2D Geometry Chapter Questions

1. Name the different types of transformations.

2. Explain how to tell which figure is the pre-image and which is the image.

3. How are dilations different from translations and how are they different from reflections?

4. How are the coordinates of a pre-image affected by a translation? By a rotation?

5. If two parallel lines are cut by a transversal, name the types of angles that are congruent.

6. Explain why a remote exterior angle is equal to the sum the opposite interior angles.

NJ Center for Teaching and Learning ~ 1 ~ www.njctl.org Chapter Problems Translations Class work Complete the translation and write the rule to describe the translation. 1. Translate the figure right 2 units and up 1 unit. a. (x, y)

4. Translate the figure right 7 units and down 3 units. a. (x, y) 2. Translate the figure up 1 unit. a. (x, y)

5. Translate the figure left 3 units. a. (x, y) 3. Translate the figure left 1 unit and down 2 units. a. (x, y)

NJ Center for Teaching and Learning ~ 2 ~ www.njctl.org 6. Translate the figure left 2 units and up 4 units. a. (x, y)

Homework Complete the translation and write the rule to describe the translation. 7. Translate the figure left 3 units and up 3 units. a. (x, y)

10. Translate the figure right 1 unit and up 8. Tra 2 units. nslate the figure right 2 units and up 2 units. a. (x, y) a. (x, y)

11. Translate the figure down 4 units. 9. Translate the figure left 3 units and a. (x, y) down 4 units. a. (x, y)

NJ Center for Teaching and Learning ~ 3 ~ www.njctl.org 12. Translate the figure right 4 units and down 2 units. a. (x, y)

Rotations Class work Describe the rotation about the origin pictured from Figure 1 to Figure 2. 13.

Complete the described rotation. 14. A rotation of 90° clockwise about the origin. 15. A rotation of 180° clockwise about the origin.

NJ Center for Teaching and Learning ~ 4 ~ www.njctl.org Describe each rotation about the origin in another way. 16. 40° clockwise 17. 125° counter-clockwise 18. 200° counter-clockwise 19. 250° clockwise 20. 180° clockwise

Name the coordinates of the point after the described rotation about the origin. 21. A (5, -2) 90° clockwise 22. H (-3, -4) 90° counter-clockwise 23. B (12, -8) half-turn 24. X (-8, 2) 90° clockwise

NJ Center for Teaching and Learning ~ 5 ~ www.njctl.org PARCC-type Questions: 25. Parallelogram A’B’C’D’ (not shown) is the image of parallelogram ABCD after a rotation of 90 clockwise about the origin. Which statements about parallelogram A’B’C’D’ are true? Select each correct statement.

26. Rhombus E’F’G’H’ (not shown) is the image of rhombus EFGH after a rotation of 90 clockwise about the origin. Which statements about rhombus E’F’G’H’ are true? Select each correct statement.

Homework 27. Describe the rotation about the origin pictured from Figure 1 to Figure 2.

NJ Center for Teaching and Learning ~ 6 ~ www.njctl.org Complete the described rotation. 28. A rotation of 90° clockwise about the origin. 29. A rotation of 180° clockwise about the origin.

Describe each rotation about the origin in another way. 30. 15° counter-clockwise 31. 210° clockwise 32. 130° counter-clockwise 33. 300° clockwise 34. 175° clockwise

Name the coordinates of the point after the described rotation about the rotation. 35. F (-4, -9) 90° counter-clockwise 36. G (8, -2) 90° clockwise 37. R (-5, 2) 90° counter-clockwise 38. T (4, -13) Half-turn

PARCC-type Questions:

NJ Center for Teaching and Learning ~ 7 ~ www.njctl.org 39. Trapezoid A’B’C’D’ (not shown) is the image of trapezoid ABCD after a rotation of 90 counterclockwise about the origin. Which statements about trapezoid A’B’C’D’ are true? Select each correct statement.

40. Rectangle E’F’G’H’ (not shown) is the image of rectangle EFGH after a rotation of 90 clockwise about the origin. Which statements about rectangle E’F’G’H’ are true? Select each correct statement.

Reflections Class work 41. Give an example in nature of a reflection.

42. Complete the described reflections. 43.

NJ Center for Teaching and Learning ~ 8 ~ www.njctl.org 47.

44. 45. 48. 46. 49. 50. 51. 52. 53. 54. 55. 56.

57.

NJ Center for Teaching and Learning ~ 9 ~ www.njctl.org 58.

63. 64. 65. Homework 66. Complete the described reflections. 67. 59. Describe the reflection shown in each 68. graph. 60.

61. 69. 62.

NJ Center for Teaching and Learning ~ 10 ~ www.njctl.org 78.

70. 71. 72. 79. 73. 74.

75. 76. 77.

NJ Center for Teaching and Learning ~ 11 ~ www.njctl.org 80. Describe the reflection shown in each 84. graph. 85. 81. 86. 82.

83. 87. 88. 89. 90.

NJ Center for Teaching and Learning ~ 12 ~ www.njctl.org 91. PARCC-type Questions 92. Three congruent figures are shown on the coordinate plane. Use the figures to answer the next two questions. Select a transformation from each group of choices to make the statement true.

93. 94. Part B: 95. Transforming figure 1 with a sequence of 2 transformations can also create figure 3. Figure 1 can be transformed to figure 3 by: a) a rotation of 90 clockwise about the origin b) a translation 8 units down c) a reflection across the x-axis 96. 97. followed by d) a rotation 90 counterclockwise about the origin e) a translation 4 units right and 3 units down f) a reflection across the y-axis 98. 99. Dilations 100. Class work 101. Given a scale factor of 3, what happens to the coordinates for the dilation of a pre- image? 102. 103. Given the coordinates of the pre-image and the scale factor, find the coordinates of the image. 104. R (-3, 6); scale factor = ½ 105. W (2, 9); scale factor = 3 106. D (4, -2); scale factor = 4 107. Q (1, -3); scale factor = 1.5 108. 109. Given the coordinates of the pre-image and image, determine the scale factor. 110. (4, -2) (8, -4) 111. (-3, -1) (-9, -3)

NJ Center for Teaching and Learning ~ 13 ~ www.njctl.org 112. (0, -5) (0, -2.5) 113. (8, 12) (2, 3) 114. (5, -4) (12.5, -10) 115. 116. Homework 117. Given a scale factor of ½, what happens to the coordinates for the dilation of a pre- image? 118. 119. Given the coordinates of the pre-image and the scale factor, find the coordinates of the image. 120. G (2, 4); scale factor = 2 121. J (-3, 9); scale factor = ⅓ 122. T (-4, 5); scale factor = 3½ 123. Y (-2, 6); scale factor = 4 124. 125. Given the coordinates of the pre-image and image, determine the scale factor. 126. (3, -5) (9, -15) 127. (-2, 7) (-4, 14) 128. (8, 0) (2, 0) 129. (3, -8) (4.5, -12) 130. (6, 3) (3, 1.5) 131. 132. Symmetry 133. Class work 134. Draw all lines of symmetry for the figure shown. 135. 76. 136. 137. 138. 139. 140. 141. Determine if the figure has rotational symmetry. If so, list the degrees where it occurs. 142. 77. 78. 143. 144. 145. 146. 147. 148. 79. 80. 149. 150. 151. 152.

NJ Center for Teaching and Learning ~ 14 ~ www.njctl.org 153. 154. Homework 155. Draw all lines of symmetry for the figure shown. 81. 82. 156. 157. 158. 159. 160. 161. 162. 163. Determine if the figure has rotational symmetry. If so, list the degrees where it occurs. 83. 84. 164. 165. 166. 167. 168. 169. 170. 85. 86. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180.

NJ Center for Teaching and Learning ~ 15 ~ www.njctl.org 181. Congruence & Similarity 182. Class work 183. For each problem:  Determine if the two figures are congruent, similar, or neither.  Explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. 87. 188. 184.

185. 189. 88. 190. 186. 90. 191.

187. 89. 192. 91.

NJ Center for Teaching and Learning ~ 16 ~ www.njctl.org 193.

194. 195. Homework 196. For each problem:  Determine if the two figures are congruent, similar, or neither.  Explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations 92. 200. 197.

201. 198. 202. 199. 94. 93.

NJ Center for Teaching and Learning ~ 17 ~ www.njctl.org 203. 206. 96. 207.

204. 95. 205. 208.

209. 210. Special Pairs of Angles 211. Class work 212. Using the figure shown, name as many of the following pairs of angles as you can find. 97. Vertical Angles 213. 98. Alternate Interior Angles 214. 99. Alternate Exterior Angles 215. 100. Same Side Interior Angles 216. 101. Adjacent Angles 217.

NJ Center for Teaching and Learning ~ 18 ~ www.njctl.org 102. Corresponding Angles 218. 219. 220. Using what you know about special pairs of angles, find the missing measure(s) in each figure. 103.

221. 222. 223. 104.

224. 225. 226. 227. 105.

NJ Center for Teaching and Learning ~ 19 ~ www.njctl.org 228. 229. 230. Homework 231. Using the figure shown, name as many of the following pairs of angles as you can find. 106. Vertical Angles 232. 107. Alternate Interior Angles 233. 108. Alternate Exterior Angles 234. 109. Same Side Interior Angles 235. 110. Adjacent Angles 236. 111. Corresponding Angles 237. 238. 239. Using what you know about special pairs of angles, find the missing measure(s) in each figure. 240. 112.

241. 242. 243.

NJ Center for Teaching and Learning ~ 20 ~ www.njctl.org 244. 245. 246. 247. 248. 113.

249. 114.

250. 251. 252. Remote Exterior Angles 253. Class work 254. Based on the Remote Exterior Angle Theorem, find the missing angle. (diagrams NOT drawn to scale) 255. 256. 115. 116.

257. 258. 259. 260. 261.

NJ Center for Teaching and Learning ~ 21 ~ www.njctl.org 262.

263.

264. 50° x°

265.

266.

267.

268. 117. 118. 269. x°

270.

NJ Center for Teaching and Learning ~ 22 ~ www.njctl.org 271. Based on the Remote Exterior Angle Theorem, find the value of x. (diagrams NOT drawn to scale) 272.

273. 119.

274.

275.

276.

277. 120.

278.

279.

280.

281.

282.

283. 121.

284.

285.

286.

287.

288. 122.

289.

290.

291.

292.

293.

NJ Center for Teaching and Learning ~ 23 ~ www.njctl.org 294. Remote Exterior Angles

295. Homework 296. 297. Based on the Remote Exterior Angle Theorem, find the missing angle. (diagrams NOT drawn to scale) 298. 299. 123. 124.

300. 301. 302. 303. 304. 305. 306.

307. 35°

308.

309.

310.

311.

312. 125. 126. 313. x°

314.

315.

316.

317.

318.

319.

320.

321. Based on the Remote Exterior Angle Theorem, find the value of x. (diagrams NOT drawn to scale) 322.

NJ Center for Teaching and Learning ~ 24 ~ www.njctl.org 323.

324. 127.

325.

326. 128.

327.

328.

329.

330.

331.

332. 129.

333.

334.

335.

336.

337. 130.

338.

339. 340.

NJ Center for Teaching and Learning ~ 25 ~ www.njctl.org 341. Unit Review 342. 343. 2D Geometry Multiple Choice Questions 344. 1. What rule describes the translation left 3 units and up 4 units? a.a. (x,y) (x + 3, y + 4) a.b. (x,y) (x + 3, y - 4) i.c. (y, x) (x – 3, y + 4) 345. d. (x,y) (x – 3, y + 4) 346. 2. What translation does the rule (x,y)(x – 7, y) describe? 347. a. Right 7 units 348. b. Left 7 units 349. c. Up 7 units 350. d. Down 7 units 351. 3. What direction are rotations unless you are told otherwise? 4. a. clockwise 352. b. south 353. c. counterclockwise 354. d. north 355. 4. What are the new coordinates of a point A (3, -2) after a 90° rotation counterclockwise about the origin? 356. a. (3, 2) 357. b. (-3, 2) 358. c. (-3, -2) 359. d. (2, 3) 360. 361. 5. What are the new coordinates of a point B (-5, 8) after a half-turn rotation about the origin? 362. a. (-8, -5) 363. b. (-5, -8) 364. c. (8, 5) 365. d. (5, -8) 366. 367. 6. What are the coordinates of a point C (-1, 6 ) after a dilation with respect to the origin with a scale factor of 1/3? 368. a. (-3, 18) 369. b. (6, -1) 370. c. (-1/3, 2) i.d. (3, -18) 371.

NJ Center for Teaching and Learning ~ 26 ~ www.njctl.org 7. What is the scale factor of a point if it changes as follows after a dilation with respect to the origin? 372. (4, 23) (20, 115) 373. a. -5 374. b. 5 375. c. 16 376. d. 1/5 377. 378. 8. How many lines of symmetry does the figure have? 379. a. 10 380. b. 3 381. c. 5 382. d. 0 383. 9. Determine if the figure has rotational symmetry. If so, list the degrees where it occurs. 384. a. Rotational Symmetry; 180° 385. b. Rotational Symmetry; 90°, 180°, 360° 386. c. Rotational Symmetry; 90°, 180°, 270°, 360° 387. d. No Rotational Symmetry 388. 389. 10. What makes two figures similar? 390. a. same size, same shape 391. b. same shape, different angles, proportional sides 392. c. same shape, congruent angles, proportional sides 393. d. same size, same shape, different angles 394. 395. 11. Which of the following terms best describes the two figures? 396. a. congruent 397. b. similar 398. c. neither congruent nor similar 399. 400.

NJ Center for Teaching and Learning ~ 27 ~ www.njctl.org 401. 12. Trapezoid B’C’D’E’ (not shown) is the image of trapezoid BCDE after a rotation of 180 about the origin. Which statements about trapezoid B’C’D’E’ are true. Select each correct statement.

402. 403. 404. 405. For 13 - 16, use the figure shown. 406. 13. What are the vertical angles? 407. a. ∠ 1 & ∠ 2 ; ∠ 3 & ∠ 4 408. b. ∠ 5 & ∠ 6 ; ∠ 7 & ∠ 8 409. c. ∠ 1 & ∠ 4 ; ∠ 5 & ∠ 8 ; ∠ 2 & ∠ 4 ; ∠ 6 & ∠ 8 Figure: Note: 410. d. ∠ 1 & ∠ 4 ; ∠ 2 & ∠ 3 ; ∠ 5 & ∠ 8 ; ∠ 6 & ∠ 7 411. 412. 14. What are the alternate interior angles? 413. a. ∠ 3 & ∠ 6 ; ∠ 4 & ∠ 5 414. b. ∠ 1 & ∠ 8 ; ∠ 2 & ∠ 7 415. c. ∠ 1 & ∠ 2 ; ∠ 5 & ∠ 6 416. d. ∠ 3 & ∠ 4 ; ∠ 7 & ∠ 8 417. 418. 15. What is the relationship between ∠2 and ∠7? 419. a. alternate exterior angles 420. b. alternate interior angles 421. c. complimentary angles 422. d. vertical angles 423. 424. 16. What is the relationship between ∠4 and ∠6? 425. a. supplementary angles 426. b. complementary angles 427. c. alternate interior angles

NJ Center for Teaching and Learning ~ 28 ~ www.njctl.org i.d. vertical angles 428. 429. 430. 431. Use the diagram to the right to answer questions 17-19. 432. Diagram NOT drawn to scale. 433. 434. 435. 17. If m1 = 67° and m2 = 38°, then what is the 436. measure of 4? 437. a. not enough information 438. b. 100° 439. c. 75° 440. d. 105° 441. 442. 18. If m1 = 72° and m4 = 108°, then what is the 443. measure of 2? 444. a. not enough information 445. b. 36° 446. c. 72° 447. d. 85° 448. 449. 19. If m1 = (3x)°, and the m2 = (5x)°, and m4 = (180 – x)° then what is x? 450. a. 30° 451. b. 20° 452. c. 40° 453. d. not enough information 454. 455. 456. 457. 458. 459. 460.

NJ Center for Teaching and Learning ~ 29 ~ www.njctl.org 461. 462. 2D Geometry Short Constructed Response Questions 463. 464. 465. 20. Translate the figure shown using the following rule: (x,y) (x – 11, y + 4) 466. 467. 468. 469. 470. 471. 472. 473. 474. 475. 476. 477. 478. 479. 480. 481. 482. 483. 484. 485. 486. 487. 488. 21. Rotate the figure 270º clockwise about the origin. 489. 490. 491. 492. 493. 494. 495. 496. 497. 498.

NJ Center for Teaching and Learning ~ 30 ~ www.njctl.org 499. 500. 501. 502. 503. 504. 505. 506. 507. 508. 22. Reflect the figure across the x-axis. 509. 510. 511. 512. 513. 514. 515. 516. 517. 518. 519. 520. 521. 522. 523. 524. 525. 526. 527. 528. 529. 530. 23. Describe the following rotations in another way: 531. a.a. 63° clockwise about the origin 532. a.b. 237° counterclockwise about the origin 533. a.c. 55° counterclockwise about the origin 534. 535.

NJ Center for Teaching and Learning ~ 31 ~ www.njctl.org 536. 24. Circle the following figures that have symmetry: 537. 538. 539. 540. 541. 542. 543. 544. 545. 546. 547. 548. 549. 550. 551. 552. 553. 25. List three different pairs of angles that are supplementary. 554. 555. 556. 557. 558. 26. List three different pairs of angles that are complementary. 559. 560. 561. 562. 563. 564. 565. 2D Geometry Extended Constructed Response Questions 566. 567. 568. 27. Draw a figure on graph paper. Label the coordinates. Write the following rules in your own words. Apply the following rules to your object. Label the coordinates of your transformed object. 569. a.a. translate: (x,y) (x + 4, y – 3) 570. 571. b. rotate 90 counterclockwise about the origin 572. 573. c. reflect across the x axis.

NJ Center for Teaching and Learning ~ 32 ~ www.njctl.org 574. 28. Draw a figure on graph paper with at least three lines of symmetry. Label 575. the lines of symmetry. 576. 577. 29. Given that , fill in all the angle measurements in the following figure: 578. 579. 580. 581. 582. 583. 584. 585. 586. 587. 588. 589. 590. 591. 592. 593. Answer Key

594. 596. 1. (x, y) (x+2, y+1) 595.

3. (x, y) (x-1, y-2)

2. (x, y) (x, y+1)

NJ Center for Teaching and Learning ~ 33 ~ www.njctl.org 597. 604.

598. 6. (x, y) (x-2, y+4) 599. 605. 600. 601. 602. 4. (x, y) (x+7, y-3) 603.

606. 607. 7. (x, y) (x-3, y+3)

5. (x, y) (x-3, y)

NJ Center for Teaching and Learning ~ 34 ~ www.njctl.org 608. 610.

8. (x, y) (x+2, y+2) 609. 611. 612. 613. 614. 10. (x, y) (x+1, y+2) 615.

9. (x, y) (x-3, y-4)

11. (x, y) (x, y-4)

NJ Center for Teaching and Learning ~ 35 ~ www.njctl.org 616. 621.

15. A rotation of 180° clockwise about the 12. (x, y) (x+4, y-2) origin 617. 622. 618.

16. 320° CCW 17. 235° CW 18. 160° CW 19. 110° CCW 619. 20. 180° CCW / Half-turn 13. 180° / Half-turn 21. A’ (-2, -5) 14. 22. H’ (4, -3) 620. 23. B’ (-12, 8) 24. X’ (2, 8) 25. b, d, e 26. b, e 27. 180° CCW / Half-turn 28. A rotation of 90° clockwise about the origin

NJ Center for Teaching and Learning ~ 36 ~ www.njctl.org 623. 631. 624. 625. 626. 627. 628. 29. A rotation of 180° clockwise about the origin 629. 632. 44.

30. 345° CW 31. 150° CCW 45. 32. 230° CW 633. 33. 60° CCW 34. 185° CCW 35. F’ (9, -4) 36. G’ (-2, -8) 37. R’ (-2, -5) 38. T’ (-4, 13) 39. b, c, e 40. a, d, f 41. Answers will vary. 42.

634. 46. Across y= -1 635. 47. Across y = x 636. 630. 637. 43.

NJ Center for Teaching and Learning ~ 37 ~ www.njctl.org 48. 655. 638. 639.

51. 49. 656. 640. 657. 641. 642. 643. 644. 645. 646. 647. 648. 649. 650. 651. 652. 50. 653. 654. 52. Across x= 2 53. Across x-axis 54. Part A: b & d 658. Part B: c & e 55. They are multiplied by 3 56. R’ (-1½, 3) 57. W’ (6, 27) 58. D’ (16, -8) 59. Q’ (1½, -4.5) 60. SF = 2 61. SF = 3 62. SF = ½

NJ Center for Teaching and Learning ~ 38 ~ www.njctl.org 1 82. 4 63. SF = 64. SF = 2½ 65. They are multiplied by ½ 66. G’ (4, 8) 67. J’ (-1, 3) 68. T’ (-14, 17.5) 69. Y’ (-8, 24) 70. SF = 3 671. 71. SF = 2 83. Yes; 72°, 144°, 216°, 288° 72. SF = ¼ 84. Yes; 180° 73. SF = 1½ 85. Yes; Infinitely Many 74. SF = ½ 86. None 2 75. 3 87. Similar; Dilation ; (x, y) (x+5, y-2) 88. Congruent; Reflect over y-axis; 672. (x, y) (x, y-3) 89. Congruent: (x, y) (x-8, y-5) 90. Congruent; Rotate 90° CCW 91. Similar; Dilation Scale Factor ¼; Rotate 659. 90° CCW 92. Congruent; Reflect over y=3; 76. 673. (x, y) (x+3, y) 93. Congruent; Rotate 90° CW, translate right 2 units, translate down 2 units 94. Similar; Dilation Scale Factor ½; Reflect over y-axis 95. Similar; Dilation S.F. 2; 674. (x, y) (x-12, y-8) 660. 96. Congruent; Rotate 90° CW, than 77. Yes; 120°, 240° Reflection over y=0 OR 78. Yes; 60°, 120°, 180°, 240°, 300° Reflection over line y = -x 79. Yes; 180° 97. 80. No 98. 661. 99. 662. 100. 663. 101. 81. 102. 664. 103. 665. 104. 666. 105. 667. 106. 668. 107. 669. 108. 670.

NJ Center for Teaching and Learning ~ 39 ~ www.njctl.org 109. 690. 8. c 675. 691. 9. c 110. 692. 10. c 676. 693. 11. b 111. 694. 12. a, c & d 112. 695. 13. d 696. 14. a 113. 697. 15. a 677. 698. 16. a 678. , unknown because lines are 699. 17. d not parallel 700. 18. b 679. , unknown because lines are 701. 19. b not parallel 702. 20. A’(-3,-1); B’(-6,-1); C’(

114. -7,1) 115. 80° 703. 21. A’(3,-2); B’(6,2); 116. 140° C’(6,7); D’(3,7) 117. 80° 118. 75° 704. 119. 60° 705. 22. A’(-3,-2); B’(-6,-2); C’(- 120. x = 20 6,-7); D’(-3,7) 121. x = 5 706. 23. a. 297° 122. x = 35 counterclockwise 123. 130° 707. b. 123° clockwise 124. 140° b. 305° clockwise 125. x = 30 708. 24. square, flower, and 126. x = 60° arrow should be circled. 127. x = 50 709. 25. answers will vary 128. x = 7 710. 26. answers will vary 129. x = 46 711. 27. answers will vary 130. x = 6 712. 28. answers will vary 680. 681. Unit Review Answers 682. 713. 29. ∠1,∠4,∠5,∠8 = 70° 683. 1. d 684. 2. b 685. 3. c 686. 4. d 714. 3,∠6,∠7 = 110° 687. 5. d ∠ 688. 6. c 689. 7. b 715.

NJ Center for Teaching and Learning ~ 40 ~ www.njctl.org