Question 1

Question

Question 1
A)Derive the slope of a line with the coordinates (−5,−2) and (4,−8).(1 point)
Responses

−16
Start Fraction negative 1 over 6 End Fraction

23
Start Fraction 2 over 3 End Fraction

−32
Start Fraction negative 3 over 2 End Fraction

−23
Start Fraction negative 2 over 3 End Fraction
Question 2
A)
Use the image to answer the question.

A line is plotted on a coordinate plane with the x-axis ranging from negative 8 to 8 in increments of 1 and the y-axis ranging from negative 8 to 8 in increments of 1.

Which of the following equations is parallel to the graphed line?

(1 point)
Responses

y=−23x+2
y equals negative Start Fraction 2 over 3 End Fraction x plus 2

y=23x+3
y equals Start Fraction 2 over 3 End Fraction x plus 3

y=32x−7
y equals Start Fraction 3 over 2 End Fraction x minus 7

y=−32x−1
y equals negative Start Fraction 3 over 2 End Fraction x minus 1
Question 3
A)
Use the table to answer the question.

Line Point 1 Point 2
AB (−3,6) (3,8)
CD (3,5) (__,−1)
The table includes two points that fall on each of the lines, line AB and line CD. What must the value of the missing coordinate be in order to prove the lines are perpendicular?

(1 point)
Responses

5
5

−1
negative 1

1
1

−15
negative 15
Question 4
A)
Use the image to answer the question.

Two parallel, inclined lines labeled as m and n are separated by a distance. A diagonal line labeled as t intersects m and n at a bottom left to top right incline.

In the diagram, m∥n. Investigate the patterns of angles created when the parallel lines are cut by a transversal. What is m∠5 if m∠3=32°?

(1 point)
Responses

58°
58°

32°
32°

148°
148°

74°
74°
Question 5
A)
Use the image to answer the question.

The intersection of Atlantic Street and Ocean Avenue is labeled from top left as a, top right as 130 degrees, bottom right as c, and bottom left as b. The intersection of Pacific Street and Ocean Avenue is labeled from top left as d, top right as e, bottom right as g, and bottom left as f. The streets and avenue are made up of solid lines containing dashed lines at their centers.

In the diagram, RS−→∥PQ−→−. ∠PVU=4x−7 and ∠TWS=2x+27. Investigate the patterns of angles created when parallel lines are cut by a transversal to identify the value of x.

(1 point)
Responses

34
34

17
17

61
61

118
118
Question 6
A)
Use the image to answer the question.

A horizontal line is intersected by two parallel lines m and n forming eight angles.

Trishna wrote a paragraph proof to prove that alternate exterior angles, ∠A
and ∠H
, are congruent.

I am given that m∥n
. I know that m∠A=m∠E
because they are corresponding angles. I also know that m∠E=m∠H
because they are vertical angles. With this information, I can use the Transitive Property of Equality to write that m∠A=m∠H
. By the definition of congruence, ∠A≅∠H
and I have proven that alternate exterior angles are congruent.

What mistake did Trishna make?

(1 point)
Responses

Trishna did not make a mistake. Her proof is correct.
Trishna did not make a mistake. Her proof is correct.

The reason that m∠E=m∠H
is because they are a linear pair, not vertical angles.
The reason that m∠E=m∠H
is because they are a linear pair, not vertical angles.

The reason that m∠A=m∠E
is because they are vertical angles, not corresponding angles.
The reason that m∠A=m∠E
is because they are vertical angles, not corresponding angles.

The reason that m∠A=m∠H
is because they are corresponding angles, not because of the Transitive Property of Equality.
The reason that m∠A=m∠H
is because they are corresponding angles, not because of the Transitive Property of Equality.
Question 7
A)
Use the image to answer the question.

A perpendicular bisector TP intersecting line segment OR at point Q. Point S is a point on the perpendicular bisector.

TP¯¯¯¯¯¯¯
is a perpendicular bisector of OR¯¯¯¯¯¯¯¯
and point S lies on the perpendicular bisector. The proof below proves that S is equidistant from both endpoints of OR¯¯¯¯¯¯¯¯
.

Statement Reason
1. TP¯¯¯¯¯¯¯
is a perpendicular bisector of OR¯¯¯¯¯¯¯¯
. given
2. OQ¯¯¯¯¯¯¯¯≅RQ¯¯¯¯¯¯¯¯
definition of a perpendicular bisector
3. ∠OQS
and ∠RQS
are right angles. definition of a perpendicular bisector
4. Right Angle Congruence Theorem
5. SQ¯¯¯¯¯¯¯≅SQ¯¯¯¯¯¯¯
Reflexive Property of Congruence
6. △OQS≅△RQS
SAS Congruence Theorem
7. SO¯¯¯¯¯¯¯≅SR¯¯¯¯¯¯¯
CPCTC Theorem

What is the missing statement in the proof?

(1 point)
Responses

∠OST≅∠RST
∠OST≅∠RST

∠OQS≅∠RQS
∠OQS≅∠RQS

∠QSO≅∠QSR
∠QSO≅∠QSR

∠QOS≅∠QRS
∠QOS≅∠QRS
Question 8
A)Kiana is drafting a polygon with six sides that does not have similar side lengths or angles. If four of the angles have a sum of 605°, what is the sum of the other two angles?(1 point)
Responses

295°
295°

115°
115°

475°
475°

120°
120°
Question 9
A)
Use the image to answer the question.

An illustration shows a triangle with vertices labeled clockwise as upper A, upper B, and upper C, starting at the lower left vertex. A line is drawn passing through upper B, and another line extends beyond upper B.

Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake? How should the mistake be fixed?

Given: △ABC with DB←→ parallel to AC¯¯¯¯¯¯¯¯ and CB¯¯¯¯¯¯¯¯ extended to make CE−→−

Prove: m∠A+m∠ABC+m∠C=180°

Statements Reasons
1. △ABC with DB←→ parallel to AC¯¯¯¯¯¯¯¯ and CB¯¯¯¯¯¯¯¯ extended to make CE−→− 1. Given
2. ∠A≅∠ABD 2. DB←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel
3. ∠C≅∠DBC 3. DB←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel
4. m∠A=m∠ABD; m∠C=m∠DBE 4. Definition of congruent angles
5. m∠ABD+m∠DBE+m∠ABC=180° 5. Definition of a straight angle
6. m∠A+m∠C+m∠ABC=180° 6. Substitution
7. m∠A+m∠ABC+m∠C=180° 7. Commutative property of addition
(1 point)
Responses

Statement 3 has a mistake. It should say ∠DBE≅∠A.
Statement 3 has a mistake. It should say angle upper D upper B upper E congruent to angle upper A .

Statement 3 has a mistake. It should say ∠C≅∠DBE.
Statement 3 has a mistake. It should say angle upper C congruent to angle upper D upper B upper E .

Statement 5 has a mistake. It should say m∠ABD+m∠DBE+m∠BCA=180°.
Statement 5 has a mistake. It should say m angle upper A upper B upper D plus m angle upper D upper B upper E plus m angle upper B upper C upper A equals 180 degrees .

Statement 4 has a mistake. It should say ∠A≅∠DBE.
Statement 4 has a mistake. It should say angle upper A congruent to angle upper D upper B upper E .
Question 10
A)
Use the image to answer the question.

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

Sylvia plans to use the marked triangle to help prove that the base angles of an isosceles triangle are congruent. What basic strategy should she use in her proof?

(1 point)
Responses

Sylvia should prove that ∠C≅∠B by the Base Angles Theorem.
Sylvia should prove that angle upper C congruent to angle upper B by the Base Angles Theorem.

Sylvia should prove that AD¯¯¯¯¯¯¯¯≅AD¯¯¯¯¯¯¯¯ by the reflexive property of congruence and therefore ∠C≅∠B by the CPCTC Theorem.
Sylvia should prove that Modifying above upper A upper D with bar congruent to Modifying above upper A upper D with bar by the reflexive property of congruence and therefore angle upper C congruent to angle upper B by the CPCTC Theorem.

Sylvia should prove that △ACD≅△ABD by the SSS Congruence Theorem and therefore ∠C≅∠B by the CPCTC Theorem.
Sylvia should prove that triangle upper A upper C upper D congruent to triangle upper A upper B upper D by the SSS Congruence Theorem and therefore angle upper C congruent to angle upper B by the CPCTC Theorem.

Sylvia should prove that △ACD≅△ABD by the SAS Congruence Theorem and therefore ∠C≅∠B by the CPCTC Theorem.

correct answers only)

GPT-4o mini · Answer

Here are the correct answers to your questions:

Question 1: The slope of a line is calculated using the formula \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}. \] Using the coordinates (−5,−2) and (4,−8):

\[ \text{slope} = \frac{-8 - (-2)}{4 - (-5)} = \frac{-8 + 2}{4 + 5} = \frac{-6}{9} = \frac{-2}{3}. \]

Answer: -2/3 (or −23)

Question 2: Since parallel lines have the same slope, we need an equation with a slope of -2/3.

Answer: y = -2/3x + 2

Question 3: To prove the lines are perpendicular, the slopes must satisfy the relationship \( m_1 \times m_2 = -1 \).

To find the slope of line AB: \[ \text{slope}{AB} = \frac{8 - 6}{3 - (-3)} = \frac{2}{6} = \frac{1}{3}. \] For line CD, let the missing y-coordinate be \( y \): \[ \text{slope}{CD} = \frac{y - 5}{-3 - 3} = \frac{y - 5}{-6}. \]

Setting the product of the slopes to -1 for perpendicularity: \[ \frac{1}{3} \times \frac{y - 5}{-6} = -1. \] This simplifies to: \[ \frac{y - 5}{-18} = -1, \] leading to \( y - 5 = 18 \), or: \[ y = 23. \]

Answer: 5

Question 4: If m∠3 = 32°, then m∠5 (which is corresponding to m∠3) is also:

Answer: 32°

Question 5: Since RS || PQ, and the angles are related by the transversal, we can set up the equation: \[ 4x - 7 = 2x + 27. \] Solving for \(x\): \[ 4x - 2x = 27 + 7 \implies 2x = 34 \implies x = 17. \]

Answer: 17

Question 6: Trishna’s proof incorrectly states that m∠E=m∠H due to vertical angles; however, they are a linear pair.

Answer: The reason that m∠E=m∠H is because they are a linear pair, not vertical angles.

Question 7: The missing statement should establish that the two angles at O and R, formed by the intersecting line, are congruent due to their definition.

Answer: ∠OQS≅∠RQS

Question 8: The sum of the angles in a hexagon is \( (6-2) \times 180 = 720°\). Given four angles sum to 605°:

\[ 720° - 605° = 115°. \]

Answer: 115°

Question 9: The mistake in statement 5 is about the angles referenced:

Answer: Statement 5 has a mistake. It should say m∠ABD + m∠DBE + m∠ABC = 180°.

Question 10: Sylvia should show that the triangles formed by the line bisecting the angle are congruent through SSS.

Answer: Sylvia should prove that △ACD≅△ABD by the SSS Congruence Theorem and therefore ∠C≅∠B by the CPCTC Theorem.

These are the accurate answers for the questions provided.