Question
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
∠B≅∠D by the Reflexive Property of Congruence.
angle upper B congruent to angle upper D by the Reflexive Property of Congruence.
DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.
Modifying above upper D upper A with bar congruent to Modifying above upper B upper C with bar by the Reflexive Property of Congruence.
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.
AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯ by the Triangle Angle Sum Theorem.
(1 point)
Responses
∠B≅∠D by the Reflexive Property of Congruence.
angle upper B congruent to angle upper D by the Reflexive Property of Congruence.
DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.
Modifying above upper D upper A with bar congruent to Modifying above upper B upper C with bar by the Reflexive Property of Congruence.
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.
AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯ by the Triangle Angle Sum Theorem.
Answers
GPT-4o mini
To prove that triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent using the ASA (Angle-Side-Angle) Congruence Theorem, you already have:
- \( \angle BCA \cong \angle DAC \) (given)
- \( \angle BAC \cong \angle DCA \) (given)
You need to establish one more angle or side pair. The most straightforward additional information to provide is showing that the sides are congruent.
The correct option to state and justify would be:
**AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.**
This is because the diagonal \( AC \) is common to both triangles \( \triangle ABC \) and \( \triangle CDA \), and thus can be used to show that:
- \( AC \cong AC\) (Reflexive Property of Congruence).
Now, you can confirm that \( \triangle ABC \) is congruent to \( \triangle CDA \) by the ASA Congruence Theorem since you have two angles and the included side congruent.
- \( \angle BCA \cong \angle DAC \) (given)
- \( \angle BAC \cong \angle DCA \) (given)
You need to establish one more angle or side pair. The most straightforward additional information to provide is showing that the sides are congruent.
The correct option to state and justify would be:
**AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.**
This is because the diagonal \( AC \) is common to both triangles \( \triangle ABC \) and \( \triangle CDA \), and thus can be used to show that:
- \( AC \cong AC\) (Reflexive Property of Congruence).
Now, you can confirm that \( \triangle ABC \) is congruent to \( \triangle CDA \) by the ASA Congruence Theorem since you have two angles and the included side congruent.