Question 1

Question

A)A 10-sided equilateral polygon is drawn. Investigate and identify a pattern to find the angle measure of one interior angle.(1 point)
Responses

36°
36°

144°
144°

180°
180°

72°
72°
Question 2
A)Noah is designing a polygon with six sides that has the same interior angle measures. At one of the vertices, he extends the line to form an exterior angle. What is the measure of the exterior angle at that vertex?(1 point)
Responses

45°
45°

90°
90°

120°
120°

60°
60°
Question 3
A)
Use the image to answer the question.

An illustration shows a triangle with angles marked as 1, 2 and 3, clockwise beginning at the top angle. A line is drawn outside of the triangle, passing through the vertex with angle 3.

Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof?

Sylvie's Proof: Given the diagram shown, ∠1≅∠5
; ∠2≅∠4
because alternate interior angles are congruent when lines are parallel. Then, I know that m∠1=m∠5
; m∠2=m∠4
because congruent angles have equal measures.

(1 point)
Responses

m∠5+m∠3+m∠4=180°
by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180°
by the Triangle Angle Sum Theorem.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the Triangle Angle Sum Theorem.

m∠1+m∠3+m∠2=180°
by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180°
by substitution.
m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the definition of a straight angle. Finally, m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by substitution.

m∠2+m∠3+m∠4=180°
by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180°
by substitution.
m angle 2 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.

m∠5+m∠3+m∠4=180°
by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180°
by substitution.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.
Question 4
A)
Use the image to answer the question.

An isosceles triangle is marked clockwise from the lower left vertex as upper A upper B upper C. The sides upper A upper B and upper B upper C are marked with single congruent tick marks.

Consider the following proof of the Base Angles Theorem. Which statement should fill in the blank?

PROOF: Given isosceles △ABC
with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
, I can construct BD←→
, the angle bisector of ∠B
. _____________________. I also know that line segments are congruent to themselves, so BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯
by the reflexive property of congruence. I now have two pairs of sides and an included angle that are congruent, so I know that △ABD≅△CBD
by the SAS Congruence Theorem. Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C
.

(1 point)
Responses

Then, by the definition of an angle bisector, I know that ∠BAC≅∠BCA
.
Then, by the definition of an angle bisector, I know that angle upper B upper A upper C congruent to angle upper B upper C upper A .

 Then, by the definition of a midpoint, I know that AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
.
 Then, by the definition of a midpoint, I know that Modifying above upper A upper D with bar congruent to Modifying above upper D upper C with bar .

Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD
.
Then, by the definition of an angle bisector, I know that angle upper A upper B upper D congruent to angle upper C upper B upper D .

Then, by the definition of an isosceles triangle, I know that AB¯¯¯¯¯¯¯¯≅CA¯¯¯¯¯¯¯¯
.
Then, by the definition of an isosceles triangle, I know that Modifying above upper A upper B with bar congruent to Modifying above upper C upper A with bar .
Question 5
A)
Use the image to answer the question.

An isosceles triangle with a sideways orientation is marked counter-clockwise from the right as upper A upper B upper C. Sides upper A upper B and upper A upper C are marked with a single congruent tick mark. A line bisects the triangle.

Fox is working to prove the Base Angles Theorem. His proof is shown below. Critique his reasoning. Which statement or reason in his proof has a mistake? How can he fix his mistake?

Given: Isosceles △ABC
with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯

Prove: ∠B≅∠C

Statements Reasons
1. Isosceles △ABC
with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯
1. Given
2. D
, the midpoint of BC¯¯¯¯¯¯¯¯

AD¯¯¯¯¯¯¯¯
through points A
and D

2. Construction
3. BD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
3. Definition of midpoint
4. AD¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯
4. Reflexive property of congruence
5. △ABD≅△ACD
5. SAS Congruence Theorem
6. ∠B≅∠C
6. CPCTC Theorem
(1 point)
Responses

Statement 6 has a mistake. It should say “∠A≅∠C
."
Statement 6 has a mistake. It should say “ angle upper A congruent to angle upper C ."

Reason 5 has a mistake. It should say “SSS Congruence Theorem.”
Reason 5 has a mistake. It should say “SSS Congruence Theorem.”

Reason 3 has a mistake. It should say “Definition of angle bisector.”
Reason 3 has a mistake. It should say “Definition of angle bisector.”

Statement 3 has a mistake. It should say “∠BAD≅∠CAD
."
pick the correct and only the correct answer)

Answer 1

Here are the correct answers to each question:

Question 1: The measure of one interior angle of a regular 10-sided polygon (decagon) is calculated as follows:
\[ \text{Interior angle} = \frac{(n-2) \times 180°}{n} = \frac{(10-2) \times 180°}{10} = \frac{8 \times 180°}{10} = 144°. \]
Response: 144°

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Question 2: For a regular hexagon, each exterior angle can be calculated as follows:
\[ \text{Exterior angle} = \frac{360°}{n} = \frac{360°}{6} = 60°. \]
Response: 60°

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Question 3: To complete Sylvie's Proof:
The correct answer is:
\[ m∠5+m∠3+m∠4=180°
\text{ by the definition of a straight angle. Finally, } m∠1+m∠3+m∠2=180°
\text{ by the Triangle Angle Sum Theorem.} \]
Response: m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by the Triangle Angle Sum Theorem.

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Question 4: To fill in the blank for the proof of the Base Angles Theorem:
The statement that should fill in the blank is:
\[ Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD. \]
Response: Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD.

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Question 5: The mistake in Fox's proof is in Statement 6. It should state:
Response: Statement 6 has a mistake. It should say “∠B≅∠C."

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These are the correct answers for all the questions provided.