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Geometric Reasoning
Circle the best answer.
1. What is the next item in the pattern?
1, 2, 4, 8, . . .
6. Given: If one angle of a triangle is a right
angle, then the other two angles are both
acute. A triangle has a 45° angle.
What conclusion can be drawn?
F One of the other two angles is 90.
A 16
C 4
B 4
G One of the other two angles is obtuse.
D 16
H All three angles are acute.
2. Which is a counterexample that shows
that the following conjecture is false: “If
1 and 2 are supplementary, then one
of the angles is obtuse”?
F m1  45 and m2  45
G m1  53 and m2  127
H m1  90 and m2  90
J m1  100 and m2  80
3. Given: All snarfs are yelbs. All yelbs are
blue. Migs can be either green or pink.
Some slokes are snarfs. What conclusion
can be drawn?
A Some migs are snarfs.
B Some snarfs are green.
J No conclusion can be drawn.
7. Which symbolic statement represents the
Law of Syllogism?
A If p  q and q  r are true statements,
then p  r is a true statement.
B If p  q and p  r are true statements,
then q  r is a true statement.
C If p  q and r  q are true statements,
then q  p is a true statement.
D If p  r and q  r are true statements,
then p  q is a true statement.
8. Which is a biconditional statement of the
conditional statement “If x3  1, then
x  1”?
C Some slokes are yelbs.
F If x  1, then x3  1.
D All slokes are migs.
G x3  1 if x  1.
4. Given the conditional statement “If it is
January, then it is winter in the United
States,” which is true?
F the converse of the conditional
G the inverse of the conditional
H the contrapositive of the conditional
J Not here
5. What is the inverse of the conditional
statement “If a number is divisible by 6,
then it is divisible by 3”?
A If a number is divisible by 3, then it is
divisible by 6.
B If a number is not divisible by 6, then
it is not divisible by 3.
C If a number is not divisible by 3, then
it is not divisible by 6.
D If a number is not divisible by 6, then
it is divisible by 3.
H x3  1 if and only if x  1.
J x  1  x3  1.
9. Which property is NOT used when
solving 15  2x  1?
A Reflex. Prop. of 
B Add. Prop. of 
C Div. Prop. of 
D Sym. Prop. of 
10. Identify the property that justifies the
statement “If B  A, then
A  B.”
F Sym. Prop. of 
G Reflex. Prop. of 
H Trans. Prop. of 
J Sym. Prop. of 
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
Geometric Reasoning
Use the partially completed two-column
proof for Exercises 11 and 12.
Given: m1  30° and m2  2m1.
Prove: 1 and 2 are complementary.
Proof:
Statements
Reasons
1. m1  30,
m2  2m1
1. Given
2.
?
2.
?
3.
?
3.
?
4.
?
4.
?
5.
?
5. Simplify.
6. 1 and 2 are
complementary.
6. Def. of comp. s
11. Each of the items listed below belongs in
one of the blanks in the Statements
column. Which belongs in Step 4?
A m2  2(30)
B m1  m2  90
C m1  m2  30  60
D m2  60
12. Which is the justification for Step 2?
F Add. Prop. of 
G Simplify.
H Subst.
J  Add. Post.
Geometric Reasoning
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Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
1. Find the next item in the pattern.
1, 2, 3, 5, 8, 13, . . .
________________________________________
2. Show that the conjecture is false by
finding a counterexample. The difference
of two odd numbers is a prime number.
________________________________________
3. Identify the hypothesis and conclusion of
the statement “If an angle has a
measure less than 90, then the angle
is acute.”
________________________________________
6. Given: If a ray bisects an angle, two
congruent angles are formed.

YW bisects XYZ.
Conjecture: XYW  WYZ
Determine whether the conjecture is valid
by the Law of Detachment.
________________________________________
7. Given: If a number is a prime number,
then it is an odd number. If a number is
not divisible by 2, then it is an odd
number. Conjecture: If a number is not
divisible by 2, then it is a prime number.
Determine whether the conjecture is valid
by the Law of Syllogism.
________________________________________
________________________________________
________________________________________
4. Write True or False. If a number is a
prime number, then it is an odd number.
________________________________________
5. Write the inverse of the conditional
statement “If the sum of two whole
numbers is even, then both addends
are even.”
8. A square is a rectangle with four
congruent sides. Write the definition
as a biconditional.
________________________________________
________________________________________
9. Solve the equation. Write a justification
for each step.
5m  3  22
________________________________________
________________________________________
________________________________________
________________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
Geometric Reasoning
10. Use the Symmetric Property of Congruence
to complete the statement “If ABC 
XYZ, then XYZ  _________.”
________________________________________
Use the partially completed two-column
proof for Exercises 11 and 12.
Given: ABC is a right angle, X is in the
interior of ABC, and mXBC  45.
Prove: BX bisects ABC.
Proof:
Statements
Reasons
1. ABC is a right
angle.
1. Given
2.
2. Def. of rt. 
?
3. X is in the interior of 3. Given
ABC.
4. mABX  mXBC
 mABC
4.  Add. Post.,
Steps 1, 3
5. mXBC  45
5. Given
6. mABX  45  90
6.
7. mABX  45
7. Subtr. Prop. of 
8. ABX  XBC
8. Def. of  s
9. BX bisects ABC.
9. Def. of 
bisector
?
11. Identify the statement that belongs in
Step 2.
________________________________________
12. Identify the justification for Step 6.
________________________________________
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Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
Answer Key Geometric
Reasoning
Free Response
1. 21
2. Possible answer: 9  3  6
Multiple Choice
3. Hypothesis: An angle has a measure
less than 90.
Conclusion: The angle is acute.
1. A
8. H
2. H
9. A
3. C
10. J
4. False
4. H
11. C
5. B
12. H
5. If the sum of two whole numbers is not
even, then both addends are not even.
6. J
7. A
6. valid
7. not valid
8. A rectangle is a square if and only if it has
four congruent sides.
9.
5m  3  22
5m  3  3  22  3
5m  25
5m 25

5
5
m5
Given equation
Add. Prop. of 
Simplify.
Div. Prop. of 
Simplify.
10. ABC
11. mABC  90
12. Substitution
13. boxes and arrows
14. Possible answer: Since 1 and 2 are
congruent, 1 and 2 have equal
measures by the definition of congruent
angles. 2 and 3 are also congruent,
so these angles also have equal
measures. By the Transitive Property of
Equality, 1 and 3 have equal
measures. Thus, by the definition of
congruent angles, 1  3.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry