closer to line or line so the measures would not be equal. What is the definition of congruent? What is the definition of congruent? Would that be a counterexample? PTS: 1 IF: asic OJ: Find counterexamples. NT: NTM RP.3 ST: TOP: Find counterexamples. KEY: ounterexamples 10

26 I: 31. NS: oplanar points always lie in the same plane. Three points are always coplanar but four are not. What does coplanar mean? What does coplanar mean? Would the points have to be in the same plane? PTS: 1 IF: asic OJ: Find counterexamples. NT: NTM RP.3 ST: TOP: Find counterexamples. KEY: ounterexamples 32. NS: Just because two angles share a common point does not mean they are vertical. They could be nearly adjacent or one could be in the interior of the other one. What is a vertical angle? What is a vertical angle? What is a vertical angle? PTS: 1 IF: asic OJ: Find counterexamples. NT: NTM RP.3 ST: TOP: Find counterexamples. KEY: ounterexamples 33. NS: If two angles are supplementary their measures total 180. F could only be supplementary to H if they are both right angles. What is the definition of supplementary? What is the definition of supplementary? What is the definition of supplementary? PTS: 1 IF: asic OJ: Find counterexamples. NT: NTM RP.3 ST: TOP: Find counterexamples. KEY: ounterexamples 11

27 I: 34. NS: The converse of a conditional statement Ê p q ˆ exchanges the hypothesis and conclusion of the conditional. It is also known as q p. heck the statement again. heck the statement again. What is the definition of converse? PTS: 1 IF: asic OJ: Write the converse of if-then statements. NT: NTM RP.3 ST: 3.0 TOP: Write the converse of if-then statements. KEY: onverse If-Then Statements 35. NS: Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. line contains at least two points. 4. plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line. oes that apply? Is that a postulate? oes that fit the situation? PTS: 1 IF: verage OJ: Identify and use basic postulates about points, lines, and planes. NT: NTM RP.1 ST: TOP: Identify and use basic postulates about points, lines, and planes. KEY: Points Lines Planes 12

28 I: 36. NS: Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. line contains at least two points. 4. plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line. Is that a postulate? oes that fit the situation? oes that apply? PTS: 1 IF: verage OJ: Identify and use basic postulates about points, lines, and planes. NT: NTM RP.1 ST: TOP: Identify and use basic postulates about points, lines, and planes. KEY: Points Lines Planes 37. NS: Planes that intersect have a common line. This plane has four lines to intersect with other planes. o they all intersect I in a line? This plane has four lines to intersect with other planes. PTS: 1 IF: asic OJ: Identify the relationships between two lines or two planes. NT: NTM GM.1 NTM GM.1a ST: 1.0 TOP: Identify the relationships between two lines or two planes. KEY: Relationship etween Two Lines Relationship etween Two Planes 13

29 I: 38. NS: oplanar segments that do not intersect are parallel. re those parallel to GF? Is that all? Is that all? PTS: 1 IF: asic OJ: Identify the relationships between two lines or two planes. NT: NTM GM.1 NTM GM.1a ST: 1.0 TOP: Identify the relationships between two lines or two planes. KEY: Relationship etween Two Lines Relationship etween Two Planes 39. NS: Skew lines do not intersect and are not coplanar. re any of those segments in the same plane as segment? Skew lines are not coplanar. o any of those segments intersect segment? PTS: 1 IF: verage OJ: Identify the relationships between two lines or two planes. NT: NTM GM.1 NTM GM.1a ST: 1.0 TOP: Identify the relationships between two lines or two planes. KEY: Relationship etween Two Lines Relationship etween Two Planes 40. NS: orresponding angles are congruent. lternate interior angles are congruent. onsecutive interior angles are supplementary. lternate exterior angles are congruent. What is the sum of supplementary angles? re those angles congruent or supplementary? What do supplementary angles add up to? PTS: 1 IF: verage OJ: Use the properties of parallel lines to determine congruent angles. NT: NTM L.2c NTM ME.1 ST: TOP: Use the properties of parallel lines to determine congruent angles. KEY: Parallel Lines ongruent ngles 14

30 I: 41. NS: orresponding angles are congruent. lternate interior angles are congruent. onsecutive interior angles are supplementary. lternate exterior angles are congruent. What do supplementary angles add up to? What do the angles of a right triangle add up to? Is that triangle isosceles? PTS: 1 IF: verage OJ: Use algebra to find angle measures. NT: NTM L.4a NTM GM.2 NTM GM.2a TOP: Use algebra to find angle measures. KEY: ngles ngle Measures 42. NS: Extend v to intersect with p. This creates a linear pair at point S with angles measuring 119 (given) and 61. The angles formed by the intersection of v and p (also linear pairs) measure 125 (corresponding angles) and 55 with the latter being one of the interior angles of the triangle formed by t, p and v. Since the sum of the angles of a triangle is 180, the angle that is vertical to <1 is 64, thus making <1 64 as well. Extend v as a transversal of q and p. Extend v as a transversal of q and p. Extend v as a transversal of q and p. PTS: 1 IF: verage OJ: Use algebra to find angle measures. NT: NTM L.4a NTM GM.2 NTM GM.2a TOP: Use algebra to find angle measures. KEY: ngles ngle Measures 43. NS: Ê y 2 y ˆ 1 The formula for slope is Ê x 2 x ˆ 1. You are not solving for the slope of the perpendicular. Remember y over x. Subtract y from y and x from x. PTS: 1 IF: asic OJ: Find slopes of lines. NT: NTM GM.1b NTM GM.2 NTM GM.2a TOP: Find slopes of lines. KEY: Slope Slope of Lines 15

31 I: 44. NS: Ê y 2 y ˆ 1 The formula for slope is Ê x 2 x ˆ. If the slopes are the same they are parallel. If the product of the two 1 slopes is 1, they are perpendicular. Parallel slopes are the same and perpendicular ones are opposite reciprocals. Parallel slopes are the same and perpendicular ones are opposite reciprocals. PTS: 1 IF: verage OJ: Use slope to identify parallel lines and perpendicular lines. NT: NTM L.2 NTM L.2c NTM RE.2 TOP: Use slope to identify parallel lines and perpendicular lines. KEY: Parallel Lines Perpendicular Lines Slope 45. NS: Postulates and theorems: If corresponding angles are congruent, then lines are parallel. If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. If alternate exterior angles are congruent, then lines are parallel. If consecutive interior angles are supplementary, then lines are parallel. If alternate interior angles are congruent, then lines are parallel. If 2 lines are perpendicular to the same line, then lines are parallel. What kind of angles are those? What kind of angles are those? Which lines are parallel? PTS: 1 IF: asic OJ: Recognize angle conditions that occur with parallel lines. NT: NTM GM.1b NTM GM.1c NTM RP.3 ST: TOP: Recognize angle conditions that occur with parallel lines. KEY: ngles Parallel Lines 16

32 I: 46. NS: Use the istance Formula to find the lengths of the sides. d = Ê x 2 x ˆ Ê y 2 y ˆ 1 2 If = or = or =, then the triangle is isosceles. If = =, then the triangle is equilateral. If neither of the above, the triangle is scalene. Use the distance formula to find the lengths of the sides. id you use the distance formula? What are the lengths of the sides? PTS: 1 IF: verage OJ: Identify and classify triangles by sides. NT: NTM GM.1 NTM GM.1b ST: 12.0 TOP: Identify and classify triangles by sides. KEY: Triangles lassify Triangles 47. NS: The ngle Sum Theorem states that the sum of the measures of the angles of a triangle is 180. What do you know about vertical angles? What do you know about vertical angles? Use the ngle Sum Theorem. PTS: 1 IF: asic OJ: pply the ngle Sum Theorem. NT: NTM GM.1 NTM GM.1b ST: TOP: pply the ngle Sum Theorem. KEY: ngle Sum Theorem 48. NS: The ngle Sum Theorem states that the sum of the measures of the angles of a triangle is 180. id you use the ngle Sum Theorem. Use the ngle Sum Theorem. Use the ngle Sum Theorem. PTS: 1 IF: verage OJ: pply the ngle Sum Theorem. NT: NTM GM.1 NTM GM.1b ST: TOP: pply the ngle Sum Theorem. KEY: ngle Sum Theorem 17

33 I: 49. NS: The Exterior ngle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. What is the sum of the measures of the angles in a triangle? id you use the Exterior ngle Theorem? Use the Exterior ngle Theorem. PTS: 1 IF: verage OJ: pply the Exterior ngle Theorem. NT: NTM GM.1 NTM GM.1b ST: TOP: pply the Exterior ngle Theorem. KEY: Exterior ngle Theorem 50. NS: Since ΔRM is isosceles, RM RM. Remember the definition of isosceles. Is that the base angle of an isosceles triangle? Remember the definition of isosceles. PTS: 1 IF: asic OJ: Use properties of isosceles triangles. NT: NTM GM.1 NTM GM.1a ST: TOP: Use properties of isosceles triangles. KEY: Isosceles Triangles 51. NS: Since ΔMX is isosceles, XM MX. Is that the base angle of an isosceles triangle? Remember the definition of isosceles. What do you know about base angles of an isosceles triangle? PTS: 1 IF: asic OJ: Use properties of isosceles triangles. NT: NTM GM.1 NTM GM.1a ST: TOP: Use properties of isosceles triangles. KEY: Isosceles Triangles 18

34 I: 52. NS: Since ΔMX is isosceles, XM MX. Subtract those from 180 to get the answer. How many degrees in a triangle? Is that the vertex angle? Subtract both base angles from 180. PTS: 1 IF: verage OJ: Use properties of isosceles triangles. NT: NTM GM.1 NTM GM.1a ST: TOP: Use properties of isosceles triangles. KEY: Isosceles Triangles 53. NS: dd m MX and m RM. That is the sum of which angles? id you add carefully? That is the sum of which angles? PTS: 1 IF: verage OJ: Use properties of isosceles triangles. NT: NTM GM.1 NTM GM.1a ST: TOP: Use properties of isosceles triangles. KEY: Isosceles Triangles 54. NS: 4y 4 = 60 3x 8 = 2x 1 id you set the two sides equal to each other? How many degrees is each angle of an equilateral triangle? How many degrees is angle H? PTS: 1 IF: asic OJ: Use properties of equilateral triangles. NT: NTM GM.2 NTM GM.2a ST: TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles 19

35 I: 55. NS: (2y + 6) + (2y + 6) + (2y + 6) = 180 x + 4 = 2x 3 What do you know about the sides of an equilateral triangle? How many degrees is each angle of an equilateral triangle? id you add or subtract when solving for y? PTS: 1 IF: verage OJ: Use properties of equilateral triangles. NT: NTM GM.2 NTM GM.2a ST: TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles 56. NS: y 2 = 30 8x + 1 = 2(5x) an x be negative? What is the measure of angle TRS? heck your math. PTS: 1 IF: verage OJ: Use properties of equilateral triangles. NT: NTM GM.2 NTM GM.2a ST: TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles 57. NS: 2y + 12 = x 2 Ê ˆ 2 = RU id you use the Pythagorean Theorem correctly? heck your math. Should you have subtracted? PTS: 1 IF: verage OJ: Use properties of equilateral triangles. NT: NTM GM.2 NTM GM.2a ST: TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles 20

36 I: 58. NS: The sum of the lengths of any two sides must be greater than the third. id you check all the sums? dd two sides and compare to the third. dd two sides and compare to the third. PTS: 1 IF: asic OJ: pply the Triangle Inequality Theorem. NT: NTM GM.2 NTM GM.2a ST: TOP: pply the Triangle Inequality Theorem. KEY: Triangles Inequality Theorem 59. NS: The sum of the lengths of any two sides must be greater than the third. Would both sides have to be longer than the base? Is the sum of the two sides longer than the base? Is that the shortest possible length? PTS: 1 IF: verage OJ: etermine the shortest distance between a point and a line. NT: NTM L.2 NTM L.2b NTM GM.1 ST: TOP: etermine the shortest distance between a point and a line. KEY: istance istance etween a Point and a Line 60. NS: Find the cross products. Multiply. ivide each side by the coefficient of the variable. Reverse the numerator and denominator. heck your cross multiplication. The left side of the proportion cannot be reduced before cross multiplying. PTS: 1 IF: verage OJ: Use properties of proportions. NT: NTM GM.1 NTM GM.1b TOP: Use properties of proportions. KEY: Proportions 21

37 I: 61. NS: = = x = 4x 12 4 = 4x 4 3 = x Is the same length as EF? heck the ratio of the corresponding sides. heck the similarity statement. PTS: 1 IF: asic OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 62. NS: Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. The ratio of the corresponding sides is 1:3. re the sides proportional? heck the similarity statement. heck the similarity statement. PTS: 1 IF: asic OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 22

38 I: 63. NS: EF RS F RT F S so ΔEF Δ. The triangles are similar. re all the sides labeled? Is there an S Similarity? PTS: 1 IF: verage OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 64. NS: F E so ΔEF Δ. heck the similarity statement. Is there an S similarity? The triangles are similar. PTS: 1 IF: verage OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 65. NS: etermine the ratio of corresponding parts. Use the ratio to find the missing information. heck your ratio. What are the values of and? heck your addition. PTS: 1 IF: verage OJ: Use similar triangles to solve problems. NT: NTM GM.1 NTM GM.1b ST: TOP: Use similar triangles to solve problems. KEY: Similar Triangles Solve Problems 23

39 I: 66. NS: etermine the ratio of corresponding parts. Use the ratio to find the missing information. Which side is? heck your operations. Which side does the question ask you to find? PTS: 1 IF: verage OJ: Use similar triangles to solve problems. NT: NTM GM.1 NTM GM.1b ST: TOP: Use similar triangles to solve problems. KEY: Similar Triangles Solve Problems 67. NS: etermine the ratio of corresponding parts. Use the ratio to find the missing information. heck your multiplication. heck your ratio. heck your multiplication. PTS: 1 IF: verage OJ: Use similar triangles to solve problems. NT: NTM GM.1 NTM GM.1b ST: TOP: Use similar triangles to solve problems. KEY: Similar Triangles Solve Problems 68. NS: etermine the ratio of corresponding parts. Use the ratio to find the missing information. heck your addition. heck the ratios. heck the ratios and your multiplication. PTS: 1 IF: verage OJ: Use similar triangles to solve problems. NT: NTM GM.1 NTM GM.1b ST: TOP: Use similar triangles to solve problems. KEY: Similar Triangles Solve Problems 24

40 I: 69. NS: In the sixth triangular number, the bottom row will contain six dots. The row above the bottom row will contain five dots. This continues until the top row contains one dot. The total number of dots is 21. This is the arrangement number not the number of dots in the arrangement. This is the number of dots in the fifth arrangement. This is the number of dots in the seventh arrangement. PTS: 1 IF: asic OJ: Recognize and describe characteristics of fractals. TOP: Recognize and describe characteristics of fractals. KEY: Fractals SHORT NSWER 70. NS: yes; Δ ΔEF by SSS Similarity. Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. PTS: 1 IF: verage OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 71. NS: yes; ΔUYW ΔVXW by Similarity. Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. PTS: 1 IF: verage OJ: Identify similar triangles. NT: NTM GM.1 NTM GM.1b ST: TOP: Identify similar triangles. KEY: Similar Triangles 25