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Question NOTE: On the bottom right corner of each question, you will see a button that says “Check Answer.” Click this to check your answer before submitting. It will not tell you the correct answer, but it will tell you whether the answer you selected is correct or not. You can use this feature ONCE per question (only for Quizzes and Sample Works). 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 SSS Congruence Theorem? (1 point) Responses AB¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯ line segment cap A cap b is congruent to line segment cap e cap d AE¯¯¯¯¯¯¯¯≅DB¯¯¯¯¯¯¯¯ line segment cap A cap e is congruent to line segment cap d cap b BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ line segment cap b cap c is congruent to line segment cap d cap c ∠ACB≅∠ECD ∠ACB≅∠ECD Skip to navigation

1 answer

To determine which additional information is needed to conclude that triangles \( \triangle ABC \) and \( \triangle EDC \) are congruent by the SSS (Side-Side-Side) Congruence Theorem, we need to ensure that all three sides of one triangle are congruent to the corresponding three sides of the other triangle.

Given that \( C \) is the midpoint of both \( \overline{BD} \) and \( \overline{AE} \), we can establish the following congruences:

  • \( AC \) corresponds to \( EC \)
  • \( BC \) corresponds to \( DC \)
  • \( AB \) corresponds to \( ED \)

Since \( C \) is the midpoint, we can already infer some of the segment lengths. However, to use the SSS theorem, we can focus on confirming that the corresponding segments are indeed equal.

Here are the potential answers you provided and what they imply:

  1. \( \overline{AB} \cong \overline{ED} \) - This establishes one pair of sides to be equal.
  2. \( \overline{AE} \cong \overline{DB} \) - This is not necessary because you already know \( C \) is the midpoint.
  3. \( \overline{BC} \cong \overline{DC} \) - This is significant because it establishes a corresponding pair of sides.
  4. \( \angle ACB \cong \angle ECD \) - This would be needed for the ASA or AAS congruence theorems, but not SSS.

To conclude whether \( \triangle ABC \cong \triangle EDC \) by SSS, you would need to know that \( \overline{AB} \cong \overline{ED} \) and that \( \overline{BC} \cong \overline{DC} \).

Therefore, the best additional information among your responses would be either:

  • \( \overline{AB} \cong \overline{ED} \) - This explicitly confirms that the first pair of corresponding sides is congruent.

If you can only choose one, I would recommend selecting \( \overline{BC} \cong \overline{DC} \) as that would facilitate confirming equal lengths of sides necessary for SSS congruency, but ultimately, confirming both pairs of corresponding sides would be ideal.

Feel free to use the "Check Answer" feature to see if your selected option aligns with this reasoning.