Geo Final Review 2014

Transcription

1 Period: ate: Geo Final Review 2014 Multiple hoice Identify the choice that best completes the statement or answers the question. 1. n angle measures 2 degrees more than 3 times its complement. Find the measure of its complement. a. 23 c. 22 b. 272 d bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = 22 c. m = 20 b. m = 3 d. m = Use the information m 1 = (3x + 30), m 2 = (5x 10), and x = 20, and the theorems you have learned to show that l Ä m. a. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. Since 1 and 2 are same-side interior angles, m 1 = m 2 = 180. y the onverse of the Same-Side Interior ngles Theorem, l Ä m. b. Since 1 and 2 are same-side interior angles, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. y substitution, m 1 = m 2 = 90. y the onverse of the lternate Interior ngles Theorem, l Ä m. c. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. y the Substitution Property of Equality, m 1 = m 2 = 90. y the onverse of the lternate Interior ngles Theorem, l Ä m. d. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. Since 1 and 2 are alternate interior angles, m 1 = m 2 = 180. y the onverse of the Same-Side Interior ngles Theorem, l Ä m. 1

2 I: 4. In a dance performance, four dancers form a diamond with vertices (2, 0), (0, 2), ( 2, 0), and (0, 2). Then, they move along the dance floor following the translation vector, 0, 4. There they pause, and then move again along the same vector. What are their coordinates after six such translations? a. (26, 0), (24, 2), (22, 0), and (24, 2) b. (2, 10), (0, 12), ( 2, 10), and (0, 8) c. (2, 24), (0, 26), ( 2, 24), and (0, 22) d. (26, 24), (24, 6), (22, 24), and (24, 22) 5. Two sides of an equilateral triangle measure (2y + 3) units and (y 2 5) units. If the perimeter of the triangle is 33 units, what is the value of y? a. y = 4 c. y = 7 b. y = 11 d. y = 15 2

3 I: 6. Use the given paragraph proof to write a two-column proof. Given: is a right angle. 1 3 Prove: 2 and 3 are complementary. Paragraph proof: Since is a right angle, m = 90 by the definition of a right angle. y the ngle ddition Postulate, m = m 1 + m 2. y substitution, m 1 + m 2 = 90. Since 1 3, m 1 = m 3 by the definition of congruent angles. Using substitution, m 3 + m 2 = 90. Thus, by the definition of complementary angles, 2 and 3 are complementary. omplete the proof. Two-column proof: Statements Reasons 1. is a right angle Given 2. m = efinition of a right angle 3. m = m 1 + m 2 3. [1] 4. m 1 + m 2 = Substitution 5. m 1 = m 3 5. [2] 6. m 3 + m 2 = Substitution 7. 2 and 3 are complementary. 7. efinition of complementary angles a. [1] ngle ddition Postulate [2] efinition of congruent angles b. [1] ngle ddition Postulate [2] efinition of equality c. [1] Substitution [2] efinition of equality d. [1] Substitution [2] efinition of congruent angles 7. Find the measure of the complement of M, where m M = 31.1 a c b d One of the acute angles in a right triangle has a measure of What is the measure of the other acute angle? a. 90 c b d

4 I: 9. raw two lines and a transversal such that 1 and 2 are alternate interior angles, 2 and 3 are corresponding angles, and 3 and 4 are alternate exterior angles. What type of angle pair is 1 and 4? a. b. 1 and 4 are corresponding angles. c. 1 and 4 are supplementary angles. d. 1 and 4 are vertical angles. 1 and 4 are alternate exterior angles. 4

5 I: 10. Write a justification for each step. m JKL = 100 m JKL = m JKM + m MKL [1] 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] x = 12 Symmetric Property of Equality a. [1] ngle ddition Postulate [2] Simplify. b. [1] ngle ddition Postulate [2] ivision Property of Equality c. [1] Segment ddition Postulate [2] Multiplication Property of Equality d. [1] Transitive Property of Equality [2] ivision Property of Equality 11. The figure shows part of the roof structure of a house. Use SS to explain why RTS RTU. omplete the explanation. It is given that [1]. Since RTS and RTU are right angles, [2] by the Right ngle ongruence Theorem. y the Reflexive Property of ongruence, [3]. Therefore, RTS RTU by SS. a. [1] ST UT [2] RTS RTU [3] RT RT b. [1] ST UT [2] RTS RTU [3] SU SU c. [1] ST UT [2] SRT URT [3] ST UT d. [1] RT RT [2] SRT URT [3] ST UT 29

6 I: 12. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. a. polygon, decagon c. polygon, dodecagon b. polygon, hexagon d. not a polygon 13. Use the onverse of the orresponding ngles Postulate and 1 2 to show that l Ä m. a. y the onverse of the orresponding ngles Postulate, 1 2. From the diagram, l Ä m. b. 1 2 is given. From the diagram, 1 and 2 are alternate interior angles. So by the onverse of the lternate Interior ngles Postulate, l Ä m. c. 1 2 is given. From the diagram, 1 and 2 are corresponding angles. So by the orresponding ngles Postulate, l Ä m. d. 1 2 is given. From the diagram, 1 and 2 are corresponding angles. So by the onverse of the orresponding ngles Postulate, l Ä m. 14. Find m, given F, E, and m E = 46. a. m = 44 c. m = 67 b. m = 134 d. m = 46 6

7 I: 15. etermine if you can use S to prove E. Explain. a. is given. E because both are right angles. y the Vertical ngles Theorem, E. Therefore, E by S. b. is given. E because both are right angles. y the djacent ngles Theorem, E. Therefore, E by S. c. is given. E because both are right angles. y the Vertical ngles Theorem, E. Therefore, E by SS. d. is given. E because both are right angles. No other congruence relationships can be determined, so S cannot be applied. 16. Translate the triangle with vertices (3, 4), (2, 1), and (4, 12) along the vector 1, 3. Find the coordinates of the new image. a. (2, 7), (1, 2), and (3, 15) b. (4, 7), (3, 2), and (5, 15) c. (6, 3), (5, 2), and (7, 11) d. ( 3, 12), ( 2, 3), and ( 4, 36) 7

8 I: 17. Given: MLN PLO, MNL POL, MO NP Prove: MLP is isosceles. omplete the proof. Proof: Statements Reasons 1. MLN PLO, MNL POL 1. Given 2. MO NP 2. Given 3. MO = NP 3. efinition of congruent line segments 4. NO = NO 4. Reflexive Property of Equality 5. MO NO = NP NO 5. Subtraction Property of Equality 6. MO NO = MN and NP NO = OP 6. Segment ddition Postulate 7. MN = OP 7. Substitution Property of Equality 8. MLN PLO 8. [1] 9. ML PL 9. [2] 10. MLP is isosceles. 10. efinition of isosceles triangle a. [1] S [2] PT b. [1] PT [2] S c. [1] S [2] PT d. [1] PT [2] S 8

9 I: 18. In a swimming pool, two lanes are represented by lines l and m. If a string of flags strung across the lanes is represented by transversal t, and x = 10, show that the lanes are parallel. a. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are same-side interior angles and they are supplementary, so the lanes are parallel by the onverse of the Same-Side Interior ngles Theorem. b. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the onverse of the lternate Interior ngles Theorem. c. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles and they are congruent, so the lanes are parallel by the lternate Interior ngles Theorem. d. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are corresponding angles and they are congruent, so the lanes are parallel by the onverse of the orresponding ngles Postulate. 9

10 I: 19. Find. a. = 10 b. Not enough information. n equiangular triangle is not necessarily equilateral. c. = 12 d. = 14 10

11 I: 20. Fill in the blanks to complete the two-column proof. Given: 1 and 2 are supplementary. m 1 = 135 Prove: m 2 = 45 Proof: Statements Reasons 1. 1 and 2 are supplementary. 1. Given 2. [1] 2. Given 3. m 1 + m 2 = [2] m 2 = Substitution Property 5. m 2 = [3] a. [1] m 1 = 135 [2] efinition of supplementary angles [3] Subtraction Property of Equality b. [1] m 1 = 135 [2] efinition of supplementary angles [3] Substitution Property c. [1] m 1 = 135 [2] efinition of complementary angles [3] Subtraction Property of Equality d. [1] m 2 = 135 [2] efinition of supplementary angles [3] Subtraction Property of Equality 11

12 I: 21. Write a justification for each step, given that EG = FH. EG = FH Given information EG = EF + FG [1] FH = FG + GH Segment ddition Postulate EF + FG = FG + GH [2] EF = GH Subtraction Property of Equality a. [1] Substitution Property of Equality [2] Transitive Property of Equality b. [1] Segment ddition Postulate [2] efinition of congruent segments c. [1] Segment ddition Postulate [2] Substitution Property of Equality d. [1] ngle ddition Postulate [2] Subtraction Property of Equality 12

13 I: 22. Given: RT SU, SRT URT, RS RU. T is the midpoint of SU. Prove: RTS RTU omplete the proof. Proof: Statements Reasons 1. RT SU 1. Given 2. RTS and RTU are right angles. 2. [1] 3. RTS RTU 3. Right ngle ongruence Theorem 4. SRT URT 4. Given 5. S U 5. [2] 6. RS RU 6. Given 7. T is the midpoint of SU. 7. Given 8. ST UT 8. efinition of midpoint 9. RT RT 9. [3] 10. RTS RTU 10. efinition of congruent triangles a. [1] efinition of perpendicular lines [2] Third ngles Theorem [3] Reflexive Property of ongruence b. [1] efinition of perpendicular lines [2] Third ngles Theorem [3] Symmetric Property of ongruence c. [1] efinition of right angles [2] Third ngles Theorem [3] Transitive Property of ongruence d. [1] efinition of perpendicular lines [2] Vertical ngles Theorem [3] Symmetric Property of ongruence 13

14 I: 23. Tell whether the transformation appears to be a reflection. Explain. a. No; the image does not appear to be flipped. b. Yes; the image appears to be flipped across a line. 24. Find m RST. a. m RST = 156 c. m RST = 108 b. m RST = 24 d. m RST = Find m 1 in the diagram. (Hint: raw a line parallel to the given parallel lines.) a. m 1 = 80 c. m 1 = 75 b. m 1 = 95 d. m 1 = 85 14

15 I: 26. billiard ball bounces off the sides of a rectangular billiards table in such a way that 1 3, 4 6, and 3 and 4 are complementary. If m 1 = 26.5, find m 3, m 4, and m 5. a. m 3 = 63.5 ; m 4 = 26.5 ; m 5 = 53 b. m 3 = 26.5 ; m 4 = ; m 5 = 26.5 c. m 3 = 26.5 ; m 4 = 63.5 ; m 5 = 63.5 d. m 3 = 26.5 ; m 4 = 63.5 ; m 5 = 53 15

16 I: 27. Use the given two-column proof to write a flowchart proof. Given: 1 4 Prove: m 2 = m 3 Two-column proof: Statements Reasons Given 2. 1 and 2 are supplementary. 3 and 4 2. efinition of linear pair are supplementary ongruent Supplements Theorem 4. m 2 = m 3 4. efinition of congruent segments omplete the proof. Flowchart proof: 1 4 Given [1] 2 3 m 2 = m 3 efinition of linear pair [2] efinition of congruent segments a. [1] 2 3 [2] efinition of congruent segments b. [1] 1 and 2 are supplements; 3 and 4 are supplementary [2] ongruent omplements Theorem c. [1] 1 and 2 are supplementary; 3 and 4 are supplementary [2] ongruent Supplements Theorem d. [1] efinition of congruent segments [2] ongruent Supplements Theorem 16

17 I: 28. Find m 1 in the diagram. (Hint: raw a line parallel to the given parallel lines.) a. m 1 = 120 c. m 1 = 125 b. m 1 = 130 d. m 1 = The point G(4, 8) is rotated 90 about point M( 7, 9) and then reflected across the line y = 6. Find the coordinates of the image G. a. ( 24, 14) c. ( 18, 20) b. (12, 14) d. ( 8, 16) 17

18 I: 30. Given: P is the midpoint of TQ and RS. Prove: TPR QPS omplete the proof. Proof: Statements 1. P is the midpoint of TQ and RS. 1. Given Reasons 2. TP QP, RP SP 2. [1] 3. [2] 3. Vertical ngles Theorem 4. TPR QPS 4. [3] a. [1] efinition of midpoint [2] RT SQ [3] SSS b. [1]. efinition of midpoint [2] TPR QPS [3] SS c. [1] efinition of midpoint [2] TPR QPS [3] SSS d. [1] efinition of midpoint [2] PRT PSQ [3] SS 31. raw and label a pair of opposite rays FG and FH. a. c. b. d. 18

19 I: 32. Using the information about John, Jason, and Julie, can you uniquely determine how they stand with respect to each other? On what basis? Statement 1: John and Jason are standing 12 feet apart. Statement 2: The angle from Julie to John to Jason measures 31º. Statement 3: The angle from John to Jason to Julie measures 49º. a. Yes. They form a unique triangle by SS. b. No. There is no unique configuration. c. Yes. They form a unique triangle by S. d. Yes. They form a unique triangle by SSS. 33. Find the measure of each exterior angle of a regular decagon. a. 18 c b. 36 d Tell whether the transformation appears to be a translation. Explain. a. Yes; all of the points have moved the same distance in the same direction. b. No; not all of the points have moved the same distance. 35. Give an example of corresponding angles. a. 3 and 6 c. 5 and 7 b. 4 and 1 d. 8 and 4 19

20 I: 36. Tell whether the transformation appears to be a rotation. Explain. a. Yes; the figure appears to be turned around a point. b. No; the figure appears to be flipped. 20

21 I: 37. Given: F G, bisects Prove: omplete the flowchart proof. Proof: F G Given bisects Given efinition of angle bisector a. 1. ongruent Supplements Theorem Reflexive Property of ongruence 4. S 5. PT b. 1. ongruent omplements Theorem Transitive Property of ongruence 4. PT 5. S c. 1. ongruent omplements Theorem Reflexive Property of ongruence 4. PT 5. S d. 1. ongruent Supplements Theorem Transitive Property of ongruence 4. S 5. PT 21

22 I: 38. Use S to prove the triangles congruent. Given: Ä GH, Ä FH, FH Prove: HGF omplete the flowchart proof. Proof: Ä GH Given G 1. Ä FH HFG HGF Given 2. S a. 1. lternate Interior ngles Theorem 2. lternate Exterior ngles Theorem b. 1. lternate Interior ngles Theorem 2. lternate Interior ngles Theorem c. 1. lternate Exterior ngles Theorem 2. lternate Interior ngles Theorem d. 1. lternate Exterior ngles Theorem 2. lternate Exterior ngles Theorem FH Given 22

23 I: 39. Rotate RSQ with vertices R(4, 1), S(5, 3), and Q(3, 1) by 90 about the origin. a. c. b. d. 40. Find the measure of each interior angle of a regular 45-gon. a. 188 c. 172 b. 176 d

24 I: 41. Write a two-column proof. Given: t l, 1 2 Prove: mä l omplete the proof. Proof: Statements 1. [1] 1. Given 2. t m 2. [2] 3. mä l 3. [3] Reasons a. [1] t l, 1 2 [2] 2 intersecting lines form linear pair of s lines. [3] Perpendicular Transversal Theorem b. [1] t l, 1 2 [2] 2 intersecting lines form linear pair of s lines. [3] 2 lines to the same line lines Ä. c. [1] t l, 1 2 [2] 2 lines to the same line lines Ä. [3] 2 intersecting lines form linear pair of s lines. d. [1] t l, 1 2 [2] Perpendicular Transversal Theorem [3] 2 lines to the same line lines Ä. 24

25 I: 42. Use the given flowchart proof to write a two-column proof of the statement F F. Flowchart proof: = ; F = F Given + F = F + + F = F F + = F Segment ddition Postulate F = F F F ddition Property of Equality omplete the proof. Substitution efinition of congruent segments Two-column proof: Statements Reasons 1. = ; F = F 1. Given 2. [1] 2. ddition Property of Equality 3. [2] 3. Segment ddition Postulate 4. F = F 4. Substitution 5. F F 5. efinition of congruent segments a. [1] + F = F ; F + = F [2] F = F b. [1] = ; F = F [2] + F = F + c. [1] F = F [2] + F = F + d. [1] + F = F + [2] + F = F ;F + = F 25

26 I: 43. Identify the transversal and classify the angle pair 11 and 7. a. The transversal is line m. The angles are corresponding angles. b. The transversal is line l. The angles are corresponding angles. c. The transversal is line l. The angles are alternate interior angles. d. The transversal is line n. The angles are alternate exterior angles. 44. Show for a = 3. omplete the proof. = a + 7 = [1] = 10 = 4a 2 = [2] = 12 2 = 10 = 6a 2 = 6(3) 2 = 18 2 = [3] = [4].. by the Reflexive Property of ongruence. So by [5]. a. [1] [2] 4(3) 2 [3] 16 [4] 16 [5] SSS b. [1] a + 7 [2] 4a 2 [3] 16 [4] 16 [5] SS c. [1] [2] 4(3) 2 [3] 26 [4] 26 [5] SSS d. [1] [2] 4(3) 2 [3] 16 [4] 16 [5] SS 26

27 I: 45. Tell whether F and 3 are only adjacent, adjacent and form a linear pair, or not adjacent. a. only adjacent b. not adjacent c. adjacent and form a linear pair 46. Find m E and m N, given m F = m P, m E = (x 2 ), and m N = (4x 2 75). a. m E = 65, m N = 65 c. m E = 65, m N = 25 b. m E = 25, m N = 65 d. m E = 25, m N = is between and E. E = 6x, = 4x + 8, and E = 27. Find E. a. E = 17.5 c. E = 57 b. E = 105 d. E = 78 27

28 I: 48. Tell whether the transformation appears to be a translation. Explain. a. Yes; all of the points have moved the same distance in the same direction. b. No; not all of the points have moved the same distance. 49. Find m. a. m = 45º c. m = 40º b. m = 50º d. m = 35º 28

29 I: 50. Use the given plan to write a two-column proof. Given: m 1 + m 2 = 90, m 3 + m 4 = 90, m 2 = m 3 Prove: m 1 = m 4 Plan: Since both pairs of angle measures add to 90, use substitution to show that the sums of both pairs are equal. Since m 2 = m 3, use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m 1 = m 4. omplete the proof. Proof: Statements Reasons 1. m 1 + m 2 = Given 2. [1] 2. Given 3. m 1 + m 2 = m 3 + m 4 3. Substitution Property 4. m 2 = m 3 4. Given 5. m 1 + m 2 = m 2 + m 4 5. [2] 6. m 1 = m 4 6. [3] a. [1] m 3 + m 4 = 90 [2] Subtraction Property of Equality [3] Substitution Property b. [1] m 3 + m 4 = 90 [2] Substitution Property [3] Subtraction Property of Equality c. [1] m 5 + m 6 = 90 [2] ddition Property of Equality [3] Substitution Property d. [1] m 5 + m 6 = 90 [2] Substitution Property [3] Subtraction Property of Equality 29

30 I: Geo Final Review 2014 nswer Section MULTIPLE HOIE 1. NS: Let m = x. Then m = (90 x). m = 3m + 2 x = 3(90 x) + 2 Substitute. x = 270 3x + 2 istribute. x = 272 3x ombine like terms. 4x = 272 dd 3x to both sides. x = ivide both sides by 4. x = 68 Simplify. The measure of is 68, so its complement is 22. heck your equation. The original angle is 2 degrees more than 3 times its complement. Simplify the terms when solving. This is the original angle. Find the measure of the complement. PTS: 1 IF: verage REF: Page 29 OJ: Using omplements and Supplements to Solve Problems NT: g ST: 6MG2.2 TOP: 1-4 Pairs of ngles KEY: complementary angles supplementary angles 1

31 I: 2. NS: Step 1 Solve for x. m = m efinition of angle bisector. (7x 1) = (4x + 8) Substitute 7x 1 for and 4x + 8 for. 7x = 4x + 9 dd 1 to both sides. 3x = 9 Subtract 4x from both sides. x = 3 ivide both sides by 3. Step 2 Find m. m = 7x 1 = 7(3) 1 = 20 heck your simplification technique. Substitute this value of x into the expression for the angle. This answer is the entire angle. ivide by two. PTS: 1 IF: verage REF: Page 23 OJ: Finding the Measure of an ngle NT: f ST: GE1.0 TOP: 1-3 Measuring and onstructing ngles KEY: angle bisectors angle measures 3. NS: m 1 = 3(20) + 30 = 90 ; m 2 = 5(20) 10 = 90 Substitute 20 for x. m 1 = m 2 = 90 Substitution Property of Equality l Ä m onverse of the lternate Interior ngles Theorem ngles 1 and 2 are alternate interior angles and are congruent. ngles 1 and 2 are alternate interior angles and are congruent. ngles 1 and 2 are alternate interior angles and are congruent. PTS: 1 IF: verage REF: Page 164 OJ: etermining Whether Lines are Parallel ST: GE7.0 TOP: 3-3 Proving Lines Parallel NT: g 2

32 I: 4. NS: The complete translation is (x, y) (x, y) + 6 (0, 4) = (x, y) + (0, 24) = (x + 0, y + 24) = (x, y + 24). (2, 0) (2, ) = (2, 24) (0, 2) (0, ) = (0, 26) ( 2, 0) ( 2, ) = ( 2, 24) (0, 2) (0, ) = (0, 22) dd 24 to each y-coordinate. dd 24 to each y-coordinate. dd 24 to each y-coordinate. PTS: 1 IF: verage REF: Page 833 OJ: pplication NT: c ST: GE22.0 TOP: 12-2 Translations 5. NS: The perimeter is 33 units and it is an equilateral triangle, so each side has length 11 units. Use this to solve for either side. 11 = 2y = y = 2y 4 = y 16 = y 2 4 = y n answer of 4 does not apply here. This is the length of each side. Now find the value of y. When solving 2y + 3 = 11, subtract 3 from both sides of the equation. The perimeter is 33 so the length of each side is 11. Set one of the sides equal to 11 and solve for y. PTS: 1 IF: dvanced NT: h ST: GE12.0 TOP: 4-1 lassifying Triangles 3

33 I: 6. NS: Two-column proof: Statements Reasons 1. is a right angle Given 2. m = efinition of a right angle 3. m = m 1 + m 2 3. ngle ddition Postulate 4. m 1 + m 2 = Substitution 5. m 1 = m 3 5. efinition of congruent angles 6. m 3 + m 2 = Substitution 7. 2 and 3 are complementary. 7. efinition of complementary angles In a paragraph proof, statements and reasons appear together. In a paragraph proof, statements and reasons appear together. In a paragraph proof, statements and reasons appear together. PTS: 1 IF: verage REF: Page 120 OJ: Reading a Paragraph Proof NT: a ST: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 7. NS: Subtract from 90º and simplify = 58.9 Find the measure of a complementary angle, not a supplementary angle. The measures of complementary angles add to 90 degrees. omplementary angles are angles whose measures have a sum of 90 degrees. PTS: 1 IF: asic REF: Page 29 OJ: Finding the Measures of omplements and Supplements NT: g ST: 6MG2.2 TOP: 1-4 Pairs of ngles KEY: complementary angles supplementary angles 4

34 I: 8. NS: Let the acute angles be M and N, with m M = m M + m N = 90 The acute angles of a right triangle are complementary m N = 90 Substitute 34.6 for m M. m N = 55.4 Subtract 34.6 from both sides. The measure of the other acute angle is less than 90 degrees. The two acute angles in a right triangle are complementary. This is the measure of the given angle. Find the measure of the other acute angle. PTS: 1 IF: asic REF: Page 225 OJ: Finding ngle Measures in Right Triangles NT: f ST: GE12.0 TOP: 4-2 ngle Relationships in Triangles 5

35 I: 9. NS: Step 1 raw two lines m, n, and a transversal p such that 1 and 2 are alternate interior angles. They should lie on opposite sides of the transversal p between lines m and n. Step 2 2 and 3 are corresponding angles. orresponding angles lie on the same side of the transversal p and on the same sides of lines m and n. dd 3 to the drawing. Step 3 3 and 4 are alternate exterior angles. They should lie on opposite sides of the transversal p and outside lines m and n. dd 4 to the drawing. 1 and 4 are corresponding angles. They lie on the same side of the transversal p and on the same sides of lines m and n. ngles 2 and 3 are corresponding angles and should lie on the same side of transversal p, on the same sides of lines m and n. ngles 2 and 3 are corresponding angles and should lie on the same side of transversal p, on the same sides of lines m and n. ngles 1 and 2 are alternate interior angles and should lie on opposite sides of transversal p, between lines m and n. PTS: 1 IF: dvanced NT: f ST: GE7.0 TOP: 3-1 Lines and ngles KEY: multi-step 6

36 I: 10. NS: m JKL = m JKM + m MKL [1] ngle ddition Postulate 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] ivision Property of Equality x = 12 Symmetric Property of Equality heck the justifications. The Segment ddition Postulate refers to segments, not angles. heck the properties. PTS: 1 IF: verage REF: Page 106 OJ: Solving an Equation in Geometry NT: e ST: GE1.0 TOP: 2-5 lgebraic Proof 11. NS: It is given that ST UT. Since RTS and RTU are right angles, RTS RTU by the Right ngle ongruence Theorem. y the Reflexive Property of ongruence, RT RT. Therefore, RTS RTU by SS. Segment SU being congruent to itself does not help in proving the triangles congruent. ngle SRT and angle URT are not right angles. heck the figure to see what is given. PTS: 1 IF: verage REF: Page 243 OJ: pplication NT: a ST: GE5.0 TOP: 4-4 Triangle ongruence: SSS and SS 12. NS: polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. polygon with 10 sides is called a decagon. hexagon has 6 sides. dodecagon has 12 sides. polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. PTS: 1 IF: asic REF: Page 382 OJ: Identifying Polygons NT: f ST: GE12.0 TOP: 6-1 Properties and ttributes of Polygons 7

37 I: 13. NS: 1 2 is given. From the diagram, 1 and 2 are corresponding angles. So by the onverse of the orresponding ngles Postulate, l Ä m. Use the given information. Use the onverse of the orresponding ngles Postulate. Use the onverse of the orresponding ngles Postulate. PTS: 1 IF: asic REF: Page 162 OJ: Using the converse of the orresponding ngles Postulate NT: g ST: GE7.0 TOP: 3-3 Proving Lines Parallel 14. NS: The Third ngles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. It is given that F and E. Therefore, E. So, m = 46. This is the complement. Use the Third ngles Theorem. This is the supplement. Use the Third ngles Theorem. The Third ngles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. PTS: 1 IF: dvanced NT: f ST: GE12.0 TOP: 4-2 ngle Relationships in Triangles 15. NS: is given. E because both are right angles. y the Vertical ngles Theorem, E. Therefore, E by S. djacent angles are angles in a plane that have their vertex and one side in common but have no interior points in common. ngle and angle E are not adjacent angles. Use S, not SS, to prove the triangles congruent. Look for vertical angles. PTS: 1 IF: asic REF: Page 253 OJ: pplying S ongruence NT: e ST: GE5.0 TOP: 4-5 Triangle ongruence: S S and HL 8

38 I: 16. NS: The image of (x,y) is (x 1, y + 3). (3, 4) (3 1, 4 + 3) = (1, 7) (2, 1) (2 1, 1 + 3) = (1, 2) (4, 12) (4 1, ) = (3, 15) Subtract 1 from each x-coordinate. dd 3 to each y-coordinate. Subtract 1 from each x-coordinate. dd 3 to each y-coordinate. Subtract 1 from each x-coordinate. dd 3 to each y-coordinate. PTS: 1 IF: verage REF: Page 833 OJ: rawing Translations in the oordinate Plane NT: c ST: GE22.0 TOP: 12-2 Translations 17. NS: [1] Steps 1 and 7 state that two angles and a nonincluded side of MLN and PLO are congruent. y S, MLN PLO. [2] Since MLN PLO, by PT, ML PL. Steps 1 and 7 state that two angles and a nonincluded side of triangle MLN and triangle PLO are congruent. Which triangle congruence theorem states that the triangles are congruent? efore using PT, you must prove that triangle MLN and triangle PLO are congruent. efore using PT, you must prove that triangle MLN and triangle PLO are congruent. Since steps 1 and 7 state that two angles and a nonincluded side are congruent, which triangle congruence theorem states that the triangles are congruent? PTS: 1 IF: verage REF: Page 261 OJ: Using PT in a Proof NT: a ST: GE5.0 TOP: 4-6 Triangle ongruence: PT 9

39 I: 18. NS: Substitute 10 for x in each expression: 3x + 4 = 3(10) + 4 = 34 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the onverse of the lternate Interior ngles Theorem. The angles are alternate interior angles. The lanes are parallel by the onverse of the lternate Interior ngles Theorem. The angles are alternate interior angles. PTS: 1 IF: verage REF: Page 165 OJ: pplication NT: a ST: GE7.0 TOP: 3-3 Proving Lines Parallel 19. NS: is equilateral. Equiangular triangles are equilateral. 2s 10 = s + 2 efinition of equilateral triangle. s = 12 Subtract s and add 10 to both sides of the equation. = 2s 10 = 2( 12) 10 Substitute 12 for s in the equation for. = 14 Simplify. = = 14 efinition of equilateral triangle. Substitute 14 for. Equiangular triangles are equilateral. Use = to solve for s, and then use = or = to find. y a corollary to the Isosceles Triangle Theorem, equiangular triangles are equilateral. Use = to solve for s, and then use = or = to find. This is s. Substitute s in the original equation to find. PTS: 1 IF: asic REF: Page 275 OJ: Using Properties of Equilateral Triangles NT: f ST: GE12.0 TOP: 4-8 Isosceles and Equilateral Triangles 10

40 I: 20. NS: Proof: Statements Reasons 1. 1 and 2 are supplementary. 1. Given 2. m 1 = Given 3. m 1 + m 2 = efinition of supplementary angles m 2 = Substitution Property 5. m 2 = Subtraction Property of Equality To get from Step 4 to Step 5, use subtraction, not substitution. The angles are supplementary, not complementary. heck to the given information. PTS: 1 IF: verage REF: Page 111 OJ: ompleting a Two-olumn Proof NT: a ST: GE2.0 TOP: 2-6 Geometric Proof 21. NS: EG = FH Given information EG = EF + FG Segment ddition Postulate FH = FG + GH Segment ddition Postulate EF + FG = FG + GH Substitution Property of Equality EF = GH Subtraction Property of Equality heck the properties. heck the steps. The ngle ddition Postulate refers to angles, not segments. PTS: 1 IF: verage REF: Page 110 OJ: Writing Justifications NT: a ST: GE2.0 TOP: 2-6 Geometric Proof 11

41 I: 22. NS: Proof: Statements Reasons 1. RT SU 1. Given 2. RTS and RTU are right angles. 2. efinition of perpendicular lines 3. RTS RTU 3. Right ngle ongruence Theorem 4. SRT URT 4. Given 5. S U 5. Third ngles Theorem 6. RS RU 6. Given 7. T is the midpoint of SU. 7. Given 8. ST UT 8. efinition of midpoint 9. RT RT 9. Reflexive Property of ongruence 10. RTS RTU 10. efinition of congruent triangles Use the correct property to show that the part is congruent to itself. Use the definition of perpendicular lines to show that the lines intersect to form right angles. ngle S and angle U are not vertical angles. Use a different justification for Reason 5. PTS: 1 IF: verage REF: Page 232 OJ: Proving Triangles ongruent NT: a ST: GE5.0 TOP: 4-3 ongruent Triangles 23. NS: reflection is a transformation that moves a figure (the preimage) by flipping it across a line. See if you can flip the image across the line to get a congruent image. PTS: 1 IF: asic REF: Page 824 OJ: Identifying Reflections NT: c ST: GE22.0 TOP: 12-1 Reflections 12

42 I: 24. NS: (3x) = (4x 24) lternate Exterior ngles Theorem x = 24 Subtract 4x from both sides. x = 24 ivide both sides by 1. m RST = 3x = 3(24) = 72 Substitute 24 for x. fter finding x, substitute to find the angle measure. Find the measure of angle RST, not the value of x. Find the measure of angle RST, not the supplement. PTS: 1 IF: verage REF: Page 156 OJ: Finding ngle Measures NT: g ST: GE7.0 TOP: 3-2 ngles Formed by Parallel Lines and Transversals 25. NS: Step 1 raw line l parallel to lines m and n. m 1 = m x + m y Step 2 Find m x. Use the orresponding ngles Postulate with lines m and l. m x = 35. Step 3 Find m y. Use the Same-Side Interior ngles Theorem with lines l and n. m y = = 50. Step 4 Find m 1. m 1 = m x + m y = = 85 Use the orresponding ngles Postulate and a theorem related to parallel lines and angle pairs. Use the orresponding ngles Postulate and a theorem related to parallel lines and angle pairs. Use the orresponding ngles Postulate and a theorem related to parallel lines and angle pairs. PTS: 1 IF: dvanced NT: f ST: GE7.0 TOP: 3-2 ngles Formed by Parallel Lines and Transversals KEY: multi-step 13

43 I: 26. NS: Since 1 3, m 1 m 3. Thus m 3 = Since 3 and 4 are complementary, m 4 = = Since 4 6, m 4 m 6. Thus m 6 = y the ngle ddition Postulate, 180 = m 4 + m 5 + m 6 = m Thus, m 5 = 53. ngle 1 and angle 3 are congruent. ongruent angles have the same measure. ngle 3 and angle 4 are complementary, not supplementary. The measure of angle 5 is 180 degrees minus the sum of the measure of angle 4 and the measure of angle 6. PTS: 1 IF: verage REF: Page 30 OJ: Problem-Solving pplication NT: g ST: 6MG2.2 TOP: 1-4 Pairs of ngles KEY: application complementary angles supplementary angles 27. NS: In a flowchart, reasons follow statements. Using the two-column proof, the statement that leads to Reason 2 is 1 and 2 are supplementary; 3 and 4 are supplementary. The reason that follows Statement 3 is ongruent Supplements Theorem. In a flowchart, reasons follow statements. ngles 1 and 2 are supplements, not complements. In a flowchart, reasons follow statements. PTS: 1 IF: verage REF: Page 119 OJ: Writing a Flowchart Proof NT: a ST: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 14

44 I: 28. NS: Step 1 raw line l parallel to lines m and n. Given: m y + m z = 90, x w, mä n Ä l Step 2 Use the lternate Interior ngles Theorem to find pairs of congruent angles. y x, z w m y = m x, m z = m w Step 3 Substitute x for y and w for z in the given m y + m z = 90. m x + m w = 90 Step 4 Use the definition of congruent angles and the given x w. m x = m w Step 5 To find m w, substitute w for x. m x + m w = 90 m w + m w = 90 2 m w = 90 m w = 45 Step 6 Find m 1. 1 and w are supplementary. m 1 + m w = 180 m = 180 m 1 = 135 raw a line parallel to the given parallel lines and use the lternate Interior ngles Theorem. raw a line parallel to the given parallel lines and use the lternate Interior ngles Theorem. raw a line parallel to the given parallel lines and use the lternate Interior ngles Theorem. PTS: 1 IF: dvanced NT: f ST: GE7.0 TOP: 3-4 Perpendicular Lines KEY: multi-step 15

45 I: 29. NS: Rotation 90 about the origin of any point (x, y) results in the image ( y, x). To rotate the point about M( 7, 9), determine h ä, the horizontal vector, and vä, the vertical vector, from G to M. Then move hä vertically from M, and move the opposite of vä horizontally from M. The result of the rotation is labeled G in the graph. The line y = 6 is a horizontal line passing through (0, 6). Reflection across a horizontal line involves movement of the point to the other side of the line, such that the image is the same distance from the line that the original point was. The x-coordinate does not change. The result of the reflection is labeled G in the graph. The point is reflected across a horizontal line, not a vertical line. Rotation occurs before reflection. Rotate around the given point, not around the origin. PTS: 1 IF: dvanced NT: c ST: GE22.0 TOP: 12-4 ompositions of Transformations 16

46 I: 30. NS: Proof: Statements 1. P is the midpoint of TQ and RS. 1. Given Reasons 2. TP QP, RP SP 2. efinition of midpoint 3. TPR QPS 3. Vertical ngles Theorem 4. TPR QPS 4. SS There is not enough information to show that segment RT is congruent to segment SQ. Use the correct postulate to prove the triangles congruent. ngle PRT and angle PSQ are not vertical angles. PTS: 1 IF: verage REF: Page 244 OJ: Proving Triangles ongruent NT: a ST: GE5.0 TOP: 4-4 Triangle ongruence: SSS and SS 31. NS: In the diagram, rays FG and FH share a common endpoint F and form the line GH. Opposite rays form a line. Opposite rays are two rays that have a common endpoint and form a line. Opposite rays form a line. PTS: 1 IF: asic REF: Page 7 OJ: rawing Segments and Rays NT: d ST: GE1.0 TOP: 1-1 Understanding Points Lines and Planes KEY: opposite rays 17

47 I: 32. NS: Statements 2 and 3 determine the measures of two angles of the triangle. Statement 1 determines the length of the included side. y S, the triangle must be unique. There is not enough information for SS. raw a diagram to help you. raw a diagram. There is enough information to determine a unique triangle. There is not enough information for SSS. raw a diagram to help you. PTS: 1 IF: verage REF: Page 252 OJ: Problem-Solving pplication NT: f ST: 7MR3.1 TOP: 4-5 Triangle ongruence: S S and HL 33. NS: decagon has 10 sides and 10 vertices. sum of exterior angle measures = 360 Polygon Exterior ngle Sum Theorem measure of one exterior angle = = 36 regular decagon has 10 congruent exterior angles, so divide the sum by 10. The measure of each exterior angle of a regular decagon is 36. ivide 360 by the number of sides. ivide 360 by the number of sides the polygon has. ivide by the number of sides the polygon has. PTS: 1 IF: verage REF: Page 384 OJ: Finding Exterior ngle Measures in Polygons NT: f ST: GE12.0 TOP: 6-1 Properties and ttributes of Polygons 18

48 I: 34. NS: translation is a transformation where all the points of a figure are moved the same distance in the same direction. This transformation is a translation because all of the points have moved the same distance in the same direction. heck where all the points have moved. PTS: 1 IF: asic REF: Page 831 OJ: Identifying Translations NT: c ST: GE22.0 TOP: 12-2 Translations 35. NS: orresponding angles lie on the same side of a transversal, on the same sides of the two lines the transversal crosses. So, 8 and 4 are corresponding angles. orresponding angles lie on the same side of a transversal, on the same sides of two lines. ngle 4 and angle 1 are supplementary angles, not corresponding angles. ngle 5 and angle 7 are vertical angles, not corresponding angles. PTS: 1 IF: asic REF: Page 147 OJ: lassifying Pairs of ngles NT: g ST: GE7.0 TOP: 3-1 Lines and ngles 19

49 I: 36. NS: : This appears to be a reflection. rotation is a transformation that turns a figure around a fixed point, called the center of rotation. If the transformation is a rotation, then the figure on the left rotates clockwise 90 about a fixed point to look like this: See if you can rotate the image around a fixed point and get a congruent image. PTS: 1 IF: asic REF: Page 839 OJ: Identifying Rotations NT: c ST: GE22.0 TOP: 12-3 Rotations 37. NS: 1a. y the Linear Pair Theorem, F and are supplementary and G and are supplementary. 1b. Given F G, by the ongruent Supplements Theorem,. 2. by the definition of an angle bisector. 3. by the Reflexive Property of ongruence 4. Two angles and a nonincluded side of and are congruent. y S,. 5. Since, by PT. For reason 1, check whether the linear pairs are complementary or supplementary. For statement 2, use the fact that line segment bisects angle, not angle. Find the correct property that states that a line segment is congruent to itself. PTS: 1 IF: verage REF: Page 260 OJ: Proving orresponding Parts ongruent ST: GE5.0 TOP: 4-6 Triangle ongruence: PT NT: a 20

50 I: 38. NS: 1. and G are alternate interior angles and Ä GH. Thus by the lternate Interior ngles Theorem, G. 2. and HFG are alternate exterior angles and Theorem, HFG. Ä FH. Thus by the lternate Exterior ngles If line is parallel to line FG, are angle and angle HFG alternate interior angles or alternate exterior angles? You switched the definitions of alternate interior and alternate exterior angles. If line segment is parallel to line segment GH, are angle and angle G alternate exterior angles or alternate interior angles? PTS: 1 IF: verage REF: Page 254 OJ: Using S to Prove Triangles ongruent NT: a ST: GE5.0 TOP: 4-5 Triangle ongruence: S S and HL 39. NS: The image of (x, y) is ( y, x). R(4, 1) R (1, 4) S(5, 3) S ( 3, 5) Q(3, 1) Q ( 1, 3) Graph the preimage and the image. This is a rotation by 180 about the origin. This is a reflection across the y-axis. The rotation is 90 counterclockwise, not clockwise. PTS: 1 IF: verage REF: Page 841 OJ: rawing Rotations in the oordinate Plane ST: GE22.0 TOP: 12-3 Rotations NT: c 21

51 I: 40. NS: Step 1 Find the sum of the interior angle measures. (n 2)180 Polygon ngle Sum Theorem = (45 2) gon has 45 sides, so substitute 45 for n. = 7740 Simplify. Step 2 Find the measure of one interior angle = 172 The interior angles are, so divide by Subtract, not add, 2 from the number of sides. Subtract 2, not 1, from the number of sides. ccording to the Polygon ngle Sum Theorem, the sum of the interior angle measures is the product of 180 and the number of sides minus 2. PTS: 1 IF: verage REF: Page 384 OJ: Finding Interior ngle Measures and Sums in Polygons NT: f ST: GE12.0 TOP: 6-1 Properties and ttributes of Polygons 41. NS: Proof: Statements Reasons 1. t l, Given 2. t m 2. If 2 intersecting lines form linear pair of s lines. 3. mä l 3. If 2 lines to the same line lines Ä. Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the lines are parallel. Switch Reason 2 and Reason 3. Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the lines are parallel. PTS: 1 IF: asic REF: Page 173 OJ: Proving Properties of Lines NT: a ST: GE2.0 TOP: 3-4 Perpendicular Lines 22

52 I: 42. NS: In a flowchart, reasons flow from the statement above. The statement above Reason 2 is + F = F +. The statement above Reason 3 is + F = F ; F + = F. Reasons flow from the statement above. Reasons flow from the statement above. Reasons flow from the statement above. PTS: 1 IF: verage REF: Page 118 OJ: Reading a Flowchart Proof NT: a ST: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 43. NS: To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. orresponding angles lie on the same side of the transversal l, on the same sides of lines n and m. To find which line is the transversal for a given angle pair, locate the line that connects the vertices. lternate interior angles lie on opposite sides of the transversal, between two lines. To find which line is the transversal for a given angle pair, locate the line that connects the vertices. PTS: 1 IF: verage REF: Page 147 OJ: Identifying ngle Pairs and Transversals ST: GE7.0 TOP: 3-1 Lines and ngles 44. NS: = a + 7 = = 10 = 4a 2 = 4(3) 2 = 12 2 = 10 = 6a 2 = 6(3) 2 = 18 2 = 16 = 16 NT: g.. by the Reflexive Property of ongruence. So by SSS. Substitute 3 for a. heck the measures of segment and segment. Use the correct postulate. PTS: 1 IF: verage REF: Page 244 OJ: Verifying Triangle ongruence NT: a ST: GE2.0 TOP: 4-4 Triangle ongruence: SSS and SS 23

53 I: 45. NS: F and 3 are adjacent angles. Their noncommon sides, F 3 also form a linear pair. and G, are opposite rays, so F and djacent angles form a linear pair if and only if their noncommon sides are opposite rays. Two angles are adjacent if they have a common vertex and a common side, but no common interior points. PTS: 1 IF: verage REF: Page 28 OJ: Identifying ngle Pairs NT: g ST: 6MG2.1 TOP: 1-4 Pairs of ngles KEY: angle pairs linear pair adjacent 46. NS: E N Third ngles Theorem m E = m N efinition of congruent angles (x 2 ) = (4x 2 75) Substitute x 2 for m E and 4x 2 75 for m N. 3x 2 = 75 Subtract 4x 2 from both sides. x 2 = 25 ivide both sides by 3. So m E = 25. Since m E = m N, m N = 25. These are the measures of angles F and P, not angles E and N. Use the Third ngles Theorem. The Third ngles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. PTS: 1 IF: verage REF: Page 226 OJ: pplying the Third ngles Theorem NT: f ST: GE12.0 TOP: 4-2 ngle Relationships in Triangles 24

54 I: 47. NS: E = + E Segment ddition Postulate 6x = ( 4x + 8) + 27 Substitute 6x for E and 4x + 8 for. 6x = 4x + 35 Simplify. 2x = 35 Subtract 4x from both sides. 2x 2 = 35 2 ivide both sides by 2. x = 35 or Simplify. E = 6x = 6( 17.5) = 105 You found the value of x. Find the length of the specified segment. heck your equation. Make sure you are not subtracting instead of adding. You found the length of a different segment. PTS: 1 IF: verage REF: Page 15 OJ: Using the Segment ddition Postulate NT: a ST: GE1.0 TOP: 1-2 Measuring and onstructing Segments KEY: segment addition postulate 48. NS: translation is a transformation where all the points of a figure are moved the same distance in the same direction. This transformation is not a translation because not all of the points have moved the same distance. heck where all the points have moved. PTS: 1 IF: asic REF: Page 831 OJ: Identifying Translations NT: c ST: GE22.0 TOP: 12-2 Translations 25

55 I: 49. NS: (x) = (3x 70) orresponding ngles Postulate 0 = 2x 70 Subtract x from both sides. 70 = 2x dd 70 to both sides. 35 = x ivide both sides by 2. m = 3x 70 m = 3(35) 70 = 35 Substitute 35 for x. Simplify. Use the orresponding ngles Postulate. First, set the measures of the corresponding angles equal to each other. Then, solve for x and substitute in the expression (3x 70). If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. PTS: 1 IF: verage REF: Page 155 OJ: Using the orresponding ngles Postulate NT: g ST: GE7.0 TOP: 3-2 ngles Formed by Parallel Lines and Transversals 50. NS: Proof: Statements Reasons 1. m 1 + m 2 = Given 2. m 3 + m 4 = Given 3. m 1 + m 2 = m 3 + m 4 3. Substitution Property 4. m 2 = m 3 4. Given 5. m 1 + m 2 = m 2 + m 4 5. Substitution Property 6. m 1 = m 4 6. Subtraction Property of Equality To get from Step 4 to Step 5, use substitution, not subtraction. To get from Step 4 to Step 5, use substitution, not addition. heck the given information. PTS: 1 IF: verage REF: Page 112 OJ: Writing a Two-olumn Proof from a Plan ST: GE2.0 TOP: 2-6 Geometric Proof NT: a 26