Download Topic 2 Review

Topic 2
Review
TOPIC VOCABULARY
Ě biconditional, p. 55
Ě GHGXFWLYHUHDVRQLQJ p. 60
Ě /DZRI6\OORJLVP p. 60
Ě 5HIOH[LYH3URSHUW\ p. 65
Ě conclusion, p. 49
Ě GLDPHWHUp. 44
Ě QHJDWLRQ p. 49
Ě 6\PPHWULF3URSHUW\ p. 65
Ě conditional, p. 49
Ě HTXLYDOHQWVWDWHPHQWV p. 49
Ě SDUDJUDSKSURRI p. 71
Ě theorem, p. 71
Ě conjecture, p. 44
Ě hypothesis, p. 49
Ě SRO\JRQ p. 44
Ě 7UDQVLWLYH3URSHUW\ p. 65
Ě contrapositive, p. 50
Ě LQGXFWLYHUHDVRQLQJ p. 44
Ě SURRI p. 66
Ě WUXWKYDOXH p. 49
Ě converse, p. 50
Ě inverse, p. 50
Ě quadrilateral, p. 55
Ě WZRFROXPQSURRI p. 66
Ě counterexample, p. 44
Ě /DZRI'HWDFKPHQW p. 60
Ě radius, p. 44
Check Your Understanding
Choose the correct vocabulary term to complete each sentence.
1. The part of a conditional that follows “then” is the ? .
2. Reasoning logically from given statements to a conclusion is ? .
3. A conditional has a(n) ? of true or false.
4. The ? of a conditional switches the hypothesis and conclusion.
5. When a conditional and its converse are true, you can write them as a single
true statement called a(n) ? .
6. A statement that you prove true is a(n) ? .
2-1 Patterns and Conjectures
Quick Review
Exercises
You use inductive reasoning when you make conclusions
based on patterns you observe. A conjecture is a conclusion
you reach using inductive reasoning. A counterexample is
an example that shows a conjecture is incorrect.
Find a pattern for each sequence. Describe the pattern
and use it to show the next two terms.
7. 1000, 100, 10, c
8. 5, -5, 5, -5, c
Example
Describe the pattern. What are the next two terms in the
sequence?
1, −3, 9, −27, . . .
Each term is -3 times the previous term. The next two
terms are -27 * ( -3) = 81 and 81 * ( -3) = -243.
9. 34, 27, 20, 13, c
10. 6, 24, 96, 384, c
Find a counterexample for each conjecture.
11. The product of any integer and 2 is greater than 2.
12. The city of Portland is in Oregon.
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2-2 Conditional Statements
Quick Review
Exercises
A conditional is an if-then statement. The symbolic form of
a conditional is p S q, where p is the hypothesis and q is
the conclusion.
Rewrite each sentence as a conditional statement.
Ě To find the converse, switch the hypothesis and
conclusion of the conditional (q S p).
13. All motorcyclists wear helmets.
14. Two nonparallel lines intersect in one point.
15. Angles that form a linear pair are supplementary.
Ě To find the inverse, negate the hypothesis and the
conclusion of the conditional (∼p S ∼q).
16. School is closed on certain holidays.
Ě To find the contrapositive, negate the hypothesis and
the conclusion of the converse (∼q S ∼p).
Write the converse, inverse, and contrapositive of the
given conditional. Then determine the truth value
of each statement.
Example
17. If an angle is obtuse, then its measure is greater than
90 and less than 180.
What is the converse of the conditional statement below?
What is its truth value?
If you are a teenager, then you are younger than 20.
Converse: If you are younger than 20, then you are a
teenager.
18. If a figure is a square, then it has four sides.
19. If you play the tuba, then you play an instrument.
20. If you baby-sit on Saturday night, then you are busy on
Saturday night.
A 7-year-old is not a teenager. The converse is false.
2-3 Biconditionals and Definitions
Quick Review
Exercises
When a conditional and its converse are true, you can
combine them as a true biconditional using the phrase if
and only if. The symbolic form of a biconditional is p 4 q.
You can write a good definition as a true biconditional.
For Exercises 21–23, determine whether each statement
is a good definition. If not, explain.
Example
Is the following definition reversible? If yes, write it as a
true biconditional.
A hexagon is a polygon with exactly six sides.
Yes. The conditional is true: If a figure is a hexagon, then it
is a polygon with exactly six sides. Its converse is also true:
If a figure is a polygon with exactly six sides, then it is a
hexagon.
Biconditional: A figure is a hexagon if and only if it is a
polygon with exactly six sides.
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Topic 2 Review
21. A newspaper has articles you read.
22. A linear pair is a pair of adjacent angles whose
noncommon sides are opposite rays.
23. An angle is a geometric figure.
24. Write the following definition as a biconditional.
An oxymoron is a phrase that contains
contradictory terms.
25. Write the following biconditional as two statements,
a conditional and its converse.
Two angles are complementary if and only if the
sum of their measures is 90.
2-4 Deductive Reasoning
Quick Review
Exercises
Deductive reasoning is the process of reasoning logically
from given statements to a conclusion.
Use the Law of Detachment to make a conclusion.
Law of Detachment: If p S q is true and p is true,
then q is true.
Law of Syllogism: If p S q and q S r are true,
then p S r is true.
26. If you practice tennis every day, then you will become
a better player. Colin practices tennis every day.
27. ∠1 and ∠2 are supplementary. If two angles are
supplementary, then the sum of their measures is 180.
Use the Law of Syllogism to make a conclusion.
Example
What can you conclude from the given information?
Given: If you play hockey, then you are on the team.
If you are on the team, then you are a varsity athlete.
The conclusion of the first statement matches the
hypothesis of the second statement. Use the Law of
Syllogism to conclude: If you play hockey, then you are
a varsity athlete.
28. If two angles are vertical, then they are congruent.
If two angles are congruent, then their measures
are equal.
29. If your father buys new gardening gloves, then he will
work in his garden. If he works in his garden, then he
will plant tomatoes.
2-5 Reasoning in Algebra and Geometry
Quick Review
Exercises
You use deductive reasoning and properties to solve
equations and justify your reasoning.
30. Fill in the reason that justifies each step.
A proof is a convincing argument that uses deductive
reasoning. A two-column proof lists each statement on
the left and the justification for each statement on the right.
Given: QS = 42
Prove: x = 13
x13
Q
Statements
2x
R
S
Reasons
1) QS = 42
1) a. ?
Example
2) QR + RS = QS
2) b. ?
What is the name of the property that justifies going from
the first line to the second line?
3) (x + 3) + 2x = 42
3) c. ?
4) 3x + 3 = 42
4) d. ?
jA @ jB and jB @ jC
jA @ jC
5) 3x = 39
5) e. ?
6) x = 13
6) f. ?
Transitive Property of Congruence
Use the given property to complete the statement.
31. Division Property of Equality:
If 2(AX) = 2(BY), then AX = ? .
32. Distributive Property: 3p - 6q = 3( ? )
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2-6 Proving Angles Congruent
Quick Review
Exercises
A statement that you prove true is a theorem. A proof
written as a paragraph is a paragraph proof. In geometry,
each statement in a proof is justified by given information, a
property, postulate, definition, or theorem.
Use the diagram for Exercises 33–36.
33. Find the value of y.
35. Find m∠BED.
Example
23
Write a paragraph proof.
Given: ∠1 ≅ ∠4
4
1
Prove: ∠2 ≅ ∠3
∠1 ≅ ∠4 because it is given. ∠1 ≅ ∠2 because vertical
angles are congruent. ∠4 ≅ ∠2 by the Transitive Property
of Congruence. ∠4 ≅ ∠3 because vertical angles are
congruent. ∠2 ≅ ∠3 by the Transitive Property of
Congruence.
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Topic 2 Review
B
A
34. Find m∠AEC.
36. Find m∠AEB.
37. Given: ∠1 and ∠2 are
complementary,
∠3 and ∠4 are
complementary,
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3
(3y 1 20)8
E
C
(5y 2 16)8
D
1
3
2
4
Topic 2
TEKS Cumulative Practice
Multiple Choice
Read each question. Then write the letter of the correct
answer on your paper.
1. What is the second step in constructing ∠S, an angle
congruent to ∠A?
4. Which counterexample shows that the following
conjecture is false?
Every perfect square number has exactly three factors.
F. The factors of 2 are 1, 2.
G. The factors of 4 are 1, 2, 4.
H. The factors of 8 are 1, 2, 4, 8.
J. The factors of 16 are 1, 2, 4, 8, 16.
A
5. How many rays are in the next two terms in the
sequence?
A. S
C.
T
R
S
B.
D.
S
R
B
A
C
2. What is the converse of the following statement?
If a whole number has 0 as its last digit, then the
number is evenly divisible by 10.
F. If a number is evenly divisible by 10, then it is a
whole number.
G. If a whole number is divisible by 10, then it is an
even number.
H. If a whole number is evenly divisible by 10, then it
has 0 as its last digit.
J. If a whole number has 0 as its last digit, then it must
be evenly divisible by 10.
3. The sum of the measures of the complement and the
supplement of ∠Y is 114. What is m∠Y ?
A. 12
C. 78
B. 66
D. 102
A. 16 and 33 rays
C. 17 and 34 rays
B. 17 and 33 rays
D. 18 and 34 rays
6. Which of the statements could be a conclusion based
on the following information?
If a polygon is a pentagon, then it has one more side
than a quadrilateral. If a polygon has one more side
than a quadrilateral, then it has two more sides than
a triangle.
F. If a polygon is a pentagon, then it has many sides.
G. If a polygon has two more sides than a
quadrilateral, then it is a hexagon.
H. If a polygon has more sides than a triangle, then it
is a pentagon.
J. If a polygon is a pentagon, then it has two more
sides than a triangle.
7. Which pair of angles must be congruent?
A. supplementary angles
B. complementary angles
C. adjacent angles
D. vertical angles
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8. Which of the following best defines a postulate?
F. a statement accepted without proof
G. a conclusion reached using inductive reasoning
H. an example that proves a conjecture false
J. a statement that you prove true
9. Which type of reasoning is based on patterns you observe?
A. deductive reasoning
B. inductive reasoning
C. detachment and syllogism
D. conclusion and hypothesis
10. Which property says that if a = b, then b = a?
F. Reflexive Property
13. Which is an undefined term?
A. ray
C. line
B. segment
D. intersection
14. Write the following statement as a true biconditional
if possible. Complementary angles are two angles
with measures that have a sum of 90.
F. Two angles are complementary if and only if the
measures of the angles have a sum of 90.
G. Two angles with measures that have a sum of 90
are complementary angles.
H. Two angles with measures that do not have a sum
of 90 are not complementary angles.
J. not reversible
15. What is the value of x?
G. Symmetric Property
A. 120
C. 30
H. Transitive Property
B. 60
D. 20
(x + 90)° 4x°
J. Substitution Property
11. Which of the following is not a postulate?
A. Through any three noncollinear points there is
exactly one plane.
B. If two distinct lines intersect, then they intersect in
exactly one point.
C. Vertical angles are congruent.
D. Through any two points there is exactly one line.
12. When constructing a perpendicular bisector, what is the
final step of the construction, after you’ve drawn two
arcs, as shown here?
X
A
B
Y
F. Draw a line through points A and X.
G. Draw a line through points A and B.
H. Draw a line through points X and B.
J. Draw a line through points X and Y.
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Topic 2 TEKS Cumulative Practice
16. Which is the biconditional of the given statement?
A hexagon is a six-sided polygon.
F. A figure is a hexagon if and only if it is a six-sided
polygon.
G. A figure is a polygon if and only if it is a six-sided
hexagon.
H. A six-sided polygon is a hexagon.
J. A hexagon is a polygon if and only if it is a six-sided
figure.
17. Given the statements below, what conclusion can be
made using the Law of Detachment?
If a student wants to go to college, then the student
must study hard.
Rashid wants to go to Rice University.
A. Rashid will go to Rice University.
B. Rashid will not have to study hard.
C. College students study hard.
D. Rashid must study hard.
18. What are two pairs of congruent angles in this figure?
11111 * 11111 = 1
11111 * 11111 = 121
11111 * 11111 = 12321
11111 * 11111 = 1234321
11111 * 11111 = ■
F
E
I
G
H
F. ∠EIF ≅ ∠FIG and ∠FIG ≅ ∠GIH
G. ∠EIF ≅ ∠GIH and ∠EIG ≅ ∠FIH
H. ∠HIG ≅ ∠EIF and ∠FIG ≅ ∠GIH
J. ∠HIG ≅ ∠EIF and ∠FIH ≅ ∠GIH
Gridded Response
19. What is the next number in the pattern?
Constructed Response
24. On a number line, P is at -4 and R is at 8. What is
the coordinate of the point Q, which is 34 the way from
R to P? Show your work.
25. Write the converse, inverse, and contrapositive
of the following statement. Determine the truth
value of each.
If you live in Oregon, then you live in the United
States.
1, -4, 9, -16, c
20. m∠BZD = 107
m∠FZE = 2x + 5
m∠CZD = x
What is the measure of ∠CZD ?
26. Determine the truth value of the conjecture below. If
false, provide a counterexample.
A perpendicular bisector of a line segment forms
three pairs of vertical angles.
27. The sequence below lists the first eight powers of 7.
B
71, 72, 73, 74, 75, 76, 77, 78, . . .
C
A
D
Z
F
E
a. Make a table that lists the digit in the ones place
for each of the first eight powers of 7. For example,
74 = 2401. The digit 1 is in the ones place.
b. What digit is in the ones place of 734 ? Explain
your reasoning.
21. The measure of an angle is three more than twice its
supplement. What is the measure of the angle?
28. Write a proof.
Given: ∠1 and ∠2 are
supplementary.
22. What is the value of x?
X 3x 1 5C8
23. Continue the pattern.
(2x 2 70)8
Prove: ∠1 and ∠2 are
right angles.
1
2
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