Sebastian, Aisha, and Conner are analyzing the statement △BGH≅△PCF . Sebastian says ∠B≅∠F because they are both on endpoints of their respective triangles. Aisha says ∠G≅∠C because both letters are in the middle of their respective triangles. Conner says ∠H≅∠P because both letters are closest to the congruency sign in their respective triangles. Which student is correct in their reasoning? (1 point) Responses None of the answers are correct. None of the answers are correct. Sebastian Sebastian Conner Conner Aisha

Answer 1

Based on the information provided, the correct reasoning about the congruent triangles \( \triangle BGH \cong \triangle PCF \) involves matching corresponding angles based on the order of the vertices in the congruence statement.

• Sebastian claims \( \angle B \cong \angle F \). This is incorrect because in the statement \( \triangle BGH \cong \triangle PCF \), \( \angle B \) corresponds to \( \angle P \), not \( \angle F \).

• Aisha claims \( \angle G \cong \angle C \). This is also incorrect because \( \angle G \) corresponds to \( \angle C \) (this part is actually correct in terms of corresponding angles based on the congruence statement).

• Conner claims \( \angle H \cong \angle P \). This is incorrect because \( \angle H \) corresponds to \( \angle F \), not \( \angle P \).

Considering that Aisha correctly noted the correspondence of \( \angle G \) and \( \angle C \), her reasoning is valid despite the fact that her statement lacks acknowledgment of the overall triangle congruence context. The correct angles based on the congruence statement are:

• \( \angle B \cong \angle P \)
• \( \angle G \cong \angle C \)
• \( \angle H \cong \angle F \)

Since Aisha correctly matches one pair of angles, the best answer is Aisha.

Answer 2

Georgia is justifying the SSS triangle congruence criteria using △ABC and △LMN (image 1). She transforms the first triangle with a translation to create △A′B′C′ (image 2), followed by a rotation to create △A"B"C" (image 3). She is surprised to see that △A"B"C" overlaps △LMN without having done a reflection. What happened? (1 point) Responses Georgia has shown that the two triangles are not congruent. If the sequence of transformations does not include a translation, rotation, and reflection, then the triangles cannot be congruent. Georgia has shown that the two triangles are not congruent. If the sequence of transformations does not include a translation, rotation, and reflection, then the triangles cannot be congruent. Georgia made a mistake. She should have done a reflection before the rotation so that her sequence of transformations included all three types of transformations. Georgia made a mistake. She should have done a reflection before the rotation so that her sequence of transformations included all three types of transformations. Georgia made a mistake. She should have translated △ABC along the vector that takes point B to point N, then done a reflection, then done a rotation. Georgia made a mistake. She should have translated triangle upper A upper B upper C along the vector that takes point upper B to point upper N , then done a reflection, then done a rotation. Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle.

Answer 3

The situation described indicates that Georgia was able to transform triangle \( \triangle ABC \) into triangle \( \triangle LMN \) through a series of transformations (translation and rotation), resulting in an overlap without needing a reflection. This outcome is significant because it highlights an important property of triangle congruence: two triangles can indeed be congruent even if the entire sequence of transformations does not explicitly include a reflection.

The correct understanding in this scenario is that congruence can be established without always requiring all three types of transformations (translation, rotation, and reflection). In fact, if two triangles can be made to overlap perfectly through any sequence of rigid transformations (isometries), they are congruent.

Given this understanding, the best answer is:

Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle.

This acknowledges that congruence can be achieved through a combination of transformations and does not necessitate a reflection in every case.

Answer 4

Patamon was given two generic triangles with one pair of congruent angles between two pairs of congruent sides. They show that a sequence of rigid transformations maps one triangle onto the other triangle such that the triangles overlap perfectly. What theorem has Patamon justified through this process?(1 point) Responses the SSS Congruence Theorem the SSS Congruence Theorem the SAS Congruence Theorem the SAS Congruence Theorem the HL Congruence Theorem the HL Congruence Theorem the CPCTC Theorem

Answer 5

In the scenario described, Patamon has two triangles with one pair of congruent angles between two pairs of congruent sides, which corresponds to the conditions of the SAS (Side-Angle-Side) Congruence Theorem. The SAS theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle between those sides is congruent, then the triangles are congruent.

Since Patamon demonstrated that a sequence of rigid transformations maps one triangle onto the other such that they overlap perfectly, it means that he has justified the SAS Congruence Theorem through this process.

Therefore, the correct response is:

the SAS Congruence Theorem.

Answer 6

If point C is the midpoint of both BD¯¯¯¯¯¯¯¯ and AE¯¯¯¯¯¯¯¯ , what additional information would you need to be able to state that △ABC≅△EDC by the SAS Congruence Theorem? (1 point) Responses BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper D upper C with bar ∠A≅∠E angle upper A congruent to angle upper E AB¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯ Modifying above upper A upper B with bar congruent to Modifying above upper E upper D with bar ∠ACB≅∠ECD

Answer 7

To apply the SAS (Side-Angle-Side) Congruence Theorem to show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle EDC \), you need to establish the equality of two sides and the angle enclosed by those two sides in both triangles.

1. Sides:

   • You already know that \( C \) is the midpoint of \( BD \) and \( AE \), which allows you to conclude that \( BC \) and \( DC \) are equal (i.e., \( BC \cong DC \)).
   • You would need to establish that another pair of corresponding sides is congruent: \( AB \) and \( ED \).

2. Angle:

   • Additionally, you need to confirm that the angles between those sides are congruent, such as \( \angle ACB \) in triangle \( ABC \) being equal to \( \angle ECD \) in triangle \( EDC \).

Given these requirements, to fulfill the conditions for SAS, you would choose one of the options provided.

Among the options, the one that would provide the necessary information that aligns with establishing the angle and corresponding sides for SAS is:

∠ACB≅∠ECD

This choice specifies that the angles between the two pairs of sides are congruent, which is needed to apply the SAS Congruence Theorem.

Answer 8

Which parallelogram is divided into two triangles that can be proven congruent by AAS? (1 point) Responses ABCD upper A upper B upper C upper D JKLM upper J upper K upper L upper M WXYZ upper W upper X upper Y upper Z PQRS

Answer 9

To determine which parallelogram can be divided into two triangles that can be proven congruent by the AAS (Angle-Angle-Side) Congruence Theorem, you need to consider the properties of the parallelogram and how it can be split into triangles.

In any parallelogram, you can draw one of its diagonals (for instance, the diagonal connecting vertices \( A \) and \( C \)) to create two triangles (e.g., \( \triangle ABC \) and \( \triangle CDA \)). The properties of a parallelogram tell us that:

1. Opposite angles are equal. Thus, if you take one angle from triangle \( ABC \) (say \( \angle A \)) and the corresponding opposite angle in triangle \( CDA \) (which is \( \angle C \)), you have two angles that are equal.
2. The diagonal that was drawn (which serves as a common side) is congruent to itself.

Therefore, both triangles share one side (the diagonal), and because you can identify two pairs of angles that are congruent, the triangles can be proven congruent by AAS.

Let's go through the options you provided systematically to determine which one meets these conditions:

1. ABCD: Dividing this parallelogram along diagonal \( AC \) results in triangles \( ABC \) and \( CDA \).
2. JKLM: Similar reasoning applies for this parallelogram as well.
3. WXYZ: Again, this parallelogram can be divided into two triangles that could be congruent.
4. PQRS: This one also can be divided similarly.

Since all the options listed (ABCD, JKLM, WXYZ, PQRS) have properties that can allow them to be divided by a diagonal into congruent triangles by AAS (since the angles are equal, and they share a side), without specific descriptions for each, we can't definitively state which specific parallelogram is being referred to.

Based on typical properties of parallelograms and the methods of proving triangle congruence by AAS, all four options can potentially fulfill the requirement when viewed abstractly. Therefore, unless specific triangles or angles are defined within those parallelograms, each of the parallelograms is equally valid for AAS.

However, if you have a specific image or geometric context for these parallelograms that defines other criteria or specific angles, that would ultimately provide the correct answer. Please confirm if additional context is available, or based on the general properties, any of them would work with two triangles proven by AAS method.