Review of the problems that are on the Chapter 4 exam

1. If BCDE is congruent to OPQR, then 𝐵𝐸 is congruent to __?__. 𝑃𝑄 𝑂𝑃 𝑄𝑅 𝑂𝑅 4-1

1. If BCDE is congruent to OPQR, then 𝐵𝐸 is congruent to __?__. 𝑂𝑅 because B is the first letter and corresponds to O and E is the last letter corresponding to R. 4-1

2. Given ΔABC ≌ ΔPQR, m∠B = 4v+1, and m∠Q = 8v-7, find m∠B and m∠Q. 26 10 9 23 4-1

2. Given ΔABC ≌ ΔPQR, m∠B = 4v+1, and m∠Q = 8v-7, find m∠B and m∠Q. Since ∠B corresponds to ∠Q 4𝑣+1=8𝑣 −7 1=4𝑣 −7 8=4𝑣 2=𝑣 Plugging in 2 4 2 +1=8+1=9 4-1

3. The two triangles are congruent as suggested by their appearance 3. The two triangles are congruent as suggested by their appearance. Find the value of c. The diagrams are not to scale. 5 57 4 3 4-1

3 is the bottom side length and c is in the corresponding part so 𝑐=3. 3. The two triangles are congruent as suggested by their appearance. Find the value of c. The diagrams are not to scale. 3 is the bottom side length and c is in the corresponding part so 𝑐=3. 4-1

4. Justify the last two steps of the proof. Symmetric Prop of ≌; SSS Reflexive Prop of ≌; SSS Symmetric Prop of ≌; SAS Reflexive Prop of ≌; SAS 4-2

4. Justify the last two steps of the proof. Reflexive Prop of ≌ SSS 4-2

5. Name the angle included by the sides 𝑃𝑁 and 𝑁𝑀 . ∠P ∠N ∠M none of these 4-2

5. Name the angle included by the sides 𝑃𝑁 and 𝑁𝑀 . ∠N because it touches both of the listed sides. (Same letter in both sides is always the included angle.) 4-2

∠𝐶𝐵𝐴≅∠𝐶𝐷𝐴 ∠𝐵𝐴𝐶≅∠𝐷𝐴𝐶 𝐴𝐵 ⊥ 𝐴𝐷 𝐴𝐵 ≌ 𝐴𝐷 6. What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate? ∠𝐶𝐵𝐴≅∠𝐶𝐷𝐴 ∠𝐵𝐴𝐶≅∠𝐷𝐴𝐶 𝐴𝐵 ⊥ 𝐴𝐷 𝐴𝐵 ≌ 𝐴𝐷 4-2

6. What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate? We know 𝐴𝐶 ≌ 𝐴𝐶 Reflexive ∠𝐶𝐵𝐴≅∠𝐶𝐷𝐴 no because that is AAS ∠𝐵𝐴𝐶≅∠𝐷𝐴𝐶 no because already marked so not additional information. 𝐴𝐵 ⊥ 𝐴𝐷 No because that is ASA So correct answer is D 𝐴𝐵 ≌ 𝐴𝐷 because side included angle side SAS. 4-2

7. What is the missing reason in the two-column proof? AAS Theorem SAS Postulate SSS Postulate ASA Postulate 4-3

7. What is the missing reason in the two-column proof? D. ASA Postulate Non-Response Grid ASA Postulate 4-3

AAS ASA SAS None of these 8. Name the theorem or postulate that lets you immediately conclude ΔABD ≌ ΔCBD. AAS ASA SAS None of these 4-3

𝐵𝐷 ≅ 𝐵𝐷 Reflexive So ΔABD ≌ ΔCBD by SAS 8. Name the theorem or postulate that lets you immediately conclude ΔABD ≌ ΔCBD. 𝐵𝐷 ≅ 𝐵𝐷 Reflexive So ΔABD ≌ ΔCBD by SAS 4-3

9. Based on the given information, what can you conclude, and why? ∆𝐸𝐹𝐺≅∆𝐺𝐼𝐻 by SAS ∆𝐸𝐹𝐺≅∆𝐼𝐻𝐺 by SAS ∆𝐸𝐹𝐺≅∆𝐺𝐼𝐻 by ASA ∆𝐸𝐹𝐺≅∆𝐼𝐻𝐺 by ASA 4-3

9. Based on the given information, what can you conclude, and why? Since ∠𝐹𝐺𝐸≅ ∠𝐹𝐺𝐸 because of vertical angles ∆𝐸𝐹𝐺≅∆𝐼𝐻𝐺 by ASA 4-3

No, the two triangles are not congruent. 10. R, S, and T are the vertices of one triangle. E, F, and D are the vertices of another triangle. ∠𝑅=60, 𝑚∠𝑆=100,𝑚∠𝐹=60, 𝑚∠𝐷=20, RS = 4, and EF = 4. Are the two triangles congruent? If yes, explain and tell which segment is congruent to 𝑅𝑇 . yes, by SAS; 𝐸𝐷 yes, by AAS; 𝐸𝐷 yes, by ASA; 𝐹𝐷 No, the two triangles are not congruent. 4-4

Since triangles add up to 180 degrees ∠T= 20 𝑜 and ∠E= 100 𝑜 . 10. R, S, and T are the vertices of one triangle. E, F, and D are the vertices of another triangle. ∠𝑅=60, 𝑚∠𝑆=100,𝑚∠𝐹=60, 𝑚∠𝐷=20, RS = 4, and EF = 4. Are the two triangles congruent? If yes, explain and tell which segment is congruent to 𝑅𝑇 . S F 100o 60o 4 4 60o 20o 100o 20o R T E D Since triangles add up to 180 degrees ∠T= 20 𝑜 and ∠E= 100 𝑜 . So yes triangles are congruent by ASA And 𝑅𝑇 will be congruent to 𝐹𝐷 . 4-4

11. Supply the missing reasons to complete the proof. AAS; Corres. parts of ≌ Δ are ≌ SAS; Corres. parts of ≌ Δ are ≌ ASA; Substitution ASA; Corres. parts of ≌ Δ are ≌ 4-4

11. Supply the missing reasons to complete the proof. ASA Corres. parts of ≌ Δ are ≌ 4-4

12. Write a two-column proof. 4-4

13. What is the value of x? 66o 71o 132o 142o 4-5

13. What is the value of x? 𝑥+48+𝑥=180 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑥+48+𝑥=180 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 2𝑥+48=180 𝐶𝑜𝑚𝑏𝑖𝑛𝑒 𝑙𝑖𝑘𝑒 𝑡𝑒𝑟𝑚𝑠 2𝑥=132 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 48 𝑓𝑟𝑜𝑚 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑥=66 𝐷𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 2 Answer is A 66o 4-5

14. The legs of an isosceles triangle have lengths 3x+2 and 2x+6 14. The legs of an isosceles triangle have lengths 3x+2 and 2x+6. The base has length 4x + 3. What is the length of the base? 4 14 19 Cannot be determined 4-5

3𝑥+2=2𝑥+6 𝐼𝑠𝑜𝑐𝑒𝑙𝑒𝑠 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 ℎ𝑎𝑣𝑒≅𝑙𝑒𝑔𝑠 14. The legs of an isosceles triangle have lengths 3x+2 and 2x+6. The base has length 4x + 3. What is the length of the base? 3𝑥+2=2𝑥+6 𝐼𝑠𝑜𝑐𝑒𝑙𝑒𝑠 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 ℎ𝑎𝑣𝑒≅𝑙𝑒𝑔𝑠 𝑥+2=6 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 2𝑥 𝑓𝑟𝑜𝑚 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑥=4 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 2 𝑓𝑟𝑜𝑚 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑃𝑙𝑢𝑔 𝑥=4 𝑖𝑛𝑡𝑜 4𝑥+3 4 4 +3=16+3=19 Correct answer is C. 19 4-5

15. Find the values of x and y. x = 24; y = 66 x = 66; y = 24 x = 90; y = 66 x = 90; y = 24 4-5

15. Find the values of x and y. Since 𝐴𝐷 bisects the vertex angle, 𝐴𝐷 is the perpendicular bisector of 𝐵𝐶 so x = 90 The angles of a triangle add up to 180 degrees 90+66+𝑦=180 156+𝑦=180 𝑦=24 The correct answer is D x = 90; y = 24 4-5

∠A ≌ ∠E m∠BCE =90 𝐴𝐶 ≌ 𝐷𝐶 𝐴𝐶 ≌ 𝐵𝐷 16. What additional information will allow you to prove the triangles congruent by the HL Theorem? ∠A ≌ ∠E m∠BCE =90 𝐴𝐶 ≌ 𝐷𝐶 𝐴𝐶 ≌ 𝐵𝐷 4-6

A. ∠A ≌ ∠E is not correct AAS 16. What additional information will allow you to prove the triangles congruent by the HL Theorem? A. ∠A ≌ ∠E is not correct AAS B. m∠BCE =90 is outside the triangle so does not help. C. 𝐴𝐶 ≌ 𝐷𝐶 Right Δ, ≌ Hypotenuses, and ≌ legs so yes HL D. 𝐴𝐶 ≌ 𝐵𝐷 does not work because 𝐵𝐷 is not a side of a triangle. 4-6

I only II only III only II and III 17. For which situation could you immediately prove Δ1 ≌ Δ2 using the HL Theorem? I only II only III only II and III 4-6

17. For which situation could you immediately prove Δ1 ≌ Δ2 using the HL Theorem? Correct answer is C. III only Would be congruent by ASA does not have hypotenuses ≌ so not HL Would be congruent by SAS again does not have hypotenuses ≌ so not HL Right Triangles, hypotenuses are ≌, and they have a leg in common so definitely HL 4-6

No, the triangles cannot be proven congruent 18. Is there enough information to conclude that the two triangles are congruent? If so, what is a correct congruence statement? Yes; ΔCAB≌ΔDAC. Yes; ΔACB≌ΔACD. Yes; ΔABC≌ΔACD. No, the triangles cannot be proven congruent 4-6

1. Given right angles we have right triangles. 18. Is there enough information to conclude that the two triangles are congruent? If so, what is a correct congruence statement? 1. Given right angles we have right triangles. 2. Since 𝐴𝐵 ≅ 𝐴𝐷 we have congruent hypotenuses. 3. AC ≅ AC so legs are ≅ 4.Yes; ΔACB≌ΔACD by HL. Correct answer is B. 4-6

19. What common side do ΔCDE and ΔFED share? 𝐸𝐷 𝐶𝐸 𝐶𝐷 𝐹𝐸 4-7

19. What common side do ΔCDE and ΔFED share? They have side 𝐸𝐷 in common correct answer is A. 4-7

20. What common angle do ΔACF and ΔBCG? 4-7

20. What common angle do ΔACF and ΔBCG? They have angle ∠C in common correct answer is C. 4-7

Chapter 4 Exam TOMORROW! Journals due Friday Portfolio Due Friday