Question

An illustration shows a triangle with angles marked as 1, 2 and 3, clockwise beginning at the top angle. A line is drawn outside of the triangle, passing through the vertex with angle 3.
Consider the following proof of the Triangle Angle Sum Theorem. Where are the mistakes in the proof?

Given: Diagram of a generic triangle with parallel lines as shown

Prove: m∠1+m∠3+m∠2=180°

Statements Reasons
1. Diagram of a generic triangle and parallel lines as shown 1. Given
2. ∠1≅∠5 ; ∠2≅∠4 2. Alternate interior angles are congruent when lines are parallel
3. m∠1=m∠5 ; m∠2=m∠4 3. Definition of congruent angles
4. m∠1+m∠3+m∠4=180° 4. Definition of a straight angle
5. m∠1+m∠3+m∠2=180° 5. Triangle Angle Sum Theorem (1 point)
There is a mistake is Statement
and there is a mistake in Reason
.

Answer 1

In the given proof of the Triangle Angle Sum Theorem, there are indeed mistakes in the statements and reasons. Let's identify where the errors occur:

### Mistake in Statement:
The mistake is in **Statement 4**:
- The statement asserts that \( m∠1 + m∠3 + m∠4 = 180° \). While this represents the sum of angles formed when extending a line through one of the triangle's vertices (and it is indeed equal to 180° since it represents a straight angle), it is actually incorrect in the context of the triangle in question. This statement should be expressing the relationship involving angle \(2\), not \(4\).

### Mistake in Reason:
The mistake is in **Reason 5**:
- The reason states that \( m∠1 + m∠3 + m∠2 = 180° \) based on the Triangle Angle Sum Theorem, but this is incorrect. The theorem is about the angles within a triangle, and this equality must be derived from the previous steps of the proof rather than assumed. Instead, the relationship in **Statement 4** should correctly lead to the conclusion involving angle \(2\) rather than rely on the Triangle Angle Sum Theorem without justification.

Overall, the correct flow of reasoning should clearly show how the angles \(1\), \(2\), and \(3\) relate through the properties of parallel lines and the geometry of the triangle itself. The proof needs to clarify how \( m∠1 + m∠3 + m∠2 = 180° \) based on the angles established and not cite the theorem prematurely.