Mrs. Rivas ∠𝑮 𝑺𝑨 ∠𝑻 𝑹𝑬 ∆𝑻𝑨𝑺 𝑻𝑺 ∆𝑨𝑻𝑺 ∠𝑨 Lesson 4-1 1. 𝑆 ≅ 2. 𝐺𝑅 ≅ ∆𝑺𝑨𝑻 ≅ ∆𝑮𝑹𝑬. Complete each congruence statement ∠𝑮 𝑺𝑨 1. 𝑆 ≅ 2. 𝐺𝑅 ≅ ∠𝑻 𝑹𝑬 3. 𝐸 ≅ 4. 𝐴𝑇 ≅ ∆𝑻𝑨𝑺 𝑻𝑺 5. ∆𝐸𝑅𝐺 ≅ 6. 𝐸𝐺 ≅ ∆𝑨𝑻𝑺 ∠𝑨 7. ∆𝑅𝐸𝐺 ≅ 8. ∠𝑅 ≅

Mrs. Rivas Yes; corresponding sides and corresponding angles are ≅. Lesson 4-1 State whether the figures are congruent. Justify your answers. 9. ∆𝐴𝐵𝐹; ∆𝐸𝐷𝐶 Yes; corresponding sides and corresponding angles are ≅.

Mrs. Rivas No; the only corresponding part that is ≅ is 𝑼𝑽 . Lesson 4-1 State whether the figures are congruent. Justify your answers. 10. ∆𝑇𝑈𝑉; ∆𝑈𝑉𝑊 No; the only corresponding part that is ≅ is 𝑼𝑽 .

Mrs. Rivas Yes; corresponding sides and corresponding angles are ≅. Lesson 4-1 State whether the figures are congruent. Justify your answers. 11. ⧠𝑋𝑌𝑍𝑉;⧠ 𝑈𝑇𝑍𝑉 Yes; corresponding sides and corresponding angles are ≅.

Mrs. Rivas Yes; corresponding sides and corresponding angles are ≅. Lesson 4-1 State whether the figures are congruent. Justify your answers. 12.∆𝐴𝐵𝐷; ∆𝐸𝐷𝐵 Yes; corresponding sides and corresponding angles are ≅.

Mrs. Rivas 𝑨𝑺𝑨 ∠𝑻≅∠𝑺 𝑻𝒀 ≅ 𝑺𝑾 ∠𝒀≅∠𝑾 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. ∠𝑻≅∠𝑺 𝑻𝒀 ≅ 𝑺𝑾 ∠𝒀≅∠𝑾 𝑨𝑺𝑨

Mrs. Rivas 𝑺𝑨𝑺 𝑬𝑨 ≅ 𝑷𝑳 𝑨𝑳 ≅ 𝑳𝑨 ∠𝑬𝑨𝑳≅∠𝑷𝑳𝑨 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.. 𝑬𝑨 ≅ 𝑷𝑳 𝑨𝑳 ≅ 𝑳𝑨 ∠𝑬𝑨𝑳≅∠𝑷𝑳𝑨 𝑺𝑨𝑺

Mrs. Rivas 𝑵𝒐 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 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. 𝑵𝒐 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆

Mrs. Rivas 𝑺𝑺𝑺 𝑪𝑵 ≅ 𝑨𝑶 𝑵𝑨 ≅ 𝑶𝑪 𝑨𝑪 ≅ 𝑪𝑨 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. 𝑪𝑵 ≅ 𝑨𝑶 𝑵𝑨 ≅ 𝑶𝑪 𝑨𝑪 ≅ 𝑪𝑨 𝑺𝑺𝑺

Mrs. Rivas 𝒁𝑷 bisects 𝑿𝒀 means that 𝑿𝑴 ≅ 𝒀𝑴 . 𝑷𝑿 ≅ 𝑷𝒀 Given Lessons 4-2 and 4-3 17. Given: 𝑃𝑋 ≅ 𝑃𝑌 , 𝑍𝑃 bisects 𝑋𝑌 . Prove: ∆𝑃𝑋𝑍  ∆𝑃𝑌𝑍 𝒁𝑷 bisects 𝑿𝒀 means that 𝑿𝑴 ≅ 𝒀𝑴 . 𝑷𝑿 ≅ 𝑷𝒀 Given 𝑷𝒁 ≅ 𝑷𝒁 Reflective property of ≅ ∆𝑷𝑿𝑴≅∆𝑷𝒀𝑴 SSS ∠𝑿𝑷𝑴≅∠𝒀𝑷𝑴 Def. of ≅. 𝑷𝒁 ≅ 𝑷𝒁 Reflective property of ≅ ∆𝑷𝑿𝒁≅∆𝑷𝒀𝒁 by SAS.

Mrs. Rivas It is given that 𝒎∠𝟏 ≅ 𝒎∠𝟐 and 𝒎∠𝟑 ≅ 𝒎∠𝟒. Lessons 4-2 and 4-3 18. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, 𝑃𝐷 ≅ 𝑃𝐶 , P is the midpoint of 𝐴𝐵 . Prove: ∆𝐴𝐷𝑃 ≅ ∆𝐵𝐶𝑃 It is given that 𝒎∠𝟏 ≅ 𝒎∠𝟐 and 𝒎∠𝟑 ≅ 𝒎∠𝟒. By the ⦨ Addition. Post., 𝒎∠𝑫𝑷𝑨 ≅ 𝒎∠𝑪𝑷𝑩. Since 𝑃 is the midpoint of 𝑨𝑩 , 𝑷𝑨 ≅ 𝑷𝑩 . It is given that 𝑷𝑫 ≅ 𝑷𝑪 , so ∆𝑨𝑫𝑷 ≅ ∆𝑩𝑪𝑷 by SAS.

Mrs. Rivas 𝑶𝑳 ∥ 𝑴𝑵 , so ∠𝑶𝑳𝑵 ≅ ∠𝑴𝑵𝑳. Lesson 4-4 21. Given: 𝐿𝑂 ≅ 𝑀𝑁 , 𝐿𝑂 ∥ 𝑀𝑁 Prove: ∠𝑀𝐿𝑁 ≅ ∠𝑂𝑁𝐿 𝑶𝑳 ∥ 𝑴𝑵 , so ∠𝑶𝑳𝑵 ≅ ∠𝑴𝑵𝑳. 𝑳𝑵 ≅ 𝑳𝑵 by the Reflexive Prop. of ≅. Since 𝑳𝑶 ≅ 𝑴𝑵 , ∆𝑴𝑳𝑵 ≅∆𝑶𝑵𝑳 by SAS, Then ∠𝑴𝑳𝑵≅∠𝑶𝑵𝑳 by CPCTC.

Mrs. Rivas 𝑶𝑺 ≅ 𝑶𝑺 Reflexive Prop. of ≅. Since ∠𝑻 ≅∠𝑬 and ∠𝑻𝑺𝑶 ≅∠𝑬𝑶𝑺 Lesson 4-4 22. Given: ∠𝑂𝑇𝑆 ≅ ∠𝑂𝐸𝑆, ∠𝐸𝑂𝑆 ≅ ∠𝑂𝑆𝑇 Prove: 𝑇𝑂 ≅ 𝐸𝑆 𝑶𝑺 ≅ 𝑶𝑺 Reflexive Prop. of ≅. Since ∠𝑻 ≅∠𝑬 and ∠𝑻𝑺𝑶 ≅∠𝑬𝑶𝑺 ∆𝑻𝑺𝑶 ≅ ∆𝑬𝑶𝑺 by AAS 𝑻𝑶 ≅ 𝑬𝑺 by CPCTC

Mrs. Rivas ∠𝟏 ≅∠𝟐,∠𝟑 ≅∠𝟒 Given 𝑴 is the midpoint of 𝑷𝑹 , 𝒔𝒐 𝑷𝑴 ≅ 𝑹𝑴 . Lesson 4-4 23. Given: ∠1 ≅ ∠2,∠3 ≅∠4, 𝑀 is the midpoint of 𝑃𝑅 Prove: ∆𝑃𝑀𝑄 ≅ ∆𝑅𝑀𝑄 ∠𝟏 ≅∠𝟐,∠𝟑 ≅∠𝟒 Given 𝑴 is the midpoint of 𝑷𝑹 , 𝒔𝒐 𝑷𝑴 ≅ 𝑹𝑴 . 𝑺𝑸 ≅ 𝑺𝑸 Reflexive Property 𝑸𝑴 ≅ 𝑸𝑴 Reflexive Property ∆𝑷𝐐𝐒≅∆𝑹𝑸𝑺 AAS ∆𝑷𝐐𝐌≅∆𝑹𝑸𝑴 SSS 𝑷𝑸 ≅ 𝑹𝑸 CPCTC ∠𝑷𝑸𝑴≅∠𝑹𝑴𝑸 CPCTC

Mrs. Rivas ∠𝟏 ≅∠𝟐,∠𝑨𝑷𝑶 ≅∠𝑩𝑸𝑶 Supplement of ≅ angles are ≅. Lesson 4-4 24. Given: 𝑃𝑂 = 𝑄𝑂, ∠1 ≅ ∠2, Prove: ∠𝐴 ≅ ∠𝐵 ∠𝟏 ≅∠𝟐,∠𝑨𝑷𝑶 ≅∠𝑩𝑸𝑶 Supplement of ≅ angles are ≅. 𝑷𝑶 ≅ 𝑸𝑶 Given ∠𝑷𝑶𝑨 ≅∠𝑩𝑶𝑸 Vertical ⦞ are ≅. ∆𝑶𝑷𝑸≅∆𝑩𝑸𝑶 ASA ∠𝑨 ≅∠𝑩 CPCTC

Mrs. Rivas 𝟐𝒙+𝟏𝟎=𝟔𝟎 𝟐𝒙=𝟓𝟎 𝒙=𝟐𝟓 𝟔𝟎 𝟔𝟎 𝟔𝟎 Lesson 4-5 Find the value of each variable 25. 𝟐𝒙+𝟏𝟎=𝟔𝟎 𝟔𝟎 𝟐𝒙=𝟓𝟎 𝒙=𝟐𝟓 𝟔𝟎 𝟔𝟎

Mrs. Rivas 𝟐𝟓+𝟗𝟎+𝒙=𝟏𝟖𝟎 𝟏𝟏𝟓+𝒙=𝟏𝟖𝟎 𝒙=𝟔𝟓 𝟐𝟓 Lesson 4-5 Find the value of each variable 26. 𝟐𝟓+𝟗𝟎+𝒙=𝟏𝟖𝟎 𝟐𝟓 𝟏𝟏𝟓+𝒙=𝟏𝟖𝟎 𝒙=𝟔𝟓

Mrs. Rivas 𝒁+𝟏𝟑𝟓=𝟏𝟖𝟎 𝒛=𝟒𝟓 𝒙+𝒚=𝟏𝟑𝟓 𝒙=𝟒𝟓 𝟒𝟓+𝒚=𝟏𝟑𝟓 𝒚=𝟗𝟎 𝑹𝒆𝒎𝒐𝒕𝒆 𝒂𝒏𝒈𝒍𝒆𝒔 Lesson 4-5 Find the value of each variable 27. 𝒁+𝟏𝟑𝟓=𝟏𝟖𝟎 𝒛=𝟒𝟓 𝑹𝒆𝒎𝒐𝒕𝒆 𝒂𝒏𝒈𝒍𝒆𝒔 𝑰𝒔𝒐𝒔𝒄𝒆𝒍𝒆𝒔 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 base angles are congruent 𝒙+𝒚=𝟏𝟑𝟓 𝒙=𝟒𝟓 𝟒𝟓+𝒚=𝟏𝟑𝟓 𝒚=𝟗𝟎

Mrs. Rivas 𝑷𝑿 ≅ 𝑷𝒀 Given ∠𝟏 ≅∠𝟐 Isosceles ∆ Theorem Lesson 4-5 28. Given: ∠5 ≅ ∠6, 𝑃𝑋 ≅ 𝑃𝑌 Prove: ∆𝑃𝐴𝐵 is isosceles. 𝑷𝑿 ≅ 𝑷𝒀 Given ∠𝟏 ≅∠𝟐 Isosceles ∆ Theorem ∠𝟑 ≅∠𝟒 ≅ Supplement Theorem ∠𝟓 ≅∠𝟔 Given ∆𝑷𝑿𝑨≅∆𝑷𝒀𝑩 ASA 𝑷𝑨 ≅ 𝑷𝑩 CPCTC and ∆𝑷𝑨𝑩 is isosceles by Isosceles ∆ Theorem

Mrs. Rivas 𝑨𝑷 ≅ 𝑩𝑷 and 𝑷𝑪 ≅ 𝑷𝑫 Given ∠𝑨𝑷𝑪 ≅∠𝑩𝑷𝑫 Vertical ⦞ are ≅. Lesson 4-5 29. Given: 𝐴𝑃 ≅ 𝐵𝑃 , 𝑃𝐶 ≅ 𝑃𝐷 Prove: ∆𝑄𝐶𝐷 is isosceles 𝑨𝑷 ≅ 𝑩𝑷 and 𝑷𝑪 ≅ 𝑷𝑫 Given ∠𝑨𝑷𝑪 ≅∠𝑩𝑷𝑫 Vertical ⦞ are ≅. ∆𝑨𝑷𝑪 ≅ ∆𝑩𝑷𝑫 SAS ∠𝑨𝑪𝑷 ≅∠𝑩𝑫𝑷 CPCTC ∠𝑷𝑪𝑫 ≅ ∠𝑷𝑫𝑪 Isosceles ∆ Theorem 𝒎∠𝑨𝑪𝑷+ 𝒎∠𝑷𝑪𝑫= 𝒎∠𝑩𝑫𝑷+ 𝒎∠𝑷𝑫𝑪 Add. Prop. of ≅ so 𝒎∠𝑨𝑪𝑫 = 𝒎∠𝑩𝑫𝑪 by the ∠ Add. Post. and substitution. 𝑸𝑫 = 𝑸𝑪 by the Conv. of the Isosceles ∆ Theorem. ∆𝑸𝑪𝑫 is isosceles by Definition of Isosceles ∆.

Mrs. Rivas ∆𝑨𝑹𝑶 ≅ ∆𝑹𝑨𝑭; 𝑯𝑳 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. ∆𝑨𝑹𝑶 ≅ ∆𝑹𝑨𝑭; 𝑯𝑳

Mrs. Rivas ∆𝑹𝑸𝑴 ≅ ∆𝑸𝑹𝑺;𝑺𝑺𝑺 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. ∆𝑹𝑸𝑴 ≅ ∆𝑸𝑹𝑺;𝑺𝑺𝑺

Mrs. Rivas ∆𝑨𝑶𝑵 ≅ ∆𝑴𝑶𝑷;𝑨𝑨𝑺 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. ∆𝑨𝑶𝑵 ≅ ∆𝑴𝑶𝑷;𝑨𝑨𝑺

Mrs. Rivas ∆𝑮𝑹𝑻 ≅ ∆𝑮𝑺𝑶;𝑨𝑺𝑨 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. ∆𝑮𝑹𝑻 ≅ ∆𝑮𝑺𝑶;𝑨𝑺𝑨

Mrs. Rivas Lessons 4-6 and 4-7 34. Given: M is the midpoint of 𝐴𝐵 , 𝑀𝐶 ⊥ 𝐴𝐶 , 𝑀𝐷 ⊥ 𝐵𝐷 , ∠1≅∠2 Prove: ∆𝐴𝐶𝑀 ≅ ∆𝐵𝐷𝑀 ∠𝟏 ≅∠𝟐 Given 𝑴𝑪 ≅ 𝑴𝑫 by the Conv. of the Isosceles ∆ Theorem. 𝑨𝑴 ≅ 𝑩𝑴 by the Def. of Midpoint 𝑴𝑪 ≅ 𝑨𝑪 means that ∠𝑨𝑪𝑴 is a right angle and ∆𝑨𝑪𝑴 is a right ∆. 𝑴𝑫 ⊥ 𝑩𝑫 means that ∠𝑩𝑫𝑴 is a right ⦨ and ∆𝑩𝑫𝑴 is a right ∆ . ∆𝑨𝑪𝑴≅∆𝑩𝑫𝑴 by HL.

Mrs. Rivas 35. The longest leg of ∆𝐴𝐵𝐶, 𝐴𝐵 , measures 10 centimeters. 𝐵𝐶 measures 8 centimeters. You measure two of the legs of ∆𝑋𝑌𝑍 and find that 𝐴𝐶 ≅ 𝑋𝑍 and 𝐵𝐶 ≅ 𝑌𝑍 . 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 don’t know that there are right angles.