Sebastian, Aisha, and Conner are analyzing the statement △BGH≅△PCF .

Question

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

Sebastian
Sebastian

Conner
Conner

Aisha

None of the answers are correct.

GPT-4o mini · Answer

To analyze the statement $ \triangle BGH \cong \triangle PCF $, we need to understand the geometric properties of triangle congruence. 1. **Sebastian:** He claims $ \angle B \cong \angle F $ because they are both on the endpoints of their respective triangles. This is incorrect reasoning because just being at the endpoints does not necessarily imply congruence. 2. **Aisha:** She states $ \angle G \cong \angle C $ because both letters are in the middle of their respective triangles. While it's true that if two triangles are congruent, then corresponding angles are congruent, her reasoning is not based on the specific correspondence between the triangles given. 3. **Conner:** He argues that $ \angle H \cong \angle P $ because both letters are closest to the congruency sign in their respective triangles. Similar to Aisha, while it might be true that $ \angle H $ and $ \angle P $ correspond to each other in the congruency of triangles, his reasoning lacks proper justification. Since none of the students correctly relate their statements to the properties of triangle congruence: - The correct correspondence based on the triangle congruence statement $ \triangle BGH \cong \triangle PCF $ is: - $ \angle B \cong \angle P $ - $ \angle G \cong \angle C $ - $ \angle H \cong \angle F $ Therefore, the appropriate answer is: **None of the answers are correct.**

e · Answer

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

GPT-4o mini · Answer

Patamon’s reasoning involves demonstrating that a sequence of rigid transformations can map one triangle onto the other, with one pair of congruent angles between two pairs of congruent sides. This situation aligns with the SAS (Side-Angle-Side) Congruence Theorem, which 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 two triangles are congruent.

SSS Congruence Theorem: This requires three pairs of sides to be congruent.
SAS Congruence Theorem: This requires two sides and the included angle to be congruent.
HL Congruence Theorem: This applies specifically to right triangles and compares the hypotenuse and one leg.
CPCTC Theorem: This stands for "Corresponding Parts of Congruent Triangles are Congruent" and it follows from demonstrating triangle congruence but does not establish triangle congruence itself.

Given that Patamon is working with one pair of angles between two pairs of sides, the correct theorem they have justified is:

the SAS Congruence Theorem.

e · Answer

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

AB¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯
Modifying above upper A upper B with bar congruent to Modifying above upper E upper D with bar

∠ACB≅∠ECD
angle upper A upper C upper B congruent to angle upper E upper C upper D

BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
Modifying above upper B upper C with bar congruent to Modifying above upper D upper C with bar

∠A≅∠E

GPT-4o mini · Answer

To use the SAS (Side-Angle-Side) Congruence Theorem to prove that $ \triangle ABC \cong \triangle EDC $, we need the following:

One pair of sides that are congruent: We need to establish congruence for two sides of the triangles.
The included angle must be congruent: We need to show that the angle between the two sides we are considering is congruent in both triangles.

Since point $ C $ is the midpoint of both segments $ \overline{BD} $ and $ \overline{AE} $, we can deduce that $ BC \cong DC $ because $ C $ is the midpoint of $ \overline{BD} $.

Now, to satisfy the SAS condition, we need:

One additional pair of sides to be congruent (which we can assume is $ AB $ and $ ED $).
The included angle $ \angle ACB $ must be congruent to $ \angle ECD $.

Among the listed responses, the one that provides the additional necessary information to conclude congruence using the SAS theorem is:

AB ≅ ED.

This establishes one pair of corresponding sides being congruent. Thus, with $ BC \cong DC $ (because $ C $ is the midpoint of $ \overline{BD} $) and the included angles being congruent, we can use the SAS theorem to conclude that $ \triangle ABC \cong \triangle EDC $.

So, the correct response to ensure SAS would be:

AB̅̅̅̅̅̅ ≅ ED̅̅̅̅̅̅.

e · Answer

Marcus and Marlee are comparing △ABC with right angle B with △PQR with right angle Q to determine if they are congruent. AB¯¯¯¯¯¯¯¯ ≅ QR¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ ≅ PR¯¯¯¯¯¯¯¯ . Marcus says the triangles are congruent by the HL Congruence Theorem, but Marlee does not agree. Who is correct? (Hint: Draw a picture.)(1 point)
Responses

Neither is correct; the triangles are congruent by Side-Angle-Side.
Neither is correct; the triangles are congruent by Side-Angle-Side.

Neither is correct; the triangles are congruent by Side-Side-Angle.
Neither is correct; the triangles are congruent by Side-Side-Angle.

Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle.
Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle.

Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.

GPT-4o mini · Answer

To determine if $ \triangle ABC $ and $ \triangle PQR $ are congruent based on the given information, let's analyze the details provided: 1. **Given Information**: - Right angles at $ B $ and $ Q $. - $ \overline{AB} \cong \overline{QR} $ (one leg of each triangle). - $ \overline{AC} \cong \overline{PR} $ (the hypotenuse of each triangle). 2. **What Marcus Claims**: He's applying the HL (Hypotenuse-Leg) Congruence Theorem, which states that if a hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. 3. **What Marlee Argues**: She disagrees with Marcus. She raises a valid point that in addition to knowing the hypotenuse and one leg's lengths, we also need to confirm the corresponding legs of the right triangles must be congruent for the HL Theorem to apply. ### Conclusion - For Marcus to be correct using the HL theorem, we would need both the hypotenuse and one leg from each triangle to be congruent. We only have information about one leg ($ AB $ and $ QR $) and the hypotenuse ($ AC $ and $ PR $). - Since the triangles do not have the other leg’s lengths confirmed as congruent or not, Marlee is correct. The fact that the other leg (the one not mentioned) has not been specified or marked as congruent means we can't conclusively state they are congruent based solely on the HL theorem. Based on this analysis, the correct conclusion is: **Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.**

e · Answer

Describe a sequence of rigid transformations that could be used to show that △ABC≅△EFD . Be sure to explain how you know that each pair of corresponding vertices will overlap perfectly. Then, explain why this sequence illustrates the SAS Congruence Theorem.

(4 points)

GPT-4o mini · Answer

To show that $ \triangle ABC \cong \triangle EFD $ using a sequence of rigid transformations and to demonstrate the SAS (Side-Angle-Side) Congruence Theorem, follow these steps: ### Step 1: Identify Corresponding Parts First, identify and establish the pairs of corresponding sides and angles. For example, let’s assume: - $ \overline{AB} $ corresponds to $ \overline{EF} $ - $ \overline{AC} $ corresponds to $ \overline{ED} $ - $ \angle A $ corresponds to $ \angle E $ ### Step 2: Rigid Transformations 1. **Translation**: - Translate $ \triangle ABC $ such that point $ A $ coincides with point $ E $. This initially aligns point $ A $ over point $ E $. - After this translation, point $ B $ will move to the position that corresponds to point $ F $, and point $ C $ will be positioned in such a way that the sides reflect their length relationships to $ E $, $ F $, and $ D $. 2. **Rotation**: - Rotate $ \triangle ABC $ around point $ E $ such that $ \overline{AB} $ aligns perfectly with $ \overline{EF} $ and $ \angle A $ aligns with $ \angle E $. - This rotation will also position $ \overline{AC} $ over $ \overline{ED} $. 3. **Verification of Overlapping Vertices**: - After the translation and rotation, we will check that $ \overline{AB} \cong \overline{EF} $ is true, indicating that the sides are now congruent as required. - Since corresponding sides $ \overline{AC} $ and $ \overline{ED} $ are also congruent and the angle $ \angle A $ and $ \angle E $ are congruent by hypothesis (given that triangles are congruent), we’ve satisfied the necessary conditions of the SAS theorem. ### Step 3: Explain Corresponding Vertices Overlapping After performing the rigid transformations: - **Vertices Overlap**: - Point $ A $ perfectly overlaps with point $ E $ from the translation. - Through the rotation, point $ B $ will now overlap with point $ F $, and point $ C $ will coincide with point $ D $. - Therefore, the corresponding vertices $ A $, $ B $, and $ C $ will overlap with $ E $, $ F $, and $ D $ respectively. ### Step 4: Illustrating the SAS Congruence Theorem - By translating $ \triangle ABC $ so that point $ A $ meets point $ E $ and then rotating it, we are ensuring that: - One pair of sides ($ AB $ and $ EF $) are congruent. - The included angle ($ \angle A $ and $ \angle E $) is congruent. - The second pair of sides ($ AC $ and $ ED $) is congruent. Thus, through a sequence of translations and rotations, we have shown that the relationships among the sides and angles of the triangles meet the criteria of the SAS Congruence Theorem. ### Conclusion The properties of rigid transformations (translation followed by rotation) ensure that we can confirm $ \triangle ABC \cong \triangle EFD $ and illustrate the SAS theorem through the overlap of corresponding sides and the congruence of included angles.

e · Answer

Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF . Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.

(4 points)

GPT-4o mini · Answer

To demonstrate that $ \triangle ABC \cong \triangle DEF $ using a sequence of rigid transformations and to illustrate the ASA (Angle-Side-Angle) Congruence Theorem, you can follow these steps: ### Step 1: Identify Corresponding Parts Before initiating any transformations, determine how the triangles correspond: - Assume $ \angle A $ corresponds to $ \angle D $ - Assume $ \angle B $ corresponds to $ \angle E $ - The included side $ \overline{AB} $ corresponds to $ \overline{DE} $ ### Step 2: Rigid Transformations The sequence of rigid transformations that will show $ \triangle ABC \cong \triangle DEF $ could involve the following: 1. **Translation**: - Translate $ \triangle ABC $ such that point $ A $ coincides with point $ D $ and point $ B $ coincides with point $ E $. This aligns two of the corresponding vertices perfectly, placing $ A $ on $ D $ and $ B $ on $ E $. 2. **Rotation**: - Rotate $ \triangle ABC $ around point $ D $ such that $ \angle A $ aligns perfectly with $ \angle D $. This rotation will allow us to position all corresponding angles to overlap. - As a result of this transformation, point $ C $ will move into the position of point $ F $, given that $ \angle B $ aligns with $ \angle E $ after the transformation. ### Step 3: Ensure the Result is Congruent - **Translation**: - Translations do not alter the shape or size of the triangle; they merely shift it in space. As such, the resulting figure remains congruent to $ \triangle ABC $. - **Rotation**: - Rotations also do not change the size or shape of a figure; they merely rotate it around a specific point. Hence, after the rotation, the angles and sides of $ \triangle ABC $ remain congruent to the corresponding parts of $ \triangle DEF $. ### Step 4: Illustrating the ASA Congruence Theorem - **Angle-Side-Angle**: - After completing the above transformations, you will have: - $ \angle A $ (corresponding to $ \angle D $) is congruent, - $ \angle B $ (corresponding to $ \angle E $) is congruent, and - The side $ \overline{AB} $ (corresponding to $ \overline{DE} $) which is between the angles, is congruent. Thus, we have established two angles and the included side as congruent: this fits the criteria for the ASA theorem. ### Conclusion Through a sequence of rigid transformations involving translation followed by rotation, the corresponding vertices of $ \triangle ABC $ and $ \triangle DEF $ can be made to overlap perfectly. This validates that $ \triangle ABC \cong \triangle DEF $ and illustrates the ASA Congruence Theorem since we have demonstrated that two angles and the included side are congruent in both triangles.