Question
consider the following proof of the traingle angle sum theorem. which statement has a mistake? how should the mistake be fixed?
given: ABC with DE || AC
prove: m<A + m<ABC + m<C = 180*
STATEMENTS | REASONS
1. ABC with DE || AC 1. Given
2.<DBA = <B; <EBC = <C 2. DE || AC and alternate interior angles are congruent whe lines are parallel
3. m<DBA = m<A; m<EBC = m<C || 3. defintion of congurent angles
4. m<DBA + m<ABC + m<EBC = 180* || 4. definition of a straight angle
5. m<A + m<ABC + m<C = 180* || 5. substitution
given: ABC with DE || AC
prove: m<A + m<ABC + m<C = 180*
STATEMENTS | REASONS
1. ABC with DE || AC 1. Given
2.<DBA = <B; <EBC = <C 2. DE || AC and alternate interior angles are congruent whe lines are parallel
3. m<DBA = m<A; m<EBC = m<C || 3. defintion of congurent angles
4. m<DBA + m<ABC + m<EBC = 180* || 4. definition of a straight angle
5. m<A + m<ABC + m<C = 180* || 5. substitution
Answers
Answered by
GPT-4o mini
The mistake in the proof lies in the transition between steps 3 and 4. Specifically, step 4 incorrectly asserts that the angles \( m<DBA + m<ABC + m<EBC = 180^\circ \) forms a straight angle. The angles being added do not necessarily create a straight angle.
Here’s how to fix the mistake:
**Correction:** Instead of asserting that \( m<DBA + m<ABC + m<EBC = 180^\circ \), the proof should establish that angles \( m<DBA \) and \( m<EBC \) are exterior angles and \( m<ABC \) is an interior angle of triangle ABC, specifically connected to the linear pair formed by \( \angle DBA \) and \( \angle ABC \).
Here's a corrected version of the proof:
### Corrected Proof:
**Given:** Triangle ABC with line DE parallel to AC.
**Prove:** \( m<A + m<ABC + m<C = 180^\circ \)
| STATEMENTS | REASONS |
|----------------------------------------------|-----------------------------------------------------------------------------|
| 1. Triangle ABC with DE || AC | Given |
| 2. \( \angle DBA = \angle B \); \( \angle EBC = \angle C \) | DE || AC and alternate interior angles are congruent when lines are parallel |
| 3. \( m<DBA = m<A \); \( m<EBC = m<C \) | Definition of congruent angles |
| 4. \( m<A + m<ABC + m<C = 180^\circ \) | Angles on a straight line (not directly stated, but implied that \( \angle DBA \) and \( \angle ABC \) are supplementary) |
| 5. \( m<A + m<ABC + m<C = 180^\circ \) | Substitution (noting the relationships established in steps 2 and 3) |
This fix clarifies the relationships between the angles in terms of being part of a linear setup, thus helping to reach the conclusion that the angles in triangle ABC sum to 180 degrees.
Here’s how to fix the mistake:
**Correction:** Instead of asserting that \( m<DBA + m<ABC + m<EBC = 180^\circ \), the proof should establish that angles \( m<DBA \) and \( m<EBC \) are exterior angles and \( m<ABC \) is an interior angle of triangle ABC, specifically connected to the linear pair formed by \( \angle DBA \) and \( \angle ABC \).
Here's a corrected version of the proof:
### Corrected Proof:
**Given:** Triangle ABC with line DE parallel to AC.
**Prove:** \( m<A + m<ABC + m<C = 180^\circ \)
| STATEMENTS | REASONS |
|----------------------------------------------|-----------------------------------------------------------------------------|
| 1. Triangle ABC with DE || AC | Given |
| 2. \( \angle DBA = \angle B \); \( \angle EBC = \angle C \) | DE || AC and alternate interior angles are congruent when lines are parallel |
| 3. \( m<DBA = m<A \); \( m<EBC = m<C \) | Definition of congruent angles |
| 4. \( m<A + m<ABC + m<C = 180^\circ \) | Angles on a straight line (not directly stated, but implied that \( \angle DBA \) and \( \angle ABC \) are supplementary) |
| 5. \( m<A + m<ABC + m<C = 180^\circ \) | Substitution (noting the relationships established in steps 2 and 3) |
This fix clarifies the relationships between the angles in terms of being part of a linear setup, thus helping to reach the conclusion that the angles in triangle ABC sum to 180 degrees.
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