Download ExamView - geometry review for final chapters 5 and 6 .tst

Name: ______________________ Class: _________________ Date: _________
Geometry - Review for Final Chapters 5 and 6
____
1. Classify
a.
b.
____
PQR by its sides. Then determine whether it is a right triangle.
scalene ; right
isoceles ; not right
c.
d.
scalene ; not right
isoceles ; right
c.
d.
101°
5°
2. Find m∠KMN .
a.
b.
20°
79°
1
ID: A
Name: ______________________
____
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.
b.
____
ID: A
45°, 45°
11.25°, 78.75°
c.
d.
22.5°, 67.5°
30°, 60°
c.
d.
QRP ≅
QRP ≅
4. Write a congruence statement for the triangles.
a.
b.
PRQ ≅
PRQ ≅
TUV
UTV
2
TUV
UTV
Name: ______________________
____
5. In the diagram,
a.
b.
____
y = 51
y = 54
CDE ≅
ID: A
GHI . Find the value of y.
c.
d.
y = 42
y = 64
c.
d.
56°
112°
6. Find m∠BDC .
a.
b.
68°
84°
3
Name: ______________________
____
7. Find the value of x.
a.
b.
____
ID: A
10
4
c.
d.
5
6
8. In the diagram, M is the midpoint of BC . Find the coordinates of M.
a.
b.
ÁÊÁ 3 ˜ˆ˜
ÁÁ ,2 ˜˜
Á2 ˜
Ë
¯
ÁÊÁ 3 4 ˜ˆ˜
ÁÁ , ˜˜
Á4 3˜
Ë
¯
c.
d.
4
ÁÊÁ 4 3 ˜ˆ˜
ÁÁ , ˜˜
Á3 4˜
Ë
¯
ÁÊÁ 3 ˜ˆ˜
ÁÁ 2, ˜˜
Á 2˜
Ë
¯
Name: ______________________
____
ID: A
9. Which statements about the diagram are true?
a.
b.
c.
BDE is obtuse.
ABC is scalene.
The value of x is 18.
d.
e.
f.
____ 10. What additional information is needed to prove that
a.
b.
∠BNA ≅ ∠COD and CO ≅ BN
AN ≅ DO and ∠ABN ≅ ∠DCO
c.
d.
ABC is obtuse.
BDE is right.
The value of y is 113.
ABN ≅
DCO using the SAS Congruence Theorem?
∠ABN ≅ ∠DCO and CO ≅ BN
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
an isosceles right triangle? Assume all variables are positive.
ÊÁ
ˆ
ÊÁ
ˆ
ÁÁ
ÁÁ 3a 2 3a 2 ˜˜˜
3a 3 ˜˜˜˜
Á
Á
˜˜
a. ÁÁ 3a,
˜
d. ÁÁ
,
2 ˜˜˜
2 ˜˜˜
ÁÁ
ÁÁ 2
Ë
¯
Ë
¯
ÁÊÁ 3a 3a ˜ˆ˜
˜˜
,
b. ÁÁÁ
e. ÊÁË 3a, 3a ˆ˜¯
˜
2
2
Ë
¯
Ê
ˆ
Á
˜
c. Ë 0, 3a ¯
5
PQR
Name: ______________________
ID: A
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.
6
no
Name: ______________________
ID: A
____ 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.
b.
c.
yes; AAS Congruence Theorem
yes; ASA Congruence Theorem
no
a.
b.
c.
yes; AAS Congruence Theorem
yes; ASA Congruence Theorem
no
____ 17.
7
Name: ______________________
ID: A
____ 18.
a.
b.
c.
yes; AAS Congruence Theorem
yes; ASA Congruence Theorem
no
____ 19. Which reason is not necessary to explain how you can find the distance across the lake?
a.
b.
c.
d.
e.
ASA Congruence Theorem
Right Angles Congruence Theorem
SAS Congruence Theorem
Corresponding parts of congruent triangles are congruent.
Vertical Angles Congruence Theorem
____ 20. Which reason is not used in a plan to prove that XW ≅ ZY ?
a.
b.
HL Congruence Theorem
Reflexive Property of Congruence
c.
d.
8
Base Angles Theorem
Corresponding parts of congruent
triangles are congruent.
Name: ______________________
ID: A
____ 21. Which reason is not used in a plan to prove that ∠1 ≅ ∠2?
a.
b.
SAS Congruence Theorem
Alternate Interior Angles Theorem
c.
d.
Vertical Angles Congruence Theorem
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.
b.
C(2m, m)
C(m, m)
c.
d.
C(m, 2m)
C(2m, 2m)
c.
d.
C(8, 0)
C(4, 0)
____ 23. an isosceles triangle
a.
b.
C(3, 0)
C(6, 0)
9
Name: ______________________
ID: A
____ 24. In the diagram, AB passes through the center C of the circle and DC ⊥AB. Name two triangles that are
congruent.
a.
b.
DCA ≅
DAC ≅
ACD
CBD
c.
d.
DCA ≅ CBD
not enough information
____ 25. Use the information in the diagram to determine which statements are true.
a.
b.
c.
d.
You can use the Vertical Angles Congruence Theorem to prove that ABC ≅ DEC .
∠CAB ≅ ∠CDE because corresponding parts of congruent triangles are congruent.
Point C is the midpoint of AD.
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.)
c.
a. The slope of AC is undefined.
ABC is a right triangle.
ÊÁ 3m m ˆ˜
, ˜˜˜˜ .
b. The midpoint of BC is ÁÁÁÁ
d.
ABC is isosceles.
2
2¯
Ë
10
Name: ______________________
ID: A
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.
b.
c.
d.
e.
f.
SAS Congruence Theorem
Converse of Alternate Interior Angles Theorem
Corresponding parts of congruent triangles are congruent.
Given
Reflexive Property of Congruence
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⊥KM
Prove
JKL ≅ JML
11
Name: ______________________
ID: A
33. Given DF ≅ DH ,FG ≅ HG
Prove
DFG ≅ DHG
34. Given B is the midpoint of AE , AC Ä DE
Prove
ABC ≅ EBD
35. Given ST ≅ UV , RS Ä TU , ∠RTS and ∠TVU are right angles.
Prove
RST ≅ TUV
12
Name: ______________________
ID: A
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.
13
Name: ______________________
ID: A
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
Prove
Coordinates of vertices of
1
MP = BC
2
ABC , M is the midpoint of AB, P is the midpoint of AC .
14
Name: ______________________
ID: A
____ 43. Find BC.
a.
b.
BC = 15
BC = 30
c.
d.
BC = 24
BC = 9
c.
d.
m∠BAD = 23°
m∠BAD = 67°
____ 44. Find m∠BAD.
a.
b.
m∠BAD = 46°
m∠BAD = 11.5°
15
Name: ______________________
ID: A


→
____ 45. In the diagram, AD bisects ∠BAC. Find BD .
a.
b.
BD = 12
BD = 1
c.
d.
BD = 9
BD = 18
____ 46. Write an equation of the perpendicular bisector of the segment with endpoints G ÊÁË −2,0 ˆ˜¯ and H ÊÁË 8,−6 ˆ˜¯ .
a.
b.
5
34
x+
3
3
5
y= x−8
3
y=
____ 47. Find the coordinates of the circumcenter of
a. ÊÁË 6,−14.5ˆ˜¯
b. ÁÊË 0,1.5 ˜ˆ¯
c.
d.
16
5
y=− x−
3
3
5
y=− x+2
3
ABC with vertices A ÊÁË −4,0ˆ˜¯ , B ÊÁË 8,3ˆ˜¯ , and C ÊÁË 4,3 ˆ˜¯ .
c. ÊÁË 2,1.5 ˆ˜¯
d. ÁÊË 6,1.1 ˜ˆ¯
16
Name: ______________________
ID: A
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
b.
x=
c.
9
2
x=4
15
2
19
e. x =
3
f. x = 5
d.
x=
____ 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.
17
Name: ______________________
ID: A
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.
18
Name: ______________________
ID: A
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.
19
ID: A
Geometry - Review for Final Chapters 5 and 6
Answer Section
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
C
C
D
D
C
D
B
A
B, C, D, E
C, D
B, C, E
26°, 64°
B
B
A
A
C
B
C
C
B
A
B
A
D
B, C, D
F
D
E
A
B
STATEMENTS
1. J is the midpoint of KM .
2. KJ ≅ MJ
3. JL⊥KM
REASONS
1. Given
2. Definition of midpoint
3. Given
4. ∠LJM and ∠LJK are right angles.
4. Definition of perpendicular lines
5. ∠LJM ≅ ∠LJK
6. LJ ≅ LJ
7. JKL ≅ JML
5. Right Angles Congruence Theorem
6. Reflexive Property of Congruence
7. SAS Congruence Theorem
1
ID: A
33.
STATEMENTS
1. DF ≅ DH
REASONS
1. Given
2. FG ≅ HG
2. Given
3. DG ≅ DG
4. DFG ≅ DHG
3. Reflexive Property of Congruence
STATEMENTS
1. B is the midpoint of AE .
2. AC Ä DE
REASONS
1. Given
2. Given
3. AB ≅ EB
4. ∠ABC ≅ ∠DBE
5. ∠CAB ≅ ∠DEB
6. ABC ≅ EBD
3. Definition of midpoint
4. SSS Congruence Theorem
34.
4. Vertical Angles Congruence Theorem
5. Alternate Interior Angles Theorem
6. ASA Congruence Theorem
35.
36.
37.
38.
39.
STATEMENTS
REASONS
1. Given
1. ST ≅ UV , RS Ä TU , ∠RTS and ∠TVU
are right angles
2. ∠RTS ≅ ∠TVU
2. Right Angles Congruence Theorem
3. ∠SRT ≅ ∠UTV
3. Corresponding Angles Theorem
5. AAS Congruence Theorem
5. RST ≅ TUV
no; The congruent angles are not the included angles.
yes; Two pairs of sides and the included angles are congruent.
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.
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 .
ÊÁ 3n ˆ˜
5n
42. Using the Midpoint Formula, M ÁÁÁÁ 0, ˜˜˜˜ and P ÁÊË 2n,0 ˜ˆ¯ . Using the Distance Formula, MP =
and BC = 5n .
2
2
Ë
¯
So, MP =
1
BC .
2
43. B
2
ID: A
44.
45.
46.
47.
48.
49.
50.
C
C
B
A
D
B
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 .
3
Geometry - Review for Final Chapters 5 and 6 [Answer Strip]
D
__________
3.
C
__________
5.
C
__________
1.
D
__________
4.
D
__________
6.
C
__________
2.
ID: A
Geometry - Review for Final Chapters 5 and 6 [Answer Strip]
B
__________
7.
B, C, D, E 9.
__________
B
__________13.
A
__________
8.
C, D
__________10.
B
__________14.
B, C, E
__________11.
ID: A
Geometry - Review for Final Chapters 5 and 6 [Answer Strip]
A
__________15.
B
__________18.
B
__________21.
C
__________19.
A
__________22.
A
__________16.
B
__________23.
C
__________17.
C
__________20.
ID: A
Geometry - Review for Final Chapters 5 and 6 [Answer Strip]
B
__________43.
A
__________24.
D
__________25.
C
__________44.
F
__________27.
D
__________28.
E
__________29.
A
__________30.
B
__________31.
B, C, D
__________26.
ID: A
Geometry - Review for Final Chapters 5 and 6 [Answer Strip]
C
__________45.
B
__________46.
D
__________48.
A
__________47.
B
__________49.
ID: A