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Math 119 Summer 2004 Exam 3A
Morrow
Name: _________________________
This exam is closed note, closed book. Each item (question/part) is worth two points unless
otherwise marked.
TRUE/FALSE. If a statement is always true, write true. If it is not, do not write false. Instead,
write a word or phrase that could replace the underlined statement to make the sentence true.
1. CPCTC stands for “corresponding parts of corresponding triangles are congruent”.
Corresponding parts of congruent triangles are congruent
2. A parallelogram is always a quadrilateral.
True
3. If one pair of opposite sides of a quadrilateral are parallel and the other pair are congruent,
then the quadrilateral is a parallelogram.
Parallel
4. If all the angles of a rhombus are congruent, then the rhombus is a square.
True
5. In a triangle, an exterior angle is larger than any angle of the triangle.
Any nonadjacent interior angle of the triangle
6. If a, b, and c are real numbers and a = b + c, then a > b.
A, b, and c are real numbers, a = b + c, and c is positive
7. In a triangle, the largest angle is formed from the longest sides.
Opposite the longest side
8. The diagonals of a trapezoid bisect each other.
Parallelogram OR Rhombus OR Square OR Rectangle
9. The diagonals of a rectangle are perpendicular.
Rhombus OR Square
10. All angles of a rhombus are congruent.
Sides
11. A parallelogram is a quadrilateral that can have angles of all different measures.
Trapezoid OR 4-sided polygon
12. If a pair of angles of a trapezoid are congruent, then the trapezoid is isosceles.
Base angles
13. The geometric mean between 5 and 15 is 10.
X where 5/x = x/15 OR √75
14. The length of the median of a trapezoid is determined by the legs of the trapezoid and is equal
to one-half the sum of the two legs.
Bases; one-half the sum of the two bases
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SHORT ANSWER. Answer each question. The more work you show, the more opportunities
there are for me to award partial credit.
15. Given two triangles ΔABC and ΔDEF (not pictured). State the reason why the triangles are
congruent for each of the following. If you cannot conclude the triangles are congruent, state
this.
(a) AB  DE, B  E, and AC  DF.
No Conclusion Possible
(b) A  D, C  F, and AC  DF.
ASA
(c) A  D, B  E, and C  F.
No Conclusion Possible
(d) Triangles ABC and DEF are right triangles with right angles B and E.
AC  DF and AB  DE.
HL
16. Suppose M is the midpoint of AB and N is the midpoint of AC. Let CB = 24.
(a) Find MN.
C
MN = ½ (CB) = ½ * 24 = 12
N
(b) Why is AMN  ABC?
MN  CB by the Midsegment Theorem.
Hence the corresponding angles are congruent.
A
M
B
(c) Why is ΔAMN ~ ΔABC?
A  A by the Reflexive Property of Congruence.
The triangles are similar by the AA Similarity Theorem.
(d) How do the sides of ΔAMN relate to the corresponding sides of ΔABC?
The sides are proportional.
(e) Given that CB = 24, is it possible for ΔABC to be an isosceles triangle with AB = 12?
Why or why not?
Possible; sides of length 24, base of length 12.
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17. Given parallelogram ABCD with diagonals AC and DB . Suppose AC  DB , AC = 6,
and DB = 8.
(a) Find AB.
32 + 42 = (AB)2
AB = 5
(b) Find the perimeter of the parallelogram.
Since the diagonals are perpendicular, this is a rhombus. Hence the perimeter =
4*(AB) = 4*5 = 20.
CONSTRUCTIONS. Use your straightedge and compass to make the following constructions.
Leave all “helper marks”. Answers with no “helper marks” will receive no credit.
18. Construct a right triangle that has an acute angle A and hypotenuse of length c.
Steps:
(1) Create a copy of angle A
(2) Make one of the sides length c
(3) Make the perpendicular line to the other
side of angle A
A
c
PROOFS. (8 points each)
19. Given:
Prove:
A
A and 3 are complements of 1
AB  CD
2 and D are right angles
AC  CE
STATEMENTS
1. A and 3 are complements of 1
AB  CD
2 and D are right angles
2. A  3
3. 2  D
4. ΔABC  ΔCDE
5. AC  CE
REASONS
1. Given
2.
3.
4.
5.
C
B
1 23
E
D
Complements to the same angle are congruent
All right angles are congruent.
ASA
CPCTC
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20. Given:
Prove:
D is the midpoint of BC
AD  BC
ΔABC is isosceles
STATEMENTS
1. D is the midpoint of BC
AD  BC
2. BD  DC
3. ADB  ADC
4. AD  AD
5. ΔADB  ΔADC
6. AB  AC
7. ΔABC is isosceles
REASONS
1. Given
2.
3.
4.
5.
6.
7.
Definition of midpoint
Definition of perpendicular lines
Reflexive Property of Congruence
SAS
CPCTC
Definition of isosceles triangle
21. Provide a formal proof for the following statement:
The opposite sides of a parallelogram are congruent.
Given:
Prove:
B
Parallelogram ABCD
AB  DC
A
STATEMENTS
1. Parallelogram ABCD
2. ΔABC  ΔCDA
3. AB  DC
C
D
REASONS
1. Given
2. The diagonal of a parallelogram forms congruent
triangles
3. CPCTC
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