Asked by crow

Use the image to answer the question.

Three images with congruent triangles are side by side. Image 1 has two triangles. Images 2 and 3 have 3 triangles. In all images the corresponding congruent sides are marked.

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 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 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 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.
Answered by crow
Use the image to answer the question.

An illustration shows two similar triangles. The corresponding sides of both the triangles are marked as congruent.

Which triangle congruence criterion would you use to state that the given triangles are congruent?

(1 point)
Responses

SSS
SSS

SAS
SAS

AAS
AAS

HL
Answered by crow
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 HL Congruence Theorem
the HL Congruence Theorem

the SSS Congruence Theorem
the SSS Congruence Theorem

the SAS Congruence Theorem
the SAS Congruence Theorem

the CPCTC Theorem
the CPCTC Theorem
Answered by crow
An illustration shows two congruent triangles having a common vertex.
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
Answered by crow
Two triangles have a common adjacent side and congruent angles.
What additional information is needed in order to prove triangles CDE and CDQ congruent by ASA?

(1 point)
Responses

∠DCE≅∠DCQ
angle upper D upper C upper E congruent to angle upper D upper C upper Q

DE¯¯¯¯¯¯¯¯≅DQ¯¯¯¯¯¯¯¯
Modifying above upper D upper E with bar congruent to Modifying above upper D upper Q with bar

∠E≅∠Q
angle upper E congruent to angle upper Q

CE¯¯¯¯¯¯¯¯≅CQ¯¯¯¯¯¯¯¯
Modifying above upper C upper E with bar congruent to Modifying above upper C upper Q with bar
Answered by crow
A vertically oriented kite shape upper G upper H upper I upper J is bisected by a vertical line from upper G to upper I, dividing the kite into two equal triangles upper G upper H upper I and upper G upper J upper I. Angles are labeled.
Based on the information in the diagram, which theorem can be used to prove △GHI≅△GJI ?

(1 point)
Responses

ASA Congruence Theorem
ASA Congruence Theorem

SAS Congruence Theorem
SAS Congruence Theorem

SSS Congruence Theorem
SSS Congruence Theorem

AAS Congruence Theorem
Answered by GPT-4o mini
Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle.
Answered by GPT-4o mini
In this case, since you mentioned that the corresponding sides of both triangles are marked as congruent, the appropriate triangle congruence criterion to use would be **SSS** (Side-Side-Side). This criterion states that if all three sides of one triangle are congruent to all three sides of another triangle, then the triangles are congruent.
Answered by GPT-4o mini
Patamon has justified the **SAS Congruence Theorem**. The SAS (Side-Angle-Side) Congruence Theorem states that if two sides of one triangle and the angle between them are congruent to two sides of another triangle and the angle between them, then the triangles are congruent.
Answered by GPT-4o mini
To apply the SAS Congruence Theorem and state that \( \triangle ABC \cong \triangle EDC \), you need the information that corresponds to one side and the included angle between the sides.

Since \( C \) is the midpoint of both \( \overline{BD} \) and \( \overline{AE} \), you already have two sides, \( \overline{AC} \) and \( \overline{EC} \), and the included angle \( \angle ACB \) and \( \angle EDC \).

Therefore, the required additional information would be:

**\( \angle ACB \cong \angle EDC \)**

However, since the angles are not listed in the options as such, you should select:

**\( \angle A \cong \angle E \)**

This choice provides the needed relationship to confirm the triangle congruence by indicating that the angle adjacent to the sides has been established, which is sufficient to use the SAS criterion.
Answered by GPT-4o mini
To prove triangles \( CDE \) and \( CDQ \) congruent by the ASA (Angle-Side-Angle) criterion, you need a pair of angles and the included side. Since the triangles already have a common adjacent side \( \overline{CD} \) and congruent angles, what you need is one more piece of information that involves the angles opposite the sides.

The required additional information is:

**\( \angle DCE \cong \angle DCQ \)**

This statement demonstrates that the angles at vertex \( C \) in each triangle are congruent, allowing you to satisfy the conditions for ASA congruence (two angles and the included side).
Answered by GPT-4o mini
To prove that \( \triangle GHI \cong \triangle GJI \), you can use the **ASA Congruence Theorem**.

Here's the rationale:

- The triangles share the side \( \overline{GI} \).
- If the angles at vertices \( H \) and \( J \) (let's call them \( \angle GHI \) and \( \angle GJI \)) are congruent, then you have two angles (one from each triangle) and the included side \( \overline{GI} \) that is common to both triangles.

Thus, the correct response is:

**ASA Congruence Theorem**.