Four triangles are labeled upper A upper B upper C, upper E upper D upper C, upper A prime upper B prime upper C, and upper A double prime upper B double prime upper C, with the point upper C common to all three.

Question

To prove △ABC≅△A′′B′′C′′ , which of the following shows how the ASA criterion follows from the definition of congruence in terms of rigid motions?

(1 point)
Responses

Rotate △ABC −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C negative 90 degrees (clockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper C upper A prime with bar .

Rotate △ABC −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C negative 90 degrees (clockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper B prime upper C with bar .

Rotate △ABC 90 degrees (counterclockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯.
Rotate triangle upper A upper B upper C 90 degrees (counterclockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper C upper A prime with bar .

Rotate △ABC 360 degrees (counterclockwise), and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.

Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent using the ASA (Angle-Side-Angle) criterion through rigid motions, we need to show that you can obtain one triangle from the other through a series of rotations and reflections.

From the information provided, the sequence of transformations needs to first align \( \triangle ABC \) with respect to the common point \( C \) and then position it correctly relative to triangle \( \triangle A''B''C'' \).

Let's consider the options provided to find the one that leads to congruence:

1. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( CA' \).
2. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( B'C' \).
3. Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( CA' \).
4. Rotate \( \triangle ABC \) 360 degrees (counterclockwise), and then reflect \( \triangle A'B'C' \) across \( B'C' \).

The most reasonable transformation that aligns \( \triangle ABC \) with \( \triangle A''B''C'' \) while keeping \( C \) fixed involves both a suitable rotation and a reflection. The rotation should orient the triangles appropriately, and the reflection should then correctly align them.

Looking through the options, the correct transformation is:

Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \), and then reflect \( \triangle A'B'C' \) across \( CA' \).

This is the first option. By rotating triangle \( ABC \) and reflecting it across \( CA' \), we effectively demonstrate the congruence condition under the specifications of ASA congruence.

Answer 2

Which of the following pairs of triangles could be proven congruent by the ASA Congruence Theorem?(1 point)
Responses

Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 column by 5 row grid. The sides upper A upper C and upper Q upper S are marked congruent by one tick mark.
Image with alt text: Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 column by 5 row grid. The sides upper A upper C and upper Q upper S are marked congruent by one tick mark.

Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 columns by 5 row grid.
Image with alt text: Two congruent triangles, upper A upper B upper C and upper Q upper R upper S are shown within an 11 columns by 5 row grid.

Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.
Image with alt text: Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.

Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.
Image with alt text: Two congruent triangles, upper X upper Y upper Z and upper A upper B upper C are shown within an 11 column by 5 row grid.
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Answer 3

To determine which pairs of triangles could be proven congruent by the ASA (Angle-Side-Angle) Congruence Theorem, we need to examine whether we can establish that two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle.

Criteria for ASA Congruence

1. We need two angles from each triangle to be congruent.
2. The side that is included between those two angles must also be congruent in both triangles.

Based on the options given, we only have a verbal description of the images (which describe congruent triangles), yet we can analyze based on congruences usually represented.

Let's analyze the descriptions:

1. First option: \( \triangle ABC \) and \( \triangle QRS \) with side \( AC \cong QS \) marked congruent. Without information about the angles, we cannot confirm ASA directly from just the information about one congruent side.

2. Second option: Similar description as the first, but it explicitly states these triangles are congruent. This does not specify ASA conditions.

3. Third option: \( \triangle XYZ \) and \( \triangle ABC \) are shown, but no specific details are given about angles or sides.

4. Fourth option: Same as the third one without specified sides or angles.

Conclusion:

In order to use the ASA theorem, we would need to see specific angles marked congruent in addition to the marked side for one of the pairs.

If any option specifies that two angles and one included side are marked congruent, that would be a suitable choice. Based on the descriptions alone and the need for angle congruences, the first option with marked side congruence would be the most promising, but without accompanying angle marks, we cannot strictly prove ASA.

Hence, if your images or descriptions show that for some pair one angle not mentioned is congruent, choose that option. If not, given the ambiguity and lack of angles confirmed, none appear directly provable by ASA. Please recheck the angles marked in the images for clarity!

Answer 4

Three triangles are labeled upper A upper B upper C, upper M upper N upper P, and upper A prime upper B prime upper C prime.
Khalil is using rigid transformations to map △ABC to △MNP . What type of transformation does ray AA′ indicate?

(1 point)
Responses

Ray AA′ indicates a rotation.
Ray upper A upper A prime indicates a rotation.

Ray AA′ indicates a translation.
Ray upper A upper A prime indicates a translation.

Ray AA′ indicates a reflection.
Ray upper A upper A prime indicates a reflection.

Ray AA′ indicates a dilation.

Answer 5

In the context of rigid transformations and the mapping of triangle \( \triangle ABC \) to triangle \( \triangle MNP \) using ray \( AA' \), we need to consider what the ray represents in terms of the geometric transformations.

Definitions of Transformations:

1. Rotation: Turning a shape around a fixed point.
2. Translation: Moving a shape in a straight line without rotating it.
3. Reflection: Flipping a shape over a line (the line of reflection).
4. Dilation: Changing the size of a shape while keeping its proportions the same, typically altering distances from a center point.

Analysis of Ray \( AA' \):

• The ray \( AA' \) indicates the direction from point \( A \) to point \( A' \). Depending on how \( A' \) is positioned relative to \( A \):
  • If \( A' \) is a translation of \( A \), then ray \( AA' \) indicates a translation.
  • If \( A' \) is located such that \( A \) is flipped over a line to land at \( A' \), it suggests reflection.
  • A rotation would typically be indicated by the vertices being arranged around a point.
  • Dilation would suggest a change in size rather than a simple shift of points.

Conclusion:

Without additional context regarding the arrangement of points \( A \) and \( A' \), the terminology describes a clear connection. Typically, ray \( AA' \) would primarily indicate either a translation (most common in rigid transformation context when moving between triangles) or possibly a reflection.

Given just the ray \( AA' \) as a reference, the most logical choice connecting with rigid transformations would be:

Ray \( AA' \) indicates a translation.

If the problem involves reflecting over a specific axis, there might be a case for reflection. However, given the context does not confirm this, the default we'd consider is translation. If you have visual cues indicating if \( A' \) results from a flip or a shift, please check those to clarify.

Answer 6

Parallelogram upper A upper D upper C upper B is graphed on a plane, with diagonal upper A upper C.
In parallelogram ABCD , diagonal AC¯¯¯¯¯¯¯¯ is drawn, and ∠BCA≅∠DAC and ∠BAC≅∠DCA . In order to prove △ABC≅△CDA by the ASA Congruence Theorem, what additional information must be stated and justified?

(1 point)
Responses

AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.
Modifying above upper A upper C with bar congruent to Modifying above upper A upper C with bar by the Reflexive Property of Congruence.

∠B≅∠D by the Reflexive Property of Congruence.
angle upper B congruent to angle upper D by the Reflexive Property of Congruence.

AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯ by the Triangle Angle Sum Theorem.
Modifying above upper A upper B with bar congruent to Modifying above upper C upper D with bar by the Triangle Angle Sum Theorem.

DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.

Answer 7

To prove that triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent using the ASA (Angle-Side-Angle) Congruence Theorem, we need to demonstrate that two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle.

From the information given:

1. \( \angle BCA \cong \angle DAC \) (Angle congruence)
2. \( \angle BAC \cong \angle DCA \) (Angle congruence)

To apply the ASA theorem, we must identify the included side, which in this case is side \( AC \) for both triangles being congruent.

Additional Information Needed:

The congruence of side \( AC \) must be stated and justified to confirm that both triangles share this side. The side \( AC \) can be proven congruent to itself through the Reflexive Property of Congruence.

Therefore, the required justification is:

AC ≅ AC by the Reflexive Property of Congruence.

This provides the necessary side to apply the ASA theorem along with the two pairs of angles that are already stated as congruent.

Correct Response:

AC ≅ AC by the Reflexive Property of Congruence.

This confirms that \( \triangle ABC \) is congruent to \( \triangle CDA \) by the ASA theorem since we now have:

• Two pairs of angles (given)
• One included side (reflexive property)