Skills Practice Skills Practice for Lesson 9.1
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1 Skills Practice Skills Practice for Lesson.1 Name ate Glass Lanterns Introduction to ongruence Vocabulary Identify all parts of the figure that are described by the given term. F E 1. corresponding angles 2. corresponding sides 3. congruent figures Problem Set List all of the terms that apply to each pair of polygons: similar, congruent, or neither cm J G 8 cm cm 6 cm H I K L hapter Skills Practice 311
2 6. 5 ft 5 ft 7. 4 m m 4 m m 6 ft 6 ft 6 ft 6 ft 70 4 m 4 m 70 4 m 4 m 5 ft 5 ft cm 7 cm etermine whether each pair of shapes is best described as similar or congruent. 10. Figures and EF are both equilateral triangles. The ratio of the lengths of the corresponding sides is 1 : Figures GHI and JKL are both equilateral triangles. The ratio of the lengths of the corresponding sides is 1 : Figures and EFGH are both squares. The ratio of the lengths of the corresponding sides is 2 : Figures IJKL and MNOP are both squares. The ratio of the lengths of the corresponding sides is 1 : hapter Skills Practice
3 Name ate Identify the corresponding angles of the congruent polygons. 14. HGEF 15. EF E F G H E F 16. GHI LKJ 17. IJLK NOPM G L M H I J K I J N K L O P Identify the corresponding sides of the congruent polygons. 18. EF 1. GHIJ KNML G K N F E H I J L M hapter Skills Practice 313
4 20. HGFE 21. EF E G H E F F etermine whether the given polygons are congruent. Explain your answer. 22. triangles and 12 cm 7 cm 7 cm 12 cm 314 hapter Skills Practice
5 Name ate 23. triangles E and E 5 mm E 5 mm 24. triangles and E 10 in. 5 in. 10 in. E hapter Skills Practice 315
6 25. triangles M and M M Given two congruent polygons, determine the measure of the unknown side or angle. 26. What is m? FE 64? 85 E F 316 hapter Skills Practice
7 Name ate 27. What is m? EF? F E 28. What is the length of? E 8 cm? 30 E hapter Skills Practice 317
8 2. What is the length of? 16 in. 12 in. 318 hapter Skills Practice
9 Skills Practice Skills Practice for Lesson.2 Name ate omputer Graphics Proving Triangles ongruent by Using SSS and SS Vocabulary efine each term in your own words. 1. congruent 2. theorem 3. two-column proof 4. paragraph proof Problem Set etermine what additional information you would need to prove that the triangles are similar. 5. What information would you need to use the Side-Side-Side ongruence Theorem to prove the triangles are congruent? 4 cm 4 cm 8 cm 10 cm hapter Skills Practice 31
10 6. What information would you need to use the Side-Side-Side ongruence Theorem to prove the triangles are congruent? 6 mm 6 mm 6 mm 2 mm 7. What information would you need to use the Side-ngle-Side ongruence Theorem to prove the triangles are congruent? 5 in. 5 in. 8. What information would you need to use the Side-ngle-Side ongruence Theorem to prove these triangles are congruent? 320 hapter Skills Practice
11 Name ate omplete the paragraph proof to show that the triangles are congruent.. Use the SSS Similarity Postulate to prove that EF. 10 ft 6 ft 8 ft E 8 ft F Use the to calculate the length of the unknown side of each triangle ft F 2 E 2 EF 2 F ft So and. Use the to show that triangle triangle EF. The figure shows that EF, and we just found that and. So by the, triangle triangle EF. If the two triangles are similar, then all of their angles are congruent, by the definition of. ecause all of the corresponding sides and angles are congruent, by the definition of, EF. 10. Use the SSS Similarity Postulate to prove that EF. 13 ft 5 ft 5 ft E 12 ft F Use the to calculate the length of the unknown side of each triangle ft F F ft So and. Use the SSS Similarity Postulate to show that. The figure shows that, and we just found that EF and F. So by the SSS Similarity Postulate,. If the two triangles are similar, then all of their angles are congruent, by the definition of. ecause all of the corresponding sides and angles are congruent, by the definition of, EF. hapter Skills Practice 321
12 11. Use the SSS Similarity Postulate to prove that. 10 m 26 m 24 m Use the to calculate the measure of. ngles and are, so they must be congruent. This fact means that is also a. Use the to calculate the length of the unknown side of each triangle m m So and. Use the SSS Similarity Postulate to show that triangle triangle. We are given that, and we just found that and. So by the, triangle triangle. If the two triangles are similar, then all of their angles are congruent, by the definition of. ecause all of the corresponding sides and angles are congruent, by the definition of,. 322 hapter Skills Practice
13 Name ate 12. Use the SSS Similarity Postulate to prove that E. in. 12 in. 12 in. E in. Use the ngles and E are means that is also a angle. to calculate the measure of., so they must be congruent. This fact Use the to calculate the length of the hypotenuse of each triangle in. E 2 2 E 2 E in. So. Use the to show that triangle EF. We are given that and E, and we just found that. So by the, E. If the two triangles are similar, then all of their angles are congruent, by the definition of. ecause all of the corresponding sides and angles are congruent, by the definition of, E. hapter Skills Practice 323
14 omplete the two-column proof to show that the triangles are congruent. 13. Prove that M M. M Statement 1. M M, M M 1. Reason 2., 2. efinition of congruence 3. M and M are vertical angles. 3. efinition of 4. M M SS Similarity Postulate 6. M M 7. 1, M M 1 6. efinition of 7. ivision Property of Equality M, M 10. efinition of similar triangles 11. M M 11. efinition of 324 hapter Skills Practice
15 Name ate 14. Prove that EFM GHM. E F M G H 1. EM GM, Statement FM HM 1. Reason 2. EM GM, FM HM 2. efinition of 3. EMF and GMH are. 3. efinition of vertical angles 4. EMF GMH SS Similarity Postulate 6. GM HM GH 6. efinition of 7. 1, 1 GM HM EF 1 GH 8.. EF GH. 10. MGH, GHM 10. efinition of similar triangles 11. EFM GHM 11. efinition of hapter Skills Practice 325
16 15. Prove that. 12 cm 5 cm 5 cm 12 cm Statement 1., 1. Given Reason 2., 2. efinition of Reflexive Property of Equality efinition of SS Similarity Postulate 7., 7. efinition of 8.. 1, 1, efinition of congruent triangles 326 hapter Skills Practice
17 Name ate 16. Prove that EFH GHF. E 20 cm F 15 cm 15 cm H 20 cm G 1. EF GH, Statement 1. Given Reason 2. EF GH, EH GF 2. efinition of 3. FH HF efinition of congruence 5. FEH HGF EFH, EHF 7. efinition of similar triangles 8. EF GH. EF GH 1, FH HF FH HF 1, HE 1 8. FG HE 1. FG efinition of congruent triangles Use a paragraph proof to show the triangles are congruent. 17. Use the SS ongruence Theorem to prove triangles and are congruent. hapter Skills Practice 327
18 18. Given that figure EFIH is a square, use the SSS ongruence Theorem to prove triangles EGH and HGI are congruent. E F G H I 1. Use the SSS ongruence Theorem to prove triangles E and FE are congruent. E F 328 hapter Skills Practice
19 20. Use the SS ongruence Theorem to prove triangles F and G are congruent. E F G hapter Skills Practice 32
20 330 hapter Skills Practice
21 Skills Practice Skills Practice for Lesson.3 Name ate Wind Triangles Proving Triangles ongruent by Using S and S Vocabulary Provide an example of each term. 1. postulate 2. theorem 3. congruent hapter Skills Practice 331
22 Problem Set omplete each two-column proof to prove that the triangles are congruent. 4. Use the S ongruence Theorem to prove that. Statement Reason 1., Reflexive Property of ongruence Use the S ongruence Theorem to prove that EHG IFG. E F G H I Statement Reason 1. EHG IFG Given 3. EGH and IGF are vertical angles. 3. efinition of 4. EGH IGF S ongruence Theorem 332 hapter Skills Practice
23 Name ate 6. Use the S ongruence Theorem to prove that EF. cm 15 cm E 12 cm F Statement 1. and EF are. 1. Reason 2. E 2 2. Pythagorean Theorem 3. F Substitution Property of Equality F F 2 5. Symmetric Property of Equality 6. F F 7. Property of square roots Given. F efinition of congruence 11. m 0º and m FE 0º m 12. Transitive Property of Equality 13. FE 13. efinition of congruence Given S ongruence Theorem hapter Skills Practice 333
24 7. Use the S ongruence Theorem to prove that GHI JKL. L G 24 in. 26 in. 10 in. H I J K Statement Reason 1. GHI and JKL are right triangles GI 2 2. Pythagorean Theorem 3. HI Substitution Property of Equality HI HI 2 5. Symmetric Property of Equality 6. HI HI 7. Property of square roots 8. KL 8. Given. HI KL efinition of congruence 11. m GHI 0º and m JKL 0º m GHI 12. Transitive Property of Equality 13. GHI 13. efinition of congruence 14. KLJ 14. Given 15. GHI JKL hapter Skills Practice
25 Name ate Use a paragraph proof to prove that the triangles are congruent. 8. Use the S ongruence Theorem to prove that E E. Use the S ongruence Theorem to prove that FGI IHF. F G H I hapter Skills Practice 335
26 10. Given that E is a parallelogram, use the S ongruence Theorem to prove that E. E 336 hapter Skills Practice
27 Name ate 11. Given that FGJI is a square, use the S ongruence Theorem to prove that FHI GKJ. F G H I J K etermine whether there is sufficient information to prove that the triangles are congruent. Explain your answer. 12. Is there sufficient information to prove that EF? F 10 ft 12 ft E hapter Skills Practice 337
28 13. Is there sufficient information to prove that GHI JLK? G J 7 in. 7 in H I K L 14. Is there sufficient information to prove that FE? 11 cm 11 cm E F 338 hapter Skills Practice
29 Name ate 15. Is there sufficient information to prove that GHI GJI? G 4 in. 4 2 in. H 4 in. I J hapter Skills Practice 33
30 340 hapter Skills Practice
31 Skills Practice Skills Practice for Lesson.4 Name ate Planting Grape Vines Proving Triangles ongruent by Using HL Vocabulary Write the term that best completes each statement. 1. In a right triangle, the is the side of the right triangle that is opposite the right angle. 2. (n) is a statement that has been proven to be true. 3. The of a right triangle are the two sides of the triangle that form the right angle. 4. Two figures are if they have the same size and the same shape. 5. (n) is a triangle that contains a right angle. 6. (n) is a proof consisting of two columns in which the left column contains mathematical statements that are organized in logical steps, and the right column contains the reasons for each mathematical statement. hapter Skills Practice 341
32 Problem Set omplete each two-column proof to prove the Hypotenuse-Leg ongruence Theorem. 7. omplete the two-column proof to prove that FE. E F Statement Reason 1. and FE are right triangles and 2. Given 3. FE and F 3. efinition of 4. 2 and F 2 4. Pythagorean Theorem 5. F 2 FE Substitution Property of Equality 6. FE 2 E 2 FE Substitution Property of Equality 7. E Property of square roots.. efinition of congruence 10. FE hapter Skills Practice
33 Name ate 8. omplete the two-column proof to prove that GHI JLK. G J H I K L Statement 1. GHI and JLK are right triangles HI and GI 2. Given Reason 3. and 3. efinition of congruence 4. GI 2 GH 2 HI 2 and JK 2 JL 2 LK JK 2 GH 2 LK 2 5. Substitution Property of Equality 6. JL 2 LK 2 GH 2 LK Subtraction Property of Equality Property of square roots. JL GH. efinition of SSS ongruence Theorem. omplete the two-column proof to prove that FE. 26 ft 26 ft 10 ft E 10 ft F Statement 1. and are right triangles. 1. Given F and 10 FE 2. Reason and Pythagorean Theorem E E Property of square roots 7. and, and 7. efinition of congruence SSS ongruence Theorem hapter Skills Practice 343
34 10. omplete the two-column proof to prove that GHI LKJ. G J K 15 cm 25 cm 25 cm 15 cm H I L Statement 1. GHI and LKJ are. 1. Given 2. GH 15 LK and GI 25 LJ 2. Reason and Pythagorean Theorem HI KJ HI 2 KJ 2 5. Subtraction Property of Equality 6. HI KJ HI KJ, GH LK, and GI LJ SSS ongruence Theorem Use a paragraph proof to prove that the two triangles are congruent. 11. Given that is a rectangle, use the S ongruence Theorem to prove that E E. E 344 hapter Skills Practice
35 Name ate 12. Given that FG and JI are congruent, use the S ongruence Theorem to prove that FGH JIH. F H I G J 13. Given that triangle KLN is an isosceles triangle, use the HL ongruence Theorem to prove that KLM NLM. K M L N hapter Skills Practice 345
36 14. Use the HL ongruence Theorem to prove that POQ RSQ. O in. 6 in. P Q 6 in. in. S R 15. Given that E is a square, use the SS ongruence Theorem to prove that FE. 36 ft 15 ft E 15 ft F 346 hapter Skills Practice
37 Name ate 16. Given that GHLK is a square, use the HL ongruence Theorem to prove that GJK LIH. G 16 cm H 12 cm I J 12 cm K L hapter Skills Practice 347
38 348 hapter Skills Practice
39 Skills Practice Skills Practice for Lesson.5 Name ate Triangle onstructions onstructing Triangles Vocabulary Match each definition to its corresponding term. 1. an angle included between two given sides a. construct 2. create an exact copy of a figure using a compass and b. included angle straightedge or patty paper 3. a point on a line and all points on the line to one side of the point c. line segment 4. a portion of a line between two points d. ray Problem Set Use a compass, a straightedge, and the three given line segments to construct a triangle hapter Skills Practice 34
40 Use a compass, a straightedge, and the two given line segments to construct a triangle hapter Skills Practice
41 Name ate Use a compass, a straightedge, and the three given angles to construct a triangle hapter Skills Practice 351
42 16. Use a compass, a straightedge, the two given angles, and the given line segment to construct a triangle etermine whether the triangles formed by each method would be congruent, similar, or neither. Explain your answer. 21. Would two triangles constructed using the same three line segments be congruent, similar, or neither? 352 hapter Skills Practice
43 Name ate 22. Would two triangles constructed using the same two line segments and the included angle be congruent, similar, or neither? 23. Would two triangles constructed using the same three angles be congruent, similar, or neither? 24. Would two triangles constructed using the same two line segments be congruent, similar, or neither? hapter Skills Practice 353
44 354 hapter Skills Practice
45 Skills Practice Skills Practice for Lesson.6 Name ate Koch Snowflake Fractals Vocabulary efine each term in your own words. 1. fractal 2. Koch Snowflake Problem Set Given one stage of each Koch Snowflake, draw the next stage. 3. stage 0: stage 1: 4. stage 0: stage 1: hapter Skills Practice 355
46 5. stage 1: stage 2: 6. stage 1: stage 2: etermine whether each pair of figures described are similar, congruent, or neither. 7. a new triangle added in stage 1 of a Koch Snowflake and the entire Koch Snowflake in stage 0 8. a new triangle added in stage 2 of a Koch Snowflake and a new triangle added in stage 3 of a Koch Snowflake. a new triangle added in stage 4 of a Koch Snowflake and another new triangle added in stage 4 of a Koch Snowflake 10. a new triangle added in stage 2, and the entire Koch Snowflake in stage 1 nswer each question about Koch Snowflakes. 11. s the stage number increases, does the number of sides increase or decrease? 12. s the stage number increases, does the length of each side increase or decrease? 13. s the stage number increases, does the area of each new triangle added increase or decrease? 14. s the stage number increases, does the area of the entire figure increase or decrease? 356 hapter Skills Practice
47 Name ate Given each stage, determine the number of sides for a Koch Snowflake. 15. How many sides does a Koch Snowflake have at stage 1? 16. How many sides does a Koch Snowflake have at stage 2? 17. How many sides does a Koch Snowflake have at stage 5? 18. How many sides does a Koch Snowflake have at stage 6? 1. How many sides does a Koch Snowflake have at stage 10? 20. How many sides does a Koch Snowflake have at stage 11? Given each stage and the length of a side at stage 0, determine the length of each side of a Koch Snowflake. 21. If the length of a side is 2 centimeters at stage 0, what is the length of a side at stage 1? 22. If the length of a side is 5 inches at stage 0, what is the length of a side at stage 2? 23. If the length of a side is 12 inches at stage 0, what is the length of a side at stage 4? hapter Skills Practice 357
48 24. If the length of a side is meters at stage 0, what is the length of a side at stage 3? 25. If the length of a side is 225 centimeters at stage 0, what is the length of a side at stage 6? 26. If the length of a side is 162 centimeters at stage 0, what is the length of a side at stage 5? Given each stage and the length of a side at stage 0, calculate the perimeter of a Koch Snowflake. 27. If the length of a side is 4 inches at stage 0, what is the perimeter at stage 2? 28. If the length of a side is 6 centimeters at stage 0, what is the perimeter at stage 3? 2. If the length of a side is 18 centimeters at stage 0, what is the perimeter at stage 5? 358 hapter Skills Practice
49 Name ate 30. If the length of a side is 36 meters at stage 0, what is the perimeter at stage 6? Given each stage and the length of a side at stage 0, calculate the area of one new triangle of a Koch Snowflake. Simplify, but do not evaluate any radicals. 31. If the length of a side is 12 millimeters at stage 0, what is the area of one new triangle at stage 1? 32. If the length of a side is 15 centimeters at stage 0, what is the area of one new triangle at stage 2? 33. If the length of a side is inches at stage 0, what is the area of one new triangle at stage 3? hapter Skills Practice 35
50 34. If the length of a side is 24 millimeters at stage 0, what is the area of one new triangle at stage 4? 360 hapter Skills Practice
4-2 Triangle Congruence Conditions. Congruent Triangles - C F. and
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