Question
Example
A lemon tree is perpendicular to the ground. The bottom of the tree is marked with point D. Point A of the lemon tree is connected to B and C on the ground with equal lengths of cable, as shown in the picture. Prove that AADB and AADC are congruent.
HL
B
C
MULTIPLE-CHOICE QUESTION
Why is AD congruent to AD
Vertical Angles
It was given to us
Reflexive Property of Congruence
12
legs are always congruent in a right triangle
A lemon tree is perpendicular to the ground. The bottom of the tree is marked with point D. Point A of the lemon tree is connected to B and C on the ground with equal lengths of cable, as shown in the picture. Prove that AADB and AADC are congruent.
HL
B
C
MULTIPLE-CHOICE QUESTION
Why is AD congruent to AD
Vertical Angles
It was given to us
Reflexive Property of Congruence
12
legs are always congruent in a right triangle
Answers
GPT-4o mini
To demonstrate why segments \(AD\) are congruent in the context you've provided, we can look at the relationship involving congruency in triangles \(AADB\) and \(AADC\).
### Proof of Congruence
1. **Definition of Triangles**:
- We have two triangles: \(AADB\) and \(AADC\).
2. **Common Side**:
- The segment \(AD\) is a side in both triangles. This is known as the **Reflexive Property of Congruence**, which states that any geometric figure is congruent to itself.
3. **Congruent Angles**:
- Since \(AB\) and \(AC\) are equal (as stated that \(A\) is connected to \(B\) and \(C\) with equal lengths of cable), we can denote \(AB = AC\).
- Additionally, the angles at points \(B\) and \(C\) (i.e., \(\angle ADB\) and \(\angle ADC\)) are right angles, since the lemon tree is perpendicular to the ground.
4. **Congruence by the HL Theorem**:
- Because we have:
- \(AD\) is common (Reflexive Property)
- \(AB = AC\) (given)
- \( \angle ADB = \angle ADC = 90^\circ\) (right angles),
- We can conclude that triangles \(AADB\) and \(AADC\) are congruent by the Hypotenuse-Leg (HL) theorem for right triangles.
### Answer to the Multiple-Choice Question
The correct choice for why \(AD\) is congruent to \(AD\) is:
**Reflexive Property of Congruence**
This aligns with our proof, as the segment \(AD\) serves as a common side in both triangles \(AADB\) and \(AADC\).
### Proof of Congruence
1. **Definition of Triangles**:
- We have two triangles: \(AADB\) and \(AADC\).
2. **Common Side**:
- The segment \(AD\) is a side in both triangles. This is known as the **Reflexive Property of Congruence**, which states that any geometric figure is congruent to itself.
3. **Congruent Angles**:
- Since \(AB\) and \(AC\) are equal (as stated that \(A\) is connected to \(B\) and \(C\) with equal lengths of cable), we can denote \(AB = AC\).
- Additionally, the angles at points \(B\) and \(C\) (i.e., \(\angle ADB\) and \(\angle ADC\)) are right angles, since the lemon tree is perpendicular to the ground.
4. **Congruence by the HL Theorem**:
- Because we have:
- \(AD\) is common (Reflexive Property)
- \(AB = AC\) (given)
- \( \angle ADB = \angle ADC = 90^\circ\) (right angles),
- We can conclude that triangles \(AADB\) and \(AADC\) are congruent by the Hypotenuse-Leg (HL) theorem for right triangles.
### Answer to the Multiple-Choice Question
The correct choice for why \(AD\) is congruent to \(AD\) is:
**Reflexive Property of Congruence**
This aligns with our proof, as the segment \(AD\) serves as a common side in both triangles \(AADB\) and \(AADC\).