1.9.4
Test (TST): Foundations of Geometry
Geometry Sem 1
Name: Vatshal Gaur
Date: 9/13/23
Test
Answer the following questions using what you've learned from this unit. Write your
answers in the space provided. Be sure to show all work.
1. Analyze the following pattern: 1, 2, 5, 10, 17, ...
Part I: Describe the pattern. (1 point)
Starting from 1, add 1 more each time but increase the addition number by 2
1+1=2
2+3=5
5 + 5 = 10
10 + 7 = 17
Part II: What is the 8th term of the pattern? Show your work. (1 point)
1 + 1 = 2 2 + 3 = 5 5 + 5 = 10
37 + 13 = 50 50 + 15 = 65
10 + 7 = 17
17 + 9 = 26
26 + 11 = 37
65
2. Using deductive reasoning, write the converse, inverse, and contrapositive of an
if-then statement.
Part I: Write an if-then statement below. (1 point)
If I like cards then I will buy cards
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Part II: Use the if-then statement you wrote in Part I to find each of the
following. (3 points)
A. Converse
If I don't like cards, then I won't buy cards.
B. Inverse
If I buy cards then I like cards
C. Contrapositive
If I don't buy cards then I don't like cards
3. Write a conditional statement that correctly describes each Venn diagram in the
space provided.
Part I: (1 point)
If m<A = 40 degrees then it is an Acute angle.
Part II: (1 point)
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If 2x = 4 then x = 2
4. Explain the main parts and uses of two kinds of proofs.
Part I: Draw a line to match each term with its meaning.
(5 points)
Term
Meaning
A. Definition
A statement that is assumed to
be true without proof
B. Postulate (axiom)
A statement that has already
been proven to be true
C. Common notion
A statement that tells exactly
what something is or means
D. Theorem
A statement that makes sense
based on another statement that
has already been proven
E. Corollary
A statement that is not officially
defined but that is understood to
be common sense
Part II: Find a Pattern.
Use the terms from Part I to answer each question.
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What is a proof? (1 point)
Evidence for a claim.
You want to prove a theorem in a two-column proof. You start with your given
statement and list deductions in the left-hand column. What are the three main
types of reasoning you will use for reasons in the right-hand column? (3
points; 1 point each)
1. Postulates
2. Corollaries
3. Common Notion
In an indirect proof, you prove an "if-then" statement is true by assuming the
statement is false (stating the inverse or converse), and then disproving the
false statement. You want to prove the statement shown below in an indirect
proof. What statement should you prove is false? (1 point)
Statement to Prove True: If a figure has exactly three sides, then it is a triangle.
Statement to Prove False: If a figure does not have exactly three sides, then it is a triangle.
5. Find AB, BC, and AC.
Part I: The length of line segment AC is 8x - 9. Use the figure below to find the
value of x. Show your work. (1 point)
8x - 9 = 3x + 1 + 4x - 5
8x -9 = 7x - 4
8x -7x = 9 - 4
x=5
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Part II: Use the solution you found for x in Part I to find AB, BC, and AC. Show
your work. (3 points)
AB: 3(5) + 1 = 16
15 + 1 = 16
BC: 4(5) - 5 = 15
AC: 8(5) - 9 = 31
20 - 5 = 15
40 - 9 = 31
6. Use the figure below to identify angles and segment lengths.
Part I: Find the measure of
and the measure of
explanation for each angle in your answer. (1 point)
. Give an
<FED is a vertical angle to <GEN so it is also 60 degrees.
<DEN is supplementary to <GEN so it is 120 degrees.
180 - 60 = 120.
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Part II: Suppose the length of AD is 10 and the length of DG is twice as long as
AD. Find the length of AG. Show your work. (1 point)
AD = 10
DG = 10 * 2
AG = AD + DG,
AG = 30
10 + 10*2
Part III: Name a pair of vertical angles, a straight angle, and two acute angles
other than
. (1 point)
Vertical Angles: <JNM and <ENK
Straight Angle: <CDH
Acute Angles: <CDA and <DKP
Part IV: Describe the term linear pair, and give an example from the diagram
above. (1 point)
Have a common vertex and are adjacent angles on the same line.
<JNM and <JNE
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7. A pair of lines may be parallel, perpendicular, or skew.
Part I: Describe parallel lines and include a sketch that shows parallel lines.
Give a real-world example of parallel lines. Use appropriate labeling and
symbols. (2 points)
Lines that will never intersect.
A bookshelf with two shelves on different levels from each other.
Part II: Describe perpendicular lines and include a sketch that shows
perpendicular lines. Give a real-world example of perpendicular lines. Use
appropriate labeling and symbols. (2 points)
Two lines which intersect at exactly one
point, forming 90-degree angles.
Windows have grids that make
perpendicular lines.
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Part III: Describe skew lines and include a sketch that shows skew lines. Give
a real-world example of skew lines. Use appropriate labeling and symbols.
(2 points)
Lines that are not on the same plane and will never intersect.
A bridge over water would be an example as they will never
intersect since they are on different planes.
8.
Part I:
||
and
||
. Use the figure above to find the value of each
angle. Give an explanation for each angle value. (7 points)
p = 117
r = 63
w = 117
x = 117
k = 117
m = 117
n = 63
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9. Look at the sequence below and form a conjecture (educated guess) about it.
Part I: This form of reasoning, where you form general ideas and rules based on
inductive
your experiences and observations, is called _______________.
(1 point)
Part II: Describe the pattern of the following sequence. (2 points)
It alternates between circle and squares when it adds one column and one row.
Part III: Draw the next item in the series. (1 point)
10. You just hung a picture twelve inches above the wall trim. Your friend thinks the
picture looks crooked. Use what you know about parallel lines and transversals to
determine if the picture is level.
Step 1: You don't have a level, but you are in luck. You know the wall trim is level.
You have a protractor and the sun is casting a shadow on the wall. Describe how
you can determine if the picture is level. (3 points)
You would measure the angle between the shadow and wall trim, then measure the angle
between the shadow and the bottom of the picture. If they are equal, then the bottom of
the picture is parallel to the wall trim making the picture level.
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Step 2: Assume you measured ∠3 and ∠5 and found them to be equal. Use the
spaces provided below to complete a formal proof that demonstrates the picture
is level with the trim. (4 points)
m<3 = m<5
alternate interior angles
<5 ≅ <7
f||t
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