Question
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 ∠ACB≅∠ECD ∠ACB≅∠ECD BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ line segment cap b cap c is congruent to line segment cap d cap c AE¯¯¯¯¯¯¯¯≅DB¯¯¯¯¯¯¯¯ line segment cap A cap e is congruent to line segment cap d cap b AB¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯ line segment cap A cap b is congruent to line segment cap e cap d
Answers
GPT-4o mini
To use the SSS (Side-Side-Side) Congruence Theorem to establish that \( \triangle ABC \cong \triangle EDC \), you need to confirm that all three corresponding sides of the triangles are congruent.
Given that point \( C \) is the midpoint of both segments \( \overline{BD} \) and \( \overline{AE} \):
1. \( \overline{AC} \) and \( \overline{EC} \) would need to be shown to be congruent.
2. \( \overline{BC} \) and \( \overline{DC} \) would also need to be shown to be congruent.
3. \( \overline{AB} \) and \( \overline{ED} \) should also be shown to be congruent.
Since \( C \) is the midpoint of \( \overline{BD} \) and \( \overline{AE} \), we can conclude:
- \( \overline{BC} \cong \overline{DC} \) (because each segment is half of \( \overline{BD} \))
- \( \overline{AC} \cong \overline{EC} \) (because each segment is half of \( \overline{AE} \))
However, you still need to confirm the congruence of \( \overline{AB} \) and \( \overline{ED} \) in order to apply the SSS theorem.
Among the response options provided, the statement that would provide the necessary additional information is:
**"AB ≅ ED"**
This would indicate that the third pair of sides are congruent and would complete the requirements for SSS congruence, confirming that \( \triangle ABC \cong \triangle EDC \).
Given that point \( C \) is the midpoint of both segments \( \overline{BD} \) and \( \overline{AE} \):
1. \( \overline{AC} \) and \( \overline{EC} \) would need to be shown to be congruent.
2. \( \overline{BC} \) and \( \overline{DC} \) would also need to be shown to be congruent.
3. \( \overline{AB} \) and \( \overline{ED} \) should also be shown to be congruent.
Since \( C \) is the midpoint of \( \overline{BD} \) and \( \overline{AE} \), we can conclude:
- \( \overline{BC} \cong \overline{DC} \) (because each segment is half of \( \overline{BD} \))
- \( \overline{AC} \cong \overline{EC} \) (because each segment is half of \( \overline{AE} \))
However, you still need to confirm the congruence of \( \overline{AB} \) and \( \overline{ED} \) in order to apply the SSS theorem.
Among the response options provided, the statement that would provide the necessary additional information is:
**"AB ≅ ED"**
This would indicate that the third pair of sides are congruent and would complete the requirements for SSS congruence, confirming that \( \triangle ABC \cong \triangle EDC \).