Download Geometry Final - 2015 - Stuyvesant High School

Name __________________________________________
Four digit ID # ______________
Stuyvesant High School
J. Zhang, Principal
MGS43 Final Exam
Mathematics Department
M. Ferrara, AP
January 2015
Form BLUE
Use of calculators is NOT permitted.
Diagrams are NOT drawn to scale.
Directions: On your scantron sheet, darken the letter preceding the word or expression that best answers the question or
completes the statement. There are 25 questions worth 4 points each.
1.
Given the premises: T  ~R , ~R  ~W , and W , which conclusion must be true?
(A) ~ R
2.
(B) ~ W
(C) T
(E) RT
(D) ~T
A
Given: Triangle ACD, AB  AE , AC  AD , which of the following must be true?
(A) BE  CD / 2
(B) BE CD
(D) B isa midpoint
(E) Triangle ACD is acute
B
(C) AC  CD
E
C
3.
4.
If A and B are supplementary angles, then
(A) A and B are vertical angles
(B) A and B are adjacent angles
(D) mA + mB = 180
(E) A and B are a linear pair and mA + mB = 180
If C is the midpoint of AB and D is the midpoint of
(A)
5.
AC  BC
7.
9.
AD  CD
(C)
AC , then
DB  AC
(D)
DB  3 CD
(E)
AD  BC
(B) 10
(C) 12
(D) 80
(E) 8
Given the diagram, which of the following must be true?
̅̅̅̅ is longer than ̅̅̅̅
(𝐴) 𝐴𝐷
𝐶𝐷
(B) ̅̅̅̅
𝐵𝐷 is the shortest segment
̅̅̅̅ is the longest segment
(C) 𝐴𝐷
(D) ̅̅̅̅
𝐴𝐵 is congruent to ̅̅̅̅
𝐵𝐷
(E) ̅̅̅̅
𝐴𝐵 is the shortest segment
Given AB  AD and CB  CD . If BD = 8 , AB = 5 , and BC = 6 , find AC.
(A)
8.
(B)
(C) A and B are a linear pair
How many sides does a convex polygon have if the sum of the measures of the interior angles is 1080 o?
(A) 108
6.
D
61
(B) 32 5
Given isosceles trapezoid ABCD, BC =
the legs of ABCD is
11
2
(A) 11
(B)
(D) 6 10
(E) 3 10
(C) 32 13
(D) 35 2
(E) 34 5
5 , AD = 9 5 , and BE = 10 . The sum of the lengths of
(C) 18 10
In the accompanying figure, triangle ABC is an equilateral triangle and ADEF is a rhombus. If D is
the midpoint of AB , and the perimeter of triangle ABC is 12, then what is the perimeter of ADEF?
(A) 8
(B) 9
(C) 10
(D) 12
(E) 16
10. If the diagonals of a parallelogram are perpendicular and not congruent, then the parallelogram is
(A) a rectangle
(B) a square
(C) a rhombus
(D) an isosceles trapezoid
(E) none of these
11. In the diagram below, 𝑃𝑄𝑅𝑆 is a trapezoid with SR PQ . TU is the median. If SR = x, PQ = 2x + 6, and TU = 15,
what is the value of 𝑥
(A) 10
(B) 12
(C) 8
(D) 3
(E) 13
12. Name the segments, if any, that are parallel if ∡4 ≅ ∡7.
(A) FA
(B) DC
EB
(D) both FA
(C) EF
EB
EB and EF
AB
(E) None
AB
13. ∠1 ≅ ∠2 and ∠3≅ ∠4. Which statement cannot be proven from the given statements?
(A) ̅̅̅̅
𝐴𝐾 bisects ∠𝐵𝐴𝐶
(B) ̅̅̅̅
𝐵𝐾 ≅ ̅̅̅̅
𝐶𝐾
(C) ∆𝐴𝐵𝐾 ≅ ∆𝐴𝐶𝐾
⃡ ⏊𝐵𝐶
̅̅̅̅
(E) 𝐴𝐾
(D) ∆𝐴𝐶𝐾 ≅ ∆𝐵𝐶𝐾
14. In the diagram, BA  AE , CD  DE , mBCD  40 , and mDCE  70 . Name the similar triangles.
(A) ABC
CDE
(B) ABC
DCE
(D) BAC
DCE
(E) CAB
CED
(C) ABC
DEC
15. If CB  CD in the figure below, it can be proved that FB  ED if it is also known that
(A) EABFAD
(B) BD
(C) BDFB
(D) CD
(E) CEAF
16. In rhombus ABCD, diagonal AC  18 , mABC = 110o, and the diagonals intersect at E. Which equation could be
used to find the length of one side of ABCD?
(A) sin 55 
x
9
(B) sin 55 
9
x
(C) sin 55 
18
x
(D) cos 55 
18
x
(E) cos 55 
9
x
17. Given the true statements “Bob is shorter than Ted” and “Ann is taller than Ted,” which property of inequality may be
used to support the conclusion “Ann is taller than Bob”?
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) Substitution
(E)
None of
these
18. Given right ABC , with right B . If AD  CD and BD = 5, find the length of AC.
(A) 15 3
(B) 10 3
(C) 15
(D) 10
(E)
15
3
2
19. In quadrilateral ABCD, B is a right angle. Diagonal AC is perpendicular CD . If AB  5 cm , BC  8 2 , and
CD  6 2 , then the perimeter of quadrilateral ABCD is
(A) 5  14 2  5 17
(B) 5  14 2  53
(C) 28
D
(D) 20  14 2
(E) 30  14 2
Use the following proof for questions 20-22:
Given: 𝛥𝐴𝐵𝐶
Prove: 𝐴𝐵 + 𝐵𝐶 > 𝐴𝐶
B
A
Statements
1. 𝛥𝐴𝐵𝐶
2. Extend ̅̅̅̅
𝐴𝐵 to D such that ̅̅̅̅
𝐵𝐷 ≅ ̅̅̅̅
𝐵𝐶
3. 𝐵𝐷 = 𝐵𝐶
4. Draw ̅̅̅̅
𝐶𝐷
5. ∠𝐷 ≅ ∠𝐵𝐶𝐷
6. 𝑚∠𝐷 = 𝑚∠𝐵𝐶𝐷
7. Question 20
8. Question 21
9. Question 22
10. 𝐴𝐷 = 𝐴𝐵 + 𝐵𝐷
11. 𝐴𝐷 = 𝐴𝐵 + 𝐵𝐶
12. 𝐴𝐵 + 𝐵𝐶 > 𝐴𝐶
C
Reasons
Given
Segment Existence Postulate
Congruent segments have equal length.
Two points determine a line.
In a triangle, angles opposite congruent sides are congruent.
Congruent angles are equal in measure.
A whole is greater than any of its parts.
Substitution Property
In a triangle, the greater side is opposite the larger angle
A whole is equal to the sum of its parts.
Substitution Property
Substitution Property
20. Which is the best statement for #7?
(A) 𝐴𝐷 > 𝐵𝐷
(B) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐵𝐶𝐷 (C) 𝑚∠𝐴𝐵𝐶 > 𝑚∠𝐷
(D) 𝐴𝐷 > 𝐴𝐵
21. Which is the best statement for #8?
(A)𝐴𝐷 > 𝐴𝐶
(B) 𝑚∠𝐴𝐵𝐶 > 𝑚∠𝐵𝐶𝐷 (C) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐴𝐶𝐵 (D) 𝐴𝐷 > 𝐵𝐶
22. Which is the best statement for #9?
(A) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐷
(B) 𝐴𝐷 > 𝐴𝐵
(C) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐵𝐴𝐶
(E) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐴𝐶𝐵
(E) 𝑚∠𝐴𝐶𝐷 > 𝑚∠𝐷
(D) 𝐴𝐷 > 𝐴𝐶
(E) 𝐴𝐶 > 𝐵𝐷
____________________________________________________________________________________________________________________________
Use the following proof for questions 23-25:
̅̅̅̅
Given:
𝐴𝑋 ≅ ̅̅̅̅
𝐶𝑌
∠𝐴 ≅ ∠𝐶
̅̅̅̅ is altitude of ∆𝐴𝐵𝐷
𝐵𝑋
̅̅̅̅
𝐷𝑌 is altitude of ∆𝐵𝐷𝐶
Prove:
𝐷
𝐶
𝑋
∠𝑋𝐷𝐵 ≅ ∠𝑌𝐵𝐷
𝑌
𝐴
Statements
Reasons
1. Given
1. AX CY
2. AC
̅̅̅̅ is altitude of ∆𝐴𝐵𝐷
3. 𝐵𝑋
̅̅̅̅
4. 𝐷𝑌 is altitude of ∆𝐵𝐷𝐶
2. Given
3. Given
4. Given
5. Question 23
5. BX  AD
6.
7.
8.
9.
𝐵
DY BC
AXB and CYD are right angles
AXBCYD
ΔAXBΔCYD
6. Question 23
7. Perpendicular lines intersect to form right angles.
8. Right angles are congruent.
9. ASA
10. Corresponding parts of congruent triangles are
congruent.
11. Perpendicular lines intersect to form right angles.
12. Reflexive property
13. Question 25
14. Corresponding parts of congruent triangles are
congruent.
10. BX DY
11. DXB and BYD are right angles
12. Question 24
13. ΔDXBΔBYD
14. XDBYBD
23. Which reason justifies statements 5 and 6?
(A) An altitude of a triangle is a line segment that connects a vertex of the triangle to its opposite side and ⏊ to it.
(B) If two lines intersect to form right angles then the lines are ⏊.
(C) If two adjacent angles are congruent and supplementary then they are right angles.
(D) The legs of a right triangle are ⏊.
(E) If a point is equidistant from the endpoints of a line segment then it is on the ⏊ bisector of that segment.
24. Which statement fits reason 12?
(A) ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐵
(B) ̅̅̅̅
𝐷𝐶 ≅ ̅̅̅̅
𝐷𝐶
25. Which reason justifies statement 13?
(A) AAS
(B) SSS
(C) ̅̅̅̅
𝐵𝑋 ≅ ̅̅̅̅
𝐵𝑋
(C) HL
(D) ̅̅̅̅
𝐷𝑌 ≅ ̅̅̅̅
𝐷𝑌
(D) SSA
(E) ̅̅̅̅
𝐷𝐵 ≅ ̅̅̅̅
𝐷𝐵
(E) SAS