Question

Statement: AD is perpendicular bisector of BC. Reason: Given. Statement: BD=DC. Reason: definition of bisector. Statement: ADB and ADC are right angles. Reason: definition of perpendicular. Statement: ADB=ADC. Reason: definition of congruent. Statement: AD=AD. Reason: reflexive property. Statement: ABD=ACD. Reason: _____

Answer 1

The reasoning for the statement "ABD = ACD" is based on the fact that triangles ADB and ADC share a common side AD, and we have already established that BD = DC (since AD is the perpendicular bisector) and that angles ADB and ADC are right angles.

Thus, you can fill in the blank as follows:

Reason: Hypotenuse-Leg (HL) theorem for right triangles.

This theorem states that if two right triangles have a congruent hypotenuse and one corresponding leg that is equal, then the two triangles are congruent. In this case, AD is the hypotenuse and BD = DC is the corresponding leg. Therefore, triangles ADB and ACD are congruent.

Answer 2

VERY WRONG

Answer 3

I apologize for the misunderstanding. Let’s clarify the reasoning more accurately.

To explain why triangles \( \triangle ADB \) and \( \triangle ADC \) are congruent given the prior statements, we can correctly state the reasoning:

For the statement: \( \triangle ADB \cong \triangle ADC \)

Reason: Side-Angle-Side (SAS) Congruence Postulate.

Justification:

1. AD = AD (reflexive property).
2. BD = DC (definition of bisector).
3. Angle ADB = Angle ADC (both are right angles).

With this information, we conclude that \( \triangle ADB \) is congruent to \( \triangle ADC \) through SAS since we have:

• One side (AD) is congruent to itself.
• The side opposite (BD = DC) is congruent.
• The included angles (right angles) are congruent.

Thank you for your patience, and I hope this correctly addresses your question.

Answer 4

SAS matches with the statements and reasons?

Answer 5

Yes, the Side-Angle-Side (SAS) Congruence Postulate aligns well with the statements and reasons you provided. Here’s how the progress can be structured:

1. Statement: AD is perpendicular bisector of BC.
   Reason: Given.

2. Statement: BD = DC.
   Reason: Definition of bisector.

3. Statement: \( \angle ADB \) and \( \angle ADC \) are right angles.
   Reason: Definition of perpendicular.

4. Statement: AD = AD.
   Reason: Reflexive property.

5. Statement: \( \triangle ADB \cong \triangle ADC \)
   Reason: Side-Angle-Side (SAS) Congruence Postulate.

Explanation of SAS:

• The side AD is congruent to itself (reflexive property).
• The length BD equals the length DC (by definition of a bisector).
• The angles \( \angle ADB \) and \( \angle ADC \) are both right angles.

Thus, by SAS, \( \triangle ADB \) is congruent to \( \triangle ADC \) because you have two sides and the included angle between them that are congruent in both triangles.

Thank you for your patience in clarifying this, and I hope this breakdown accurately reflects how the statements and reasons correspond to the conclusion of triangle congruence!