examview - geometry review for final chapters 5 and 6 · geometry - review for final chapters 5 and...

Name: ______________________ Class: _________________ Date: _________ ID: A

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Geometry - Review for Final Chapters 5 and 6

____ 1. Classify PQR by its sides. Then determine whether it is a right triangle.

a. scalene ; right c. scalene ; not rightb. isoceles ; not right d. isoceles ; right

____ 2. Find m∠KMN .

a. 20° c. 101°b. 79° d. 5°

Name: ______________________ ID: A

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____ 3. A ramp is designed with the profile of a right triangle. The measure of one acute angle is 2 times the measure of the other acute angle. Find the measure of each acute angle.

a. 45°, 45° c. 22.5°, 67.5°b. 11.25°, 78.75° d. 30°, 60°

____ 4. Write a congruence statement for the triangles.

a. PRQ ≅ TUV c. QRP ≅ TUVb. PRQ ≅ UTV d. QRP ≅ UTV

Name: ______________________ ID: A

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____ 5. In the diagram, CDE ≅ GHI . Find the value of y.

a. y = 51 c. y = 42b. y = 54 d. y = 64

____ 6. Find m∠BDC.

a. 68° c. 56°b. 84° d. 112°

Name: ______________________ ID: A

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____ 7. Find the value of x.

a. 10 c. 5b. 4 d. 6

____ 8. In the diagram, M is the midpoint of BC . Find the coordinates of M.

a.32

,2Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ c.

43

,34

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

b.34

,43

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ d. 2,

32

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

Name: ______________________ ID: A

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____ 9. Which statements about the diagram are true?

a. BDE is obtuse. d. ABC is obtuse.b. ABC is scalene. e. BDE is right.c. The value of x is 18. f. The value of y is 113.

____ 10. What additional information is needed to prove that ABN ≅ DCO using the SAS Congruence Theorem?

a. ∠BNA ≅ ∠COD and CO ≅ BN c. ∠ABN ≅ ∠DCO and CO ≅ BN

b. AN ≅ DO and ∠ABN ≅ ∠DCO d. AN ≅ DO and ∠BAN ≅ ∠CDN

____ 11. In a coordinate plane, PQR has vertices P(0, 0) and Q(3a, 0). Which coordinates of vertex R make PQR an isosceles right triangle? Assume all variables are positive.

a. 3a,3a 3

2

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜̃̃

d.3a 2

2,

3a 22

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜̃̃

b.3a2

,3a2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ e. 3a, 3aÊ

ËÁÁˆ¯̃̃

c. 0, 3aÊËÁÁ

ˆ¯̃̃

Name: ______________________ ID: A

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12. Find the measure of each acute angle.

____ 13. In the diagram of the basketball hoop below, can you use the Hypotenuse-Leg Congruence Theorem to prove that WXY ≅ WZY?

a. yes b. no

____ 14. Are the triangles shown in the Brazilian flag congruent by the ASA Congruence Theorem?

a. yes b. no

Name: ______________________ ID: A

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____ 15. Are the triangles shown in the four-sided die congruent by the AAS Congruence Theorem?

a. yes b. no

Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use.

____ 16.

a. yes; AAS Congruence Theoremb. yes; ASA Congruence Theoremc. no

____ 17.


Name: ______________________ ID: A

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____ 18.


____ 19. Which reason is not necessary to explain how you can find the distance across the lake?

a. ASA Congruence Theoremb. Right Angles Congruence Theoremc. SAS Congruence Theoremd. Corresponding parts of congruent triangles are congruent.e. Vertical Angles Congruence Theorem

____ 20. Which reason is not used in a plan to prove that XW ≅ ZY?

a. HL Congruence Theorem c. Base Angles Theoremb. Reflexive Property of Congruence d. Corresponding parts of congruent

triangles are congruent.

Name: ______________________ ID: A

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____ 21. Which reason is not used in a plan to prove that ∠1 ≅ ∠2?

a. SAS Congruence Theorem c. Vertical Angles Congruence Theoremb. Alternate Interior Angles Theorem d. Corresponding parts of congruent

triangles are congruent.

Find the coordinates of vertex C for the figure placed in a coordinate plane.

____ 22. a rectangle with width m and length twice its width

a. C(2m, m) c. C(m, 2m)b. C(m, m) d. C(2m, 2m)

____ 23. an isosceles triangle

a. C(3, 0) c. C(8, 0)b. C(6, 0) d. C(4, 0)

Name: ______________________ ID: A

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____ 24. In the diagram, AB passes through the center C of the circle and DC⊥AB. Name two triangles that are congruent.

a. DCA ≅ ACD c. DCA ≅ CBDb. DAC ≅ CBD d. not enough information

____ 25. Use the information in the diagram to determine which statements are true.

a. You can use the Vertical Angles Congruence Theorem to prove that ABC ≅ DEC .b. ∠CAB ≅ ∠CDE because corresponding parts of congruent triangles are congruent.

c. Point C is the midpoint of AD. d. You cannot make a conclusion using congruent triangles.

____ 26. Which statements are true for ABC with vertices A(0, 0), B(m, m), and C(2m, 0)? (Assume m is positive.)

a. The slope of AC is undefined. c. ABC is a right triangle.

b. The midpoint of BC is 3m2

,m2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ . d. ABC is isosceles.

Name: ______________________ ID: A

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Match the numbered statement below with its reason to prove that the sides of the flag are parallel.

Given QP ≅ RS , QP Ä RS

Prove QR Ä SP

a. SAS Congruence Theoremb. Converse of Alternate Interior Angles Theoremc. Corresponding parts of congruent triangles are congruent.d. Givene. Reflexive Property of Congruencef. Alternate Interior Angles Theorem

____ 27. 2. ∠QPR ≅ ∠SRP

____ 28. 3. QP ≅ RS

____ 29. 4. PR ≅ RP

____ 30. 5. QPR ≅ SRP

____ 31. 7. QR Ä SP

Write a proof.

32. Given J is the midpoint of KM ,JL⊥KMProve JKL ≅ JML

Name: ______________________ ID: A

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33. Given DF ≅ DH ,FG ≅ HGProve DFG ≅ DHG

34. Given B is the midpoint of AE, AC Ä DEProve ABC ≅ EBD

35. Given ST ≅ UV , RS Ä TU , ∠RTS and ∠TVU are right angles.Prove RST ≅ TUV

Name: ______________________ ID: A

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Decide whether there is enough information given in the picture to prove that the triangles are congruent using the SAS Congruence Theorem. Explain your reasoning.

36. ABD, CBD

37. KLN, MNL

38. In the diagram of the house, the length of AB is 15 feet. Explain why the length of BC is the same.

39. Find PF. Explain your reasoning.

Name: ______________________ ID: A

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40. A sandwich cut diagonally forms two right triangles, with JK ≅ LM . Prove that the two triangles are congruent.

41. Determine whether each statement is true or false. If true, explain your reasoning. If false, give a counterexample.

a. If two triangles are congruent, then their perimeters are the same.b. If two triangles have the same perimeter, then the triangles are congruent.

Write a coordinate proof.

42. Given Coordinates of vertices of ABC, M is the midpoint of AB, P is the midpoint of AC.

Prove MP = 12

BC

Name: ______________________ ID: A

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____ 43. Find BC.

a. BC = 15 c. BC = 24b. BC = 30 d. BC = 9

____ 44. Find m∠BAD.

a. m∠BAD = 46° c. m∠BAD = 23°b. m∠BAD = 11.5° d. m∠BAD = 67°

Name: ______________________ ID: A

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____ 45. In the diagram, AD→

bisects ∠BAC. Find BD .

a. BD = 12 c. BD = 9b. BD = 1 d. BD = 18

____ 46. Write an equation of the perpendicular bisector of the segment with endpoints G −2,0ÊËÁÁ

ˆ¯̃̃ and H 8,−6Ê

ËÁÁˆ¯̃̃ .

a. y = 53

x + 343

c. y = −53

x − 163

b. y = 53

x − 8 d. y = −53

x + 2

____ 47. Find the coordinates of the circumcenter of ABC with vertices A −4,0ÊËÁÁ

ˆ¯̃̃ , B 8,3Ê

ËÁÁˆ¯̃̃ , and C 4,3Ê

ËÁÁˆ¯̃̃ .

a. 6,−14.5ÊËÁÁ

ˆ¯̃̃ c. 2,1.5Ê

ËÁÁˆ¯̃̃

b. 0,1.5ÊËÁÁ

ˆ¯̃̃ d. 6,1.1Ê

ËÁÁˆ¯̃̃

Name: ______________________ ID: A

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In MNQ, BP = 5x + 4, AP = 7x − 11, MP = 6x − 6, and QP = 8x − 15. Match the point of concurrency P below with its value of x in the diagram.

a. x = 10 d. x = 152

b. x = 92

e. x = 193

c. x = 4 f. x = 5

____ 48. incenter P

____ 49. circumcenter P

Is there enough information given in the diagram to conclude that point P lies on the perpendicular bisector of LM ? Explain your reasoning.

50.

Name: ______________________ ID: A

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51.

52. The diagram shows fire hydrants located at points A and B. Line PQ→←

coincides with Pont Road and is the

perpendicular bisector of AB. Will a firefighter at the scene of the car fire have to walk farther to connect the hose to fire hydrant A or fire hydrant B? Explain your reasoning.

53. A pet owner plans to tie up the dog so it can reach the places shown in the diagram. Explain how the owner can determine where to place a stake so that it is the same distance from the dog’s food, the doghouse, and the door to the owner’s house.

Name: ______________________ ID: A

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54. You are drilling a hole in the triangular birdhouse shown. You want the hole to be the same distance from all three sides. Where should you drill the hole? Explain your reasoning.

55. The diagram shows a piece of wood for one side of a shelf. Point P is the center of a hole for a dowel rod

that is to be drilled 5 centimeters from sides AB and BC . You draw segment BP . Can you conclude that the segment bisects ∠ABC? Explain.

ID: A

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Geometry - Review for Final Chapters 5 and 6Answer Section

1. C 2. C 3. D 4. D 5. C 6. D 7. B 8. A 9. B, C, D, E 10. C, D 11. B, C, E 12. 26°, 64° 13. B 14. B 15. A 16. A 17. C 18. B 19. C 20. C 21. B 22. A 23. B 24. A 25. D 26. B, C, D 27. F 28. D 29. E 30. A 31. B 32.

STATEMENTS REASONS

1. J is the midpoint of KM . 1. Given

2. KJ ≅ MJ 2. Definition of midpoint

3. JL⊥KM 3. Given

4. ∠LJM and ∠LJK are right angles. 4. Definition of perpendicular lines

5. ∠LJM ≅ ∠LJK 5. Right Angles Congruence Theorem

6. LJ ≅ LJ 6. Reflexive Property of Congruence

7. JKL ≅ JML 7. SAS Congruence Theorem

ID: A

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33. STATEMENTS REASONS

1. DF ≅ DH 1. Given

2. FG ≅ HG 2. Given

3. DG ≅ DG 3. Reflexive Property of Congruence

4. DFG ≅ DHG 4. SSS Congruence Theorem


1. B is the midpoint of AE. 1. Given

2. AC Ä DE 2. Given

3. AB ≅ EB 3. Definition of midpoint

4. ∠ABC ≅ ∠DBE 4. Vertical Angles Congruence Theorem5. ∠CAB ≅ ∠DEB 5. Alternate Interior Angles Theorem

6. ABC ≅ EBD 6. ASA Congruence Theorem


1. ST ≅ UV , RS Ä TU , ∠RTS and ∠TVU are right angles

1. Given

2. ∠RTS ≅ ∠TVU 2. Right Angles Congruence Theorem3. ∠SRT ≅ ∠UTV 3. Corresponding Angles Theorem5. RST ≅ TUV 5. AAS Congruence Theorem

36. no; The congruent angles are not the included angles. 37. yes; Two pairs of sides and the included angles are congruent. 38. By the Linear Pair Postulate and the definition of supplementary angles, m∠BCA = 180° − 140° = 40°.

Because m∠A = 40° = m∠BCA, by definition of congruent angles, ∠A ≅ ∠BCA. So, by the Converse of the

Base Angles Theorem, AB ≅ BC . So, AB = BC = 15 feet. 39. 12 ft; Using the Exterior Angle Theorem, m∠PFY = m∠ZPF − m∠PYF = 60° − 30° = 30°. Because

m∠PYF = 30° = m∠PFY, by the definition of congruent angles, ∠PYF ≅ ∠PFY. So, by the Converse of the

Base Angles Theorem, PF ≅ YP. So, PF = YP = 12 feet.

40. You are given that JK ≅ LM and that the triangles are right triangles. So, one pair of corresponding legs is

congruent. By the Reflexive Property of Congruence, JL ≅ JL. So, by the HL Congruence Theorem, JLM ≅ LJK .

41. a. true; Because the triangles are congruent, the corresponding side lengths are congruent. The side lengths have the same measures by the definition of congruence. So, the triangles will have the same perimeter.b. false; Sample answer: ABC with AB = 3 meters, BC = 4 meters, and AC = 6 meters has a perimeter of 13 meters, and DEF with DE = 4 meters, EF = 5 meters, and DF = 4 meters has a perimeter of 13 meters.

ABC is not congruent to DEF.

42. Using the Midpoint Formula, M 0,3n2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ and P 2n,0Ê

ËÁÁˆ¯̃̃ . Using the Distance Formula, MP = 5n

2 and BC = 5n .

So, MP = 12

BC.

43. B

ID: A

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44. C 45. C 46. B 47. A 48. D 49. B 50. no; You would need to know that either LN = MN or LP = MP .

51. yes; Because point P is equidistant from L and M , point P is on the perpendicular bisector of LM by the

Converse of the Perpendicular Bisector Theorem. Also, LN ≅ MN , so PN is a bisector of LM . Because P

can only be on one of the bisectors, PN is the perpendicular bisector of LM . 52. The firefighter will walk the same distance to connect the hose to either fire hydrant; Point P is on the

perpendicular bisector of AB, so AP = BP by the Perpendicular Bisector Theorem. 53. The pet owner can copy the positions of the three places, connect the points to draw a triangle, and draw

three perpendicular bisectors of the triangle. The point where the perpendicular bisectors meet, the circumcenter, should be the location of the stake.

54. You should drill the hole at the incenter of the birdhouse because the incenter is equidistant from the sides of the triangle by the Incenter Theorem.

55. yes; Point P is in the interior of ∠ABC and it is equidistant from sides AB and BC . So, P lies on the bisector of ∠ABC.

ID: A Geometry - Review for Final Chapters 5 and 6 [Answer Strip]

__________ 1.C

__________ 2.C

__________ 3.D

__________ 4.D

__________ 5.C

__________ 6.D


__________ 7.B

__________ 8.A

__________ 9.B, C, D, E

__________10.C, D

__________11.B, C, E

__________13.B

__________14.B


__________15.A

__________16.A

__________17.C

__________18.B

__________19.C

__________20.C

__________21.B

__________22.A

__________23.B


__________24.A

__________25.D

__________26.B, C, D

__________27.F

__________28.D

__________29.E

__________30.A

__________31.B

__________43.B

__________44.C


__________45.C

__________46.B

__________47.A

__________48.D

__________49.B

examview - geometry review for final chapters 5 and 6 · geometry - review for final chapters 5 and...

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