Use the image to answer the question.

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, given PQ−→−∥RS−→, m∠PVT=109°. Investigate the patterns of angles created when parallel lines are cut by a transversal to identify m∠RWU.

(1 point)
Responses

54.5°
54.5°

109°
109°

71°
71°

35.5°
35.5°
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Question
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, given m∥n , ∠8 and ∠1 are alternate exterior angles. If m∠1=9x−12 and m∠8=105° , what is the value of x?

(1 point)
Responses

52.5°
52.5°

75
75

9.7
9.7

13
13
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Question
Use the image to answer the question.

A horizontal line is intersected by two parallel lines lower s and lower t that slant from left to right forming eight angles.

If lines s and t are parallel, which pair of angles is supplementary and why?

(1 point)
Responses

∠A
and ∠D
because they are consecutive exterior angles
∠A
and ∠D
because they are consecutive exterior angles

∠A
and ∠B
because they are corresponding angles
∠A
and ∠B
because they are corresponding angles

∠B
and ∠G
because they are consecutive interior angles
∠B
and ∠G
because they are consecutive interior angles

∠G
and ∠D
because they are a linear pair
∠G
and ∠D
because they are a linear pair
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Question
Line segment AC¯¯¯¯¯¯¯¯ is a perpendicular bisector of segment BD¯¯¯¯¯¯¯¯ , with the two segments meeting at point E . What is true of segment BE¯¯¯¯¯¯¯¯ ? (1 point)
Responses

It must be the same length as segment EA¯¯¯¯¯¯¯¯.
It must be the same length as segment upper C upper D .

It must be the same length as segment DE¯¯¯¯¯¯¯¯.
It must be the same length as segment upper D upper E .

It must be the same length as segment AB¯¯¯¯¯¯¯¯.
It must be the same length as segment upper A upper B .

It must be the same length as segment BC¯¯¯¯¯¯¯¯.
It must be the same length as segment upper B upper C .
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Question
Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof?

HK¯¯¯¯¯¯¯¯¯
is a perpendicular bisector of IJ¯¯¯¯¯¯
, and L is the midpoint of IJ¯¯¯¯¯¯
. M is a point on the perpendicular bisector, HK¯¯¯¯¯¯¯¯¯
. By the definition of a perpendicular bisector, I know that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯
. By the definition of a perpendicular bisector, I also know that ∠MLI
and ∠MLJ
are right angles. ∠MLI≅∠MLJ because of the Right Angle Congruence Theorem. I can also say that ML¯¯¯¯¯¯¯¯¯≅ML¯¯¯¯¯¯¯¯¯
by the Reflexive Property of Congruence. With this information, I know that △MLI≅△MLJ
by the SAS Congruence Theorem. Since the triangles are congruent, the CPCTC Theorem allows me to know that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯
. Knowing that these segments are congruent proves the Perpendicular Bisector Theorem.

(1 point)
Responses

The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯
, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯
.
The definition of a perpendicular bisector tells you that line segment cap i liters is congruent to line segment cap j liters, not that line segment cap i cap m is congruent to line segment cap j cap m.

The SAS Congruence Theorem tells you that △MLI≅△JLM
, not that △MLI≅△MLJ
.
The SAS Congruence Theorem tells you that △MLI≅△JLM
, not that △MLI≅△MLJ
.

The definition of a perpendicular bisector tells you that∠LMI
and ∠LMJ
are right angles, not that ∠MLI
and ∠MLJ
are right angles.
The definition of a perpendicular bisector tells you that ∠LMI
and ∠LMJ
are right angles, not that ∠MLI
and ∠MLJ
are right angles.

Lenny did not make a mistake. His proof is correct.
Lenny did not make a mistake. His proof is correct.
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Answer 1

Sure! Here are the answers to the questions based on the scenarios provided:

1. For the first question regarding the angles formed by parallel lines cut by a transversal, since m∠PVT is given as 109°, m∠RWU would correspond to alternate interior angles. Thus, if these angles are equal, m∠RWU would also be 109°.

   • Response: 109°

2. In the second question about alternate exterior angles, if m∠1 = 9x - 12 and m∠8 = 105°, we set them equal to each other because alternate exterior angles are equal: \[ 9x - 12 = 105 \] Solving for x: \[ 9x = 117 \quad → \quad x = 13 \]

   • Response: 13

3. In the third question regarding supplementary angles formed by two parallel lines, consecutive interior angles are supplementary. Therefore, ∠B and ∠G would be classified under that condition.

   • Response: ∠B and ∠G because they are consecutive interior angles

4. For the question about the perpendicular bisector theorem, since AC is the perpendicular bisector of BD and meets at point E, it implies that BE is equal to AE. This means: \[ BE = AE \]

   • Response: It must be the same length as segment EA.

5. Regarding Lenny's proof of the Perpendicular Bisector Theorem, the mistake found was in identifying which segments are congruent based on the definition of a perpendicular bisector. Lenny incorrectly stated that IM is congruent to JM rather than stating that IL is congruent to JL.

   • Response: The definition of a perpendicular bisector tells you that IL≅JL, not that IM≅JM.

If you have any more questions or need further assistance, feel free to ask!