Chapter 2 Review
Formal Geometry
Name: _________________________
____
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1.
2.
3.
Determine if the conjecture is true or false. If false, give a counterexample. If
n is a composite number, then n+1 is also a composite number.
A.
True.
B.
False; 8 is a composite number and 8+1 is not a composite number.
C.
False; 2 is a composite number and 2+1 is not a composite number.
D.
False; 4 is a composite number and 4+1 is not a composite number.
Which of the following are logically equivalent?
A.
A statement and its converse
B.
A statement and its inverse
C.
A statement and its contrapositive
D.
A statement, its converse, its inverse and its contrapositive
A four legged animal is called a dog.
Which of the following best describes a counterexample to the assertion
above?
4.
A.
German Shepherd
B.
Bull Dog
C.
A Shark
D.
A Cat
Determine which statement follows logically from the given statements.
If your book falls, then you must pick it up. Fallen books are most likely
damaged.
A.
If I drop my book, then I am a klutz.
B.
If I damage my book, it is because I left it outside.
C.
Sometimes people drop their textbooks.
D.
If I drop my book, then it will most likely be damaged.
Chapter 2 Review
5.
Formal Geometry
Name: _________________________
____
Determine whether the conjecture is true or false. Give a counterexample for
any false conjecture.
Given: Two angles are congruent.
Conjecture: They are both right angles.
6.
A.
False; two acute angles can have the same degree measure.
B.
True
C.
False; two adjacent angles can be congruent.
D.
False; they must be vertical angles.
Write the statement in if-then form.
A counterexample invalidates a statement.
7.
A.
If there is a counterexample, then it invalidates the statement.
B.
If there is a counterexample, then it is true.
C.
If it is true, then there is a counterexample.
D.
If it invalidates the statement, then there is a counterexample.
Determine whether a valid conclusion can be reached. If so, state the
conclusion. Explain your reasoning.
Given: If you practice basketball every day, then you will make it to the NBA!
If you make it to the NBA, then you will make a lot of money.
A.
B.
C.
D.
Valid; conclusion: If you practice basketball every day, then you can
buy a big house because you will have a lot of money. Law of Syllogism.
Invalid. I am not given the hypothesis.
Valid; conclusion: If you practice basketball every day, then you will
make a lot of money. Law of Syllogism.
Valid; conclusion: If you practice basketball every day, then you will
make a lot of money. Law of Detachment.
Chapter 2 Review
8.
Formal Geometry
Name: _________________________
____
Determine whether a valid conclusion can be reached. If so, state the
conclusion. Explain your reasoning.
Given: Old Navy is selling sweaters on Monday at 15% off. Vanessa buys a
sweater for 15% off.
A.
Valid; conclusion: Vanessa bought a sweater from Old Navy on
Monday. Law of Detachment.
9.
B.
Invalid; Vanessa bought the sweater on Tuesday.
C.
Valid; conclusion: Vanessa bought a sweater from Old Navy on
Monday. Law of Detachment.
D.
Invalid; I am not given the hypothesis.
Find the value of 𝑥 and 𝐴𝐶 if 𝐵 is between points 𝐴 and 𝐶.
𝐴𝐵 = 𝑥 − 1, 𝐵𝐶 = 5𝑥, 𝐴𝐶 = 4𝑥 + 5
A.
𝑥 = −1; 𝐴𝐶 = 1
B.
𝑥 = 3; 𝐴𝐶 = 17
C.
𝑥 = 2; 𝐴𝐶 = 13
D.
𝑥 = 9; 𝐴𝐶 = 41
⃗⃗⃗⃗⃗⃗ are
Use the figure below to answer questions 10-11. In the figure, ⃗⃗⃗⃗⃗⃗
𝑿𝑨 𝒂𝒏𝒅 𝑿𝑬
opposite rays, and ∠𝑨𝑿𝑪 is bisected by ⃗⃗⃗⃗⃗⃗
𝑿𝑩.
10.
11.
If 𝑚∠𝐴𝑋𝐶 = 50𝑥 + 16 𝑎𝑛𝑑 𝑚∠𝐴𝑋𝐵 = 30𝑥 − 2, find 𝑚∠𝐴𝑋𝐶.
A.
𝑚∠𝐴𝑋𝐶 = 80𝑥°
B.
𝑚∠𝐴𝑋𝐶 = 2°
C.
𝑚∠𝐴𝑋𝐶 = 116°
D.
𝑚∠𝐴𝑋𝐶 = 118°
If 𝑚∠𝐷𝑋𝐶 = 41, 𝑚∠𝐷𝑋𝐸 = 6𝑥 − 1, and 𝑚∠𝐶𝑋𝐸 = 15𝑥 + 4°, find 𝑚∠𝐷𝑋𝐸.
A.
𝑚∠𝐷𝑋𝐸 = 64°
B.
𝑚∠𝐷𝑋𝐸 = 4°
C.
𝑚∠𝐷𝑋𝐸 = 20°
D.
𝑚∠𝐷𝑋𝐸 = 23°
Chapter 2 Review
12-14.
Given:
Prove:
Formal Geometry
−2(𝑥 − 5) = 2𝑥 + 7
3
𝑥=4
Statements
−2(𝑥 − 5) = 2𝑥 + 7
−2𝑥 + 10 = 2𝑥 + 7
13.
3 = 4𝑥
3
=𝑥
4
3
𝑥=
4
12.
13.
14.
Name: _________________________
____
Reasons
Given
12.
Addition Property of equality
Subtraction Property of equality
14.
Symmetric Property of equality
Choose one of the following to complete the proof.
A.
Distributive Property
B.
Multiplication Property of Equality
C.
Addition Property of Equality
D.
Substitution Property of Equality
Choose one of the following to complete the proof.
A.
−2𝑥 = 2𝑥 − 3
B.
−4𝑥 = 17
C.
10 = 4𝑥 + 7
D.
𝑥 − 10 = 7
Choose one of the following to complete the proof.
A.
Distributive Property
B.
Multiplication Property of Equality
C.
Addition Property of Equality
D.
Division Property of Equality
Chapter 2 Review
15-16.
Given:
Prove:
16.
Reasons
Given
15.
If K is the midpoint of ̅̅̅̅
𝐻𝐿, then ̅̅̅̅
𝐻𝐾 ≅
̅̅̅̅. Midpoint Theorem.
𝐾𝐿
Transitive property of congruence.
̅̅̅̅
𝐺𝐻 ≅ ̅̅̅̅
𝐾𝐿
16.
Name: _________________________
____
𝐻 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 ̅̅̅̅
𝐺𝐾
̅̅̅̅
𝐾 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐻𝐿
̅̅̅̅ ≅ 𝐾𝐿
̅̅̅̅
𝐺𝐻
Statements
𝐻 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 ̅̅̅̅
𝐺𝐾
̅̅̅̅
𝐾 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐻𝐿
̅̅̅̅ ≅ 𝐻𝐾
̅̅̅̅
𝐺𝐻
15.
Formal Geometry
Choose one of the following to complete the proof.
A.
If a point passes through the midpoint of a segment, then the segment
is bisected at that point. Definition of segment bisector.
B.
If H is the midpoint of ̅̅̅̅
𝐺𝐾 , then ̅̅̅̅
𝐺𝐻 ≅ ̅̅̅̅
𝐻𝐾 . Midpoint Theorem.
C.
̅̅̅̅ ≅ 𝐻𝐾
̅̅̅̅. Definition of Congruence.
If 𝐺𝐻 = 𝐻𝐾, then 𝐺𝐻
D.
̅̅̅̅ , then H bisects the segment. Definition of
If H is the midpoint of 𝐺𝐾
segment bisector.
Choose one of the following to complete the proof.
A.
B.
C.
D.
̅̅̅̅
𝐻𝐾 ≅ ̅̅̅̅
𝐾𝐿
̅̅̅̅
̅̅̅̅
𝐺𝐾 ≅ 𝐻𝐿
̅̅̅̅ ≅ 𝐻𝐾
̅̅̅̅
𝐻𝐾
̅̅̅̅
̅̅̅̅
𝐺𝐻 ≅ 𝐾𝐿
Chapter 2 Review
17-18.
Given:
Prove:
18.
Name: _________________________
____
⃗⃗⃗⃗⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐶𝐹𝐴
𝐹𝐵
𝑚∠1 = 𝑚∠3
Statements
⃗⃗⃗⃗⃗
𝐹𝐵 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐶𝐹𝐴
𝑚∠1 = 𝑚∠2
𝑚∠2 = 𝑚∠3
𝑚∠1 = 𝑚∠3
17.
Formal Geometry
Reasons
Given
11.
12.
Substitution property of equality
Choose one of the following to complete the proof.
A.
If part of a line has one endpoint and extends infinitely in one
direction, then it is a ray. Definition of ray.
B.
If a line passes through the midpoint of a segment, then it intersects
that segment. Definition of segment bisector.
C.
If a ray divides an angle into two congruent angles, then the ray is an
angle bisector. Definition of angle bisector.
D.
If a point is the endpoint of two collinear rays, then the rays are
opposite rays. Definition of opposite rays.
Choose one of the following to complete the proof.
A.
If two angles are supplementary to a third angle, then the two angles
are congruent. Supplemental Angle Theorem.
B.
If two angles have a sum of 90 degrees, then the angles are
complementary. Definition of complementary angles.
C.
If two angles are vertical angles, then they have equal measures.
Vertical Angle Theorem.
D.
If two angles have a sum of 180 degrees, then the angles are
supplementary. Definition of supplementary angles.
Chapter 2 Review
Formal Geometry
Name: _________________________
____
Free Response. Partial credit will be awarded. Please write your proof on the back.
19. Given:
Prove:
𝐴𝐶 = 𝐵𝐷
𝐴𝐵 = 𝐶𝐷
20. Given:
Prove:
⃗⃗⃗⃗⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑆𝐾𝐼
𝐾𝑋
𝑃𝑟𝑜𝑣𝑒 ∠𝑌𝐾𝑋 ≅ ∠𝐸𝐾𝑋
Extra Practice. Given: ∠1 ≅ ∠4, ∠𝐴𝐹𝐶 ≅ ∠𝐸𝐹𝐶
Prove:
∠2 ≅ ∠3