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Mrs. Rivas โˆ 

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1 Mrs. Rivas โˆ ๐‘ฎ ๐‘บ๐‘จ โˆ ๐‘ป ๐‘น๐‘ฌ โˆ†๐‘ป๐‘จ๐‘บ ๐‘ป๐‘บ โˆ†๐‘จ๐‘ป๐‘บ โˆ ๐‘จ Lesson 4-1 1. ๏ƒ๐‘† โ‰… 2. ๐บ๐‘… โ‰…
โˆ†๐‘บ๐‘จ๐‘ป โ‰… โˆ†๐‘ฎ๐‘น๐‘ฌ. Complete each congruence statement โˆ ๐‘ฎ ๐‘บ๐‘จ 1. ๏ƒ๐‘† โ‰… 2. ๐บ๐‘… โ‰… โˆ ๐‘ป ๐‘น๐‘ฌ 3. ๏ƒ๐ธ โ‰… 4. ๐ด๐‘‡ โ‰… โˆ†๐‘ป๐‘จ๐‘บ ๐‘ป๐‘บ 5. โˆ†๐ธ๐‘…๐บ โ‰… 6. ๐ธ๐บ โ‰… โˆ†๐‘จ๐‘ป๐‘บ โˆ ๐‘จ 7. โˆ†๐‘…๐ธ๐บ โ‰… 8. โˆ ๐‘… โ‰…

2 Mrs. Rivas Yes; corresponding sides and corresponding angles are โ‰….
Lesson 4-1 State whether the figures are congruent. Justify your answers. 9. โˆ†๐ด๐ต๐น; โˆ†๐ธ๐ท๐ถ Yes; corresponding sides and corresponding angles are โ‰….

3 Mrs. Rivas No; the only corresponding part that is โ‰… is ๐‘ผ๐‘ฝ .
Lesson 4-1 State whether the figures are congruent. Justify your answers. 10. โˆ†๐‘‡๐‘ˆ๐‘‰; โˆ†๐‘ˆ๐‘‰๐‘Š No; the only corresponding part that is โ‰… is ๐‘ผ๐‘ฝ .

4 Mrs. Rivas Yes; corresponding sides and corresponding angles are โ‰….
Lesson 4-1 State whether the figures are congruent. Justify your answers. 11. โง ๐‘‹๐‘Œ๐‘๐‘‰;โง  ๐‘ˆ๐‘‡๐‘๐‘‰ Yes; corresponding sides and corresponding angles are โ‰….

5 Mrs. Rivas Yes; corresponding sides and corresponding angles are โ‰….
Lesson 4-1 State whether the figures are congruent. Justify your answers. 12.โˆ†๐ด๐ต๐ท; โˆ†๐ธ๐ท๐ต Yes; corresponding sides and corresponding angles are โ‰….

6 Mrs. Rivas ๐‘จ๐‘บ๐‘จ โˆ ๐‘ปโ‰…โˆ ๐‘บ ๐‘ป๐’€ โ‰… ๐‘บ๐‘พ โˆ ๐’€โ‰…โˆ ๐‘พ Lessons 4-2 and 4-3 13.
Can you prove the two triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not possible and tell what other information you would need. 13. โˆ ๐‘ปโ‰…โˆ ๐‘บ ๐‘ป๐’€ โ‰… ๐‘บ๐‘พ โˆ ๐’€โ‰…โˆ ๐‘พ ๐‘จ๐‘บ๐‘จ

7 Mrs. Rivas ๐‘บ๐‘จ๐‘บ ๐‘ฌ๐‘จ โ‰… ๐‘ท๐‘ณ ๐‘จ๐‘ณ โ‰… ๐‘ณ๐‘จ โˆ ๐‘ฌ๐‘จ๐‘ณโ‰…โˆ ๐‘ท๐‘ณ๐‘จ Lessons 4-2 and 4-3 14..
Can you prove the two triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not possible and tell what other information you would need. 14.. ๐‘ฌ๐‘จ โ‰… ๐‘ท๐‘ณ ๐‘จ๐‘ณ โ‰… ๐‘ณ๐‘จ โˆ ๐‘ฌ๐‘จ๐‘ณโ‰…โˆ ๐‘ท๐‘ณ๐‘จ ๐‘บ๐‘จ๐‘บ

8 Mrs. Rivas ๐‘ต๐’ ๐’‘๐’๐’”๐’”๐’Š๐’ƒ๐’๐’† Lessons 4-2 and 4-3 15.
Can you prove the two triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not possible and tell what other information you would need. 15. ๐‘ต๐’ ๐’‘๐’๐’”๐’”๐’Š๐’ƒ๐’๐’†

9 Mrs. Rivas ๐‘บ๐‘บ๐‘บ ๐‘ช๐‘ต โ‰… ๐‘จ๐‘ถ ๐‘ต๐‘จ โ‰… ๐‘ถ๐‘ช ๐‘จ๐‘ช โ‰… ๐‘ช๐‘จ Lessons 4-2 and 4-3 16.
Can you prove the two triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not possible and tell what other information you would need. 16. ๐‘ช๐‘ต โ‰… ๐‘จ๐‘ถ ๐‘ต๐‘จ โ‰… ๐‘ถ๐‘ช ๐‘จ๐‘ช โ‰… ๐‘ช๐‘จ ๐‘บ๐‘บ๐‘บ

10 Mrs. Rivas ๐’๐‘ท bisects ๐‘ฟ๐’€ means that ๐‘ฟ๐‘ด โ‰… ๐’€๐‘ด . ๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€ Given
Lessons 4-2 and 4-3 17. Given: ๐‘ƒ๐‘‹ โ‰… ๐‘ƒ๐‘Œ , ๐‘๐‘ƒ bisects ๐‘‹๐‘Œ . Prove: โˆ†๐‘ƒ๐‘‹๐‘ ๏€ โˆ†๐‘ƒ๐‘Œ๐‘ ๐’๐‘ท bisects ๐‘ฟ๐’€ means that ๐‘ฟ๐‘ด โ‰… ๐’€๐‘ด . ๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€ Given ๐‘ท๐’ โ‰… ๐‘ท๐’ Reflective property of โ‰… โˆ†๐‘ท๐‘ฟ๐‘ดโ‰…โˆ†๐‘ท๐’€๐‘ด SSS โˆ ๐‘ฟ๐‘ท๐‘ดโ‰…โˆ ๐’€๐‘ท๐‘ด Def. of โ‰…. ๐‘ท๐’ โ‰… ๐‘ท๐’ Reflective property of โ‰… โˆ†๐‘ท๐‘ฟ๐’โ‰…โˆ†๐‘ท๐’€๐’ by SAS.

11 Mrs. Rivas It is given that ๐’Žโˆ ๐Ÿ โ‰… ๐’Žโˆ ๐Ÿ and ๐’Žโˆ ๐Ÿ‘ โ‰… ๐’Žโˆ ๐Ÿ’.
Lessons 4-2 and 4-3 18. Given: โˆ 1 โ‰… โˆ 2, โˆ 3 โ‰… โˆ 4, ๐‘ƒ๐ท โ‰… ๐‘ƒ๐ถ , P is the midpoint of ๐ด๐ต . Prove: โˆ†๐ด๐ท๐‘ƒ โ‰… โˆ†๐ต๐ถ๐‘ƒ It is given that ๐’Žโˆ ๐Ÿ โ‰… ๐’Žโˆ ๐Ÿ and ๐’Žโˆ ๐Ÿ‘ โ‰… ๐’Žโˆ ๐Ÿ’. By the โฆจ Addition. Post., ๐’Žโˆ ๐‘ซ๐‘ท๐‘จ โ‰… ๐’Žโˆ ๐‘ช๐‘ท๐‘ฉ. Since ๐‘ƒ is the midpoint of ๐‘จ๐‘ฉ , ๐‘ท๐‘จ โ‰… ๐‘ท๐‘ฉ . It is given that ๐‘ท๐‘ซ โ‰… ๐‘ท๐‘ช , so โˆ†๐‘จ๐‘ซ๐‘ท โ‰… โˆ†๐‘ฉ๐‘ช๐‘ท by SAS.

12 Mrs. Rivas ๐‘ถ๐‘ณ โˆฅ ๐‘ด๐‘ต , so โˆ ๐‘ถ๐‘ณ๐‘ต โ‰… โˆ ๐‘ด๐‘ต๐‘ณ.
Lesson 4-4 21. Given: ๐ฟ๐‘‚ โ‰… ๐‘€๐‘ , ๐ฟ๐‘‚ โˆฅ ๐‘€๐‘ Prove: โˆ ๐‘€๐ฟ๐‘ โ‰… โˆ ๐‘‚๐‘๐ฟ ๐‘ถ๐‘ณ โˆฅ ๐‘ด๐‘ต , so โˆ ๐‘ถ๐‘ณ๐‘ต โ‰… โˆ ๐‘ด๐‘ต๐‘ณ. ๐‘ณ๐‘ต โ‰… ๐‘ณ๐‘ต by the Re๏ฌ‚exive Prop. of โ‰…. Since ๐‘ณ๐‘ถ โ‰… ๐‘ด๐‘ต , โˆ†๐‘ด๐‘ณ๐‘ต โ‰…โˆ†๐‘ถ๐‘ต๐‘ณ by SAS, Then โˆ ๐‘ด๐‘ณ๐‘ตโ‰…โˆ ๐‘ถ๐‘ต๐‘ณ by CPCTC.

13 Mrs. Rivas ๐‘ถ๐‘บ โ‰… ๐‘ถ๐‘บ Re๏ฌ‚exive Prop. of โ‰…. Since โˆ ๐‘ป โ‰…โˆ ๐‘ฌ and โˆ ๐‘ป๐‘บ๐‘ถ โ‰…โˆ ๐‘ฌ๐‘ถ๐‘บ
Lesson 4-4 22. Given: โˆ ๐‘‚๐‘‡๐‘† โ‰… โˆ ๐‘‚๐ธ๐‘†, โˆ ๐ธ๐‘‚๐‘† โ‰… โˆ ๐‘‚๐‘†๐‘‡ Prove: ๐‘‡๐‘‚ โ‰… ๐ธ๐‘† ๐‘ถ๐‘บ โ‰… ๐‘ถ๐‘บ Re๏ฌ‚exive Prop. of โ‰…. Since โˆ ๐‘ป โ‰…โˆ ๐‘ฌ and โˆ ๐‘ป๐‘บ๐‘ถ โ‰…โˆ ๐‘ฌ๐‘ถ๐‘บ โˆ†๐‘ป๐‘บ๐‘ถ โ‰… โˆ†๐‘ฌ๐‘ถ๐‘บ by AAS ๐‘ป๐‘ถ โ‰… ๐‘ฌ๐‘บ by CPCTC

14 Mrs. Rivas โˆ ๐Ÿ โ‰…โˆ ๐Ÿ,โˆ ๐Ÿ‘ โ‰…โˆ ๐Ÿ’ Given ๐‘ด is the midpoint of ๐‘ท๐‘น , ๐’”๐’ ๐‘ท๐‘ด โ‰… ๐‘น๐‘ด .
Lesson 4-4 23. Given: โˆ 1 โ‰… โˆ 2,โˆ 3 โ‰…โˆ 4, ๐‘€ is the midpoint of ๐‘ƒ๐‘… Prove: โˆ†๐‘ƒ๐‘€๐‘„ โ‰… โˆ†๐‘…๐‘€๐‘„ โˆ ๐Ÿ โ‰…โˆ ๐Ÿ,โˆ ๐Ÿ‘ โ‰…โˆ ๐Ÿ’ Given ๐‘ด is the midpoint of ๐‘ท๐‘น , ๐’”๐’ ๐‘ท๐‘ด โ‰… ๐‘น๐‘ด . ๐‘บ๐‘ธ โ‰… ๐‘บ๐‘ธ Reflexive Property ๐‘ธ๐‘ด โ‰… ๐‘ธ๐‘ด Reflexive Property โˆ†๐‘ท๐๐’โ‰…โˆ†๐‘น๐‘ธ๐‘บ AAS โˆ†๐‘ท๐๐Œโ‰…โˆ†๐‘น๐‘ธ๐‘ด SSS ๐‘ท๐‘ธ โ‰… ๐‘น๐‘ธ CPCTC โˆ ๐‘ท๐‘ธ๐‘ดโ‰…โˆ ๐‘น๐‘ด๐‘ธ CPCTC

15 Mrs. Rivas โˆ ๐Ÿ โ‰…โˆ ๐Ÿ,โˆ ๐‘จ๐‘ท๐‘ถ โ‰…โˆ ๐‘ฉ๐‘ธ๐‘ถ Supplement of โ‰… angles are โ‰….
Lesson 4-4 24. Given: ๐‘ƒ๐‘‚ = ๐‘„๐‘‚, โˆ 1 โ‰… โˆ 2, Prove: โˆ ๐ด โ‰… โˆ ๐ต โˆ ๐Ÿ โ‰…โˆ ๐Ÿ,โˆ ๐‘จ๐‘ท๐‘ถ โ‰…โˆ ๐‘ฉ๐‘ธ๐‘ถ Supplement of โ‰… angles are โ‰…. ๐‘ท๐‘ถ โ‰… ๐‘ธ๐‘ถ Given โˆ ๐‘ท๐‘ถ๐‘จ โ‰…โˆ ๐‘ฉ๐‘ถ๐‘ธ Vertical โฆž are โ‰…. โˆ†๐‘ถ๐‘ท๐‘ธโ‰…โˆ†๐‘ฉ๐‘ธ๐‘ถ ASA โˆ ๐‘จ โ‰…โˆ ๐‘ฉ CPCTC

16 Mrs. Rivas ๐Ÿ๐’™+๐Ÿ๐ŸŽ=๐Ÿ”๐ŸŽ ๐Ÿ๐’™=๐Ÿ“๐ŸŽ ๐’™=๐Ÿ๐Ÿ“ ๐Ÿ”๐ŸŽ ๐Ÿ”๐ŸŽ ๐Ÿ”๐ŸŽ Lesson 4-5
Find the value of each variable 25. ๐Ÿ๐’™+๐Ÿ๐ŸŽ=๐Ÿ”๐ŸŽ ๐Ÿ”๐ŸŽ ๐Ÿ๐’™=๐Ÿ“๐ŸŽ ๐’™=๐Ÿ๐Ÿ“ ๐Ÿ”๐ŸŽ ๐Ÿ”๐ŸŽ

17 Mrs. Rivas ๐Ÿ๐Ÿ“+๐Ÿ—๐ŸŽ+๐’™=๐Ÿ๐Ÿ–๐ŸŽ ๐Ÿ๐Ÿ๐Ÿ“+๐’™=๐Ÿ๐Ÿ–๐ŸŽ ๐’™=๐Ÿ”๐Ÿ“ ๐Ÿ๐Ÿ“ Lesson 4-5
Find the value of each variable 26. ๐Ÿ๐Ÿ“+๐Ÿ—๐ŸŽ+๐’™=๐Ÿ๐Ÿ–๐ŸŽ ๐Ÿ๐Ÿ“ ๐Ÿ๐Ÿ๐Ÿ“+๐’™=๐Ÿ๐Ÿ–๐ŸŽ ๐’™=๐Ÿ”๐Ÿ“

18 Mrs. Rivas ๐’+๐Ÿ๐Ÿ‘๐Ÿ“=๐Ÿ๐Ÿ–๐ŸŽ ๐’›=๐Ÿ’๐Ÿ“ ๐’™+๐’š=๐Ÿ๐Ÿ‘๐Ÿ“ ๐’™=๐Ÿ’๐Ÿ“ ๐Ÿ’๐Ÿ“+๐’š=๐Ÿ๐Ÿ‘๐Ÿ“ ๐’š=๐Ÿ—๐ŸŽ ๐‘น๐’†๐’Ž๐’๐’•๐’† ๐’‚๐’๐’ˆ๐’๐’†๐’”
Lesson 4-5 Find the value of each variable 27. ๐’+๐Ÿ๐Ÿ‘๐Ÿ“=๐Ÿ๐Ÿ–๐ŸŽ ๐’›=๐Ÿ’๐Ÿ“ ๐‘น๐’†๐’Ž๐’๐’•๐’† ๐’‚๐’๐’ˆ๐’๐’†๐’” ๐‘ฐ๐’”๐’๐’”๐’„๐’†๐’๐’†๐’” ๐‘ป๐’“๐’Š๐’‚๐’๐’ˆ๐’๐’†๐’” base angles are congruent ๐’™+๐’š=๐Ÿ๐Ÿ‘๐Ÿ“ ๐’™=๐Ÿ’๐Ÿ“ ๐Ÿ’๐Ÿ“+๐’š=๐Ÿ๐Ÿ‘๐Ÿ“ ๐’š=๐Ÿ—๐ŸŽ

19 Mrs. Rivas ๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€ Given โˆ ๐Ÿ โ‰…โˆ ๐Ÿ Isosceles โˆ† Theorem
Lesson 4-5 28. Given: โˆ 5 โ‰… โˆ 6, ๐‘ƒ๐‘‹ โ‰… ๐‘ƒ๐‘Œ Prove: โˆ†๐‘ƒ๐ด๐ต is isosceles. ๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€ Given โˆ ๐Ÿ โ‰…โˆ ๐Ÿ Isosceles โˆ† Theorem โˆ ๐Ÿ‘ โ‰…โˆ ๐Ÿ’ โ‰… Supplement Theorem โˆ ๐Ÿ“ โ‰…โˆ ๐Ÿ” Given โˆ†๐‘ท๐‘ฟ๐‘จโ‰…โˆ†๐‘ท๐’€๐‘ฉ ASA ๐‘ท๐‘จ โ‰… ๐‘ท๐‘ฉ CPCTC and โˆ†๐‘ท๐‘จ๐‘ฉ is isosceles by Isosceles โˆ† Theorem

20 Mrs. Rivas ๐‘จ๐‘ท โ‰… ๐‘ฉ๐‘ท and ๐‘ท๐‘ช โ‰… ๐‘ท๐‘ซ Given โˆ ๐‘จ๐‘ท๐‘ช โ‰…โˆ ๐‘ฉ๐‘ท๐‘ซ Vertical โฆž are โ‰….
Lesson 4-5 29. Given: ๐ด๐‘ƒ โ‰… ๐ต๐‘ƒ , ๐‘ƒ๐ถ โ‰… ๐‘ƒ๐ท Prove: โˆ†๐‘„๐ถ๐ท is isosceles ๐‘จ๐‘ท โ‰… ๐‘ฉ๐‘ท and ๐‘ท๐‘ช โ‰… ๐‘ท๐‘ซ Given โˆ ๐‘จ๐‘ท๐‘ช โ‰…โˆ ๐‘ฉ๐‘ท๐‘ซ Vertical โฆž are โ‰…. โˆ†๐‘จ๐‘ท๐‘ช โ‰… โˆ†๐‘ฉ๐‘ท๐‘ซ SAS โˆ ๐‘จ๐‘ช๐‘ท โ‰…โˆ ๐‘ฉ๐‘ซ๐‘ท CPCTC โˆ ๐‘ท๐‘ช๐‘ซ โ‰… โˆ ๐‘ท๐‘ซ๐‘ช Isosceles โˆ† Theorem ๐’Žโˆ ๐‘จ๐‘ช๐‘ท+ ๐’Žโˆ ๐‘ท๐‘ช๐‘ซ= ๐’Žโˆ ๐‘ฉ๐‘ซ๐‘ท+ ๐’Žโˆ ๐‘ท๐‘ซ๐‘ช Add. Prop. of โ‰… so ๐’Žโˆ ๐‘จ๐‘ช๐‘ซ = ๐’Žโˆ ๐‘ฉ๐‘ซ๐‘ช by the โˆ  Add. Post. and substitution. ๐‘ธ๐‘ซ = ๐‘ธ๐‘ช by the Conv. of the Isosceles โˆ† Theorem. โˆ†๐‘ธ๐‘ช๐‘ซ is isosceles by Definition of Isosceles โˆ†.

21 Mrs. Rivas โˆ†๐‘จ๐‘น๐‘ถ โ‰… โˆ†๐‘น๐‘จ๐‘ญ; ๐‘ฏ๐‘ณ Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. 30. โˆ†๐‘จ๐‘น๐‘ถ โ‰… โˆ†๐‘น๐‘จ๐‘ญ; ๐‘ฏ๐‘ณ

22 Mrs. Rivas โˆ†๐‘น๐‘ธ๐‘ด โ‰… โˆ†๐‘ธ๐‘น๐‘บ;๐‘บ๐‘บ๐‘บ Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. 31. โˆ†๐‘น๐‘ธ๐‘ด โ‰… โˆ†๐‘ธ๐‘น๐‘บ;๐‘บ๐‘บ๐‘บ

23 Mrs. Rivas โˆ†๐‘จ๐‘ถ๐‘ต โ‰… โˆ†๐‘ด๐‘ถ๐‘ท;๐‘จ๐‘จ๐‘บ Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. 32. โˆ†๐‘จ๐‘ถ๐‘ต โ‰… โˆ†๐‘ด๐‘ถ๐‘ท;๐‘จ๐‘จ๐‘บ

24 Mrs. Rivas โˆ†๐‘ฎ๐‘น๐‘ป โ‰… โˆ†๐‘ฎ๐‘บ๐‘ถ;๐‘จ๐‘บ๐‘จ Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. 33. โˆ†๐‘ฎ๐‘น๐‘ป โ‰… โˆ†๐‘ฎ๐‘บ๐‘ถ;๐‘จ๐‘บ๐‘จ

25 Mrs. Rivas Lessons 4-6 and 4-7 34. Given: M is the midpoint of ๐ด๐ต , ๐‘€๐ถ โŠฅ ๐ด๐ถ , ๐‘€๐ท โŠฅ ๐ต๐ท , โˆ 1โ‰…โˆ 2 Prove: โˆ†๐ด๐ถ๐‘€ โ‰… โˆ†๐ต๐ท๐‘€ โˆ ๐Ÿ โ‰…โˆ ๐Ÿ Given ๐‘ด๐‘ช โ‰… ๐‘ด๐‘ซ by the Conv. of the Isosceles โˆ† Theorem. ๐‘จ๐‘ด โ‰… ๐‘ฉ๐‘ด by the Def. of Midpoint ๐‘ด๐‘ช โ‰… ๐‘จ๐‘ช means that โˆ ๐‘จ๐‘ช๐‘ด is a right angle and โˆ†๐‘จ๐‘ช๐‘ด is a right โˆ†. ๐‘ด๐‘ซ โŠฅ ๐‘ฉ๐‘ซ means that โˆ ๐‘ฉ๐‘ซ๐‘ด is a right โฆจ and โˆ†๐‘ฉ๐‘ซ๐‘ด is a right โˆ† . โˆ†๐‘จ๐‘ช๐‘ดโ‰…โˆ†๐‘ฉ๐‘ซ๐‘ด by HL.

26 Mrs. Rivas 35. The longest leg of โˆ†๐ด๐ต๐ถ, ๐ด๐ต , measures 10 centimeters. ๐ต๐ถ measures 8 centimeters. You measure two of the legs of โˆ†๐‘‹๐‘Œ๐‘ and find that ๐ด๐ถ โ‰… ๐‘‹๐‘ and ๐ต๐ถ โ‰… ๐‘Œ๐‘ . Can you conclude that two triangles to be congruent by the HL Theorem? Explain why or why not. No; you only know that two sides (SS) are congruent, and you donโ€™t know that there are right angles.


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