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Name____________________________________________ Period____________________ Review for Geometry Midterm 2015: Chapters 1-5 Short Answer 1. What is the length of AC ? 2. Tell whether a triangle can have sides with lengths 1, 2, and 3. 3. Danny and Dana start hiking from the same base camp and head in opposite directions. Danny walks 6 miles due west, then changes direction and walks for 5 miles to point C. Dana hikes 6 miles due east, then changes direction and walks for 5 miles to point S. Use the diagram to find which hiker is farther from the base camp. 6. What is the slope of the line shown? 4. Given: ∆ABC ≅ ∆MNO Identify all pairs of congruent corresponding parts. 5. Apply the transformation M to the triangle with the given vertices. Identify and describe the transformation. M: (x, y) → (x – 6, y + 2) E(3, 0), F(1, –2), G(5, –4) 1 Name: ________________________ ID: A 7. What is the value of x? Identify the missing justifications. m∠PQR = x − 11, m∠SQR = x + 1, and m∠PQS = 100. m∠PQR + m∠SQR = m∠PQS x – 11 + x + 1 = 100 2x – 10 = 100 2x = 110 x = 55 a. __________ b. Substitution Property c. Simplify d. __________ e. Division Property of Equality 8. Is the line through points P(–8, –1) and Q(–5, 8) parallel to the line through points R(3, 0) and S(1, –4)? Explain. 9. Compare m∠ABC and m∠CBD. 10. Where is the circumcenter of any given triangle? 11. Find the coordinates of the midpoint of the segment whose endpoints are H(10, 1) and K(8, 3). 2 Name: ________________________ ID: A 12. What is the missing reason in the two-column proof? → → Given: QS bisects ∠TQR and SQ bisects ∠TSR Prove: ∆TQS ≅ ∆RQS Statements Reasons → 1. QS bisects ∠TQR 2. ∠TQS ≅ ∠RQS 3. QS ≅ QS 1. Given 2. Definition of angle bisector 3. Reflexive property → 4. SQ bisects ∠TSR 5. ∠TSQ ≅ ∠RSQ 6. ∆TQS ≅ ∆RQS 4. Given 5. Definition of angle bisector 6. ? 13. What is the value of x? 15. Write an equation in slope-intercept form of the line through point P(1, 4) with slope –3. 16. Complete the two-column proof. Given: x + 9 = 11 5 Prove: x = 10 x + 9 = 11 5 x =2 5 14. Find the value of x for which l is parallel to m. The diagram is not to scale. x = 10 3 a. ________ b. ________ c. ________ Name: ________________________ ID: A 18. ∠NPM ≅ 17. Find the value of x. The diagram is not to scale. ? 19. Tom is wearing his favorite bow tie to the school dance. The bow tie is in the shape of two triangles. Given: AB ≅ ED , BC ≅ DC , AC ≅ EC , ∠A ≅ ∠E Prove: ∆ABC ≅ ∆EDC Complete the proof. Proof: Statements 1. AB ≅ ED , BC ≅ DC , AC ≅ EC 2. ∠A ≅ ∠E 3. ∠BCA ≅ ∠DCE 4. ∠B ≅ ∠D 5. [3] Reasons 1. Given 2. Given 3. [1] 4. [2] 5. Definition of congruent triangles 4 Name: ________________________ ID: A 20. Given: P is the midpoint of TQ and RS . Prove: ∆TPR ≅ ∆QPS Complete the proof. Proof: Statements 1. P is the midpoint of TQ and RS . 2. [1] 2. TP ≅ QP , RP ≅ SP 3. [2] 4. ∆TPR ≅ ∆QPS 21. Determine whether triangles are congruent. Reasons 1. Given 3. Vertical Angles Theorem 4. [3] EFG and 23. Find m∠K . PQR 24. If EF = 8x + 13, FG = 16, and EG = 85 , find the value of x. The drawing is not to scale. 22. If Z is the midpoint of RT , what are x, RZ, and RT? 5 Name: ________________________ ID: A 25. The diagram shows the approximate distances from Houston to Dallas and from Austin to Dallas. What is the range of distances, d, from Austin to Houston? Use the diagram to find the following. 26. What are the measures of ∠ABD and ∠ABC ? Classify each angle as acute, right, obtuse, or straight. 28. Identify a pair of alternate exterior angles. 29. Find the value of x. The diagram is not to scale. 30. What additional information do you need to prove ∆ABC ≅ ∆ADC by the SAS Postulate? 27. Find the value of x. The diagram is not to scale. 6 Name: ________________________ ID: A 36. Justify the last two steps of the proof. Given: PQ ≅ SR and PR ≅ SQ Prove: ∆PQR ≅ ∆SRQ 31. In ∆ACE, G is the centroid and BE = 15. Find BG and GE. Proof: 1. PQ ≅ SR 32. Write the sides of ∆IJK in order from shortest to longest. 2. PR ≅ SQ 3. QR ≅ RQ 4. ∆PQR ≅ ∆SRQ 1. Given 2. Given 3. ? 4. ? 37. Supplementary angles are two angles whose measures have a sum of ____. Complementary angles are two angles whose measures have a sum of ____. 33. Tell whether a triangle can have sides with lengths 5, 11, and 7. 38. The legs of an isosceles triangle have lengths 3x + 2 and −x + 26. The base has length 2x + 2. What is the length of the base? → 34. MO bisects ∠LMN, m∠LMO = 8x − 28, and m∠NMO = 2x + 38. Solve for x and find m∠LMN. The diagram is not to scale. 39. For these triangles, select the triangle congruence statement and the postulate or theorem that supports it. 35. The lengths of two sides of a triangle are 3 inches and 8 inches. Find the range of possible lengths for the third side, s. 7 Name: ________________________ ID: A 40. ∆ABC is an isosceles triangle. AB is the longest side with length 10x + 3. BC = 5x + 5 and CA = 4x + 11. Find AB. 41. Name the line and plane shown in the diagram. 42. What are the names of four coplanar points? 43. Find the value of x. 8 Name: ________________________ ID: A 45. Find the value of k. The diagram is not to scale. 44. Name the angle included by the sides MP and PN . Multiple Choice Identify the choice that best completes the statement or answers the question. 46. Write an equation in point-slope form & slope- intercept form of the line through point J(10, –2) with slope 7. c. y − 2 = 7 (x + 10 ) a. y + 2 = 7 (x + 10 ) b. y + 2 = 7 (x − 10 ) d. y + 2 = −7 (x − 10 ) 47. Which two lines are parallel? 5y = 4x − 5 I. 7y = 5 − 5x II. III. 7y + 5x = −1 a. b. I and III I and II c. d. II and III No, two of the lines are parallel. 48. Where can the bisectors of the angles of an obtuse triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle a. I only b. III only c. I or III only 9 d. I, II, or II Name: ________________________ ID: A 49. Supply the missing reasons to complete the proof. Given: ∠A ≅ ∠D and AC ≅ DC Prove: BC ≅ EC Statement 1. ∠A ≅ ∠D and Reasons 1. Given AC ≅ DC 2. ∠BCA ≅ ∠ECD 2. Vertical angles are congruent. 3. ∆BCA ≅ ∆ECD 3. ? 4. BC ≅ EC 4. ? a. b. ASA; Corresp. parts of ≅ ∆ are ≅. ASA; Substitution c. d. AAS; Corresp. parts of ≅ ∆ are ≅. SAS; Corresp. parts of ≅ ∆ are ≅. 50. Which statement can you conclude is true from the given information? ← → Given: AB is the perpendicular bisector of IK . a. b. A is the midpoint of IK . IJ = JK c. d. AJ = BJ ∠IAJ is a right angle. 10 ID: A Review for Geometry Midterm 2015: Chapters 1-5 Answer Section SHORT ANSWER 1. 8 2. No 3. Danny is farther from the base camp than Dana. 4. ∠A ≅ ∠M , ∠B ≅ ∠N , ∠C ≅ ∠O, AB ≅ MN , BC ≅ NO, AC ≅ MO 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. This is a translation 6 units left and 2 units up. 1 Angle Addition Postulate; Addition Property of Equality No; the lines have unequal slopes. m∠ABC > m∠CBD the point of concurrency of the bisectors of the angles of the triangle (9, 2) ASA Postulate 68° 95 y = –3x + 7 a. Given b. Subtraction Property of Equality c. Multiplication Property of Equality 11 ∠BCA [1] Vertical Angles Theorem [2] Third Angles Theorem [3] ∆ABC ≅ ∆EDC [1]. Definition of midpoint [2] ∠TPR ≅ ∠QPS [3] SAS 1 ID: A 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. The triangles are congruent because x = 14, RZ = 88, and RT = 176 m∠K = 63° x=7 40 < d < 440 m∠ABD = 16°; ∠ABD is acute. m∠ABC = 180°; ∠ABC is straight. 56 ∠2 and ∠6 58 ∠ACB ≅ ∠ACD BG = 5, GE = 10 32. 33. 34. 35. 36. 37. 38. 39. 40. JK , IK , IJ Yes x = 11, m∠LMN = 120 5 < s < 11 Reflexive Property of ≅ ; SSS 180; 90 14 ∆ABC ≅ ∆JKL, HL AB = 63 EFG can be mapped to ← → 41. 42. 43. 44. 45. MN and plane M NP Points D, A, B, and J are coplanar. 9 ∠P 82 MULTIPLE CHOICE 46. 47. 48. 49. 50. B C A A B 2 PQR by a reflection: (x,y) → (x,−y).