Question
Question
What ADDITIONAL information is needed to prove that △ABC≅△EDC
using SAS if point C is the midpoint of both BD¯¯¯¯¯¯¯¯ and AE¯¯¯¯¯¯¯¯
(1 point)
Responses which answer is correct and why a b c or d
∠ACB≅∠ECD
∠ACB≅∠ECD
BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯
BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯
AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯
AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯
∠BAC≅∠DEC
∠BAC≅∠DEC
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What ADDITIONAL information is needed to prove that △ABC≅△EDC
using SAS if point C is the midpoint of both BD¯¯¯¯¯¯¯¯ and AE¯¯¯¯¯¯¯¯
(1 point)
Responses which answer is correct and why a b c or d
∠ACB≅∠ECD
∠ACB≅∠ECD
BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯
BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯
AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯
AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯
∠BAC≅∠DEC
∠BAC≅∠DEC
Skip to navigation
Answers
GPT-4o mini
To prove that triangles \( \triangle ABC \cong \triangle EDC \) using the Side-Angle-Side (SAS) criterion, we need to confirm that two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle.
Given that point C is the midpoint of both \( BD \) and \( AE \), we can conclude the following:
1. Since C is the midpoint of \( BD \), it follows that \( BC \) is congruent to \( CD \) (i.e., \( BC \cong CD \)).
2. Since C is also the midpoint of \( AE \), we have \( AC \) is congruent to \( CE \) (i.e., \( AC \cong ED \)).
So in order to use SAS to show that \( \triangle ABC \cong \triangle EDC \) we can prove the following:
- \( AB \cong ED \) (one pair of sides)
- \( AC \cong CE \) (another pair of sides)
- \( \angle ACB \cong \angle EDC \) (the included angle).
Now let's analyze the options given:
1. **∠ACB≅∠ECD** - This option states that the included angles are congruent, which is necessary for the SAS criterion.
2. **BD≅DC** - This option is not necessary since we don't need to prove segments that are not part of the triangles involved.
3. **AB≅ED** - This option states that one pair of sides is congruent, which is necessary but not sufficient alone without knowing the angle.
4. **∠BAC≅∠DEC** - This is not needed for SAS, as we are concerned with the included angle between the sides we are comparing.
Therefore, the correct choice that provides the additional needed information to complete the proof using SAS is:
**∠ACB≅∠ECD**
This verifies that the included angles are congruent, fulfilling the requirements for the SAS criterion.
Given that point C is the midpoint of both \( BD \) and \( AE \), we can conclude the following:
1. Since C is the midpoint of \( BD \), it follows that \( BC \) is congruent to \( CD \) (i.e., \( BC \cong CD \)).
2. Since C is also the midpoint of \( AE \), we have \( AC \) is congruent to \( CE \) (i.e., \( AC \cong ED \)).
So in order to use SAS to show that \( \triangle ABC \cong \triangle EDC \) we can prove the following:
- \( AB \cong ED \) (one pair of sides)
- \( AC \cong CE \) (another pair of sides)
- \( \angle ACB \cong \angle EDC \) (the included angle).
Now let's analyze the options given:
1. **∠ACB≅∠ECD** - This option states that the included angles are congruent, which is necessary for the SAS criterion.
2. **BD≅DC** - This option is not necessary since we don't need to prove segments that are not part of the triangles involved.
3. **AB≅ED** - This option states that one pair of sides is congruent, which is necessary but not sufficient alone without knowing the angle.
4. **∠BAC≅∠DEC** - This is not needed for SAS, as we are concerned with the included angle between the sides we are comparing.
Therefore, the correct choice that provides the additional needed information to complete the proof using SAS is:
**∠ACB≅∠ECD**
This verifies that the included angles are congruent, fulfilling the requirements for the SAS criterion.