Question

An illustration shows a triangle with vertices labeled clockwise as upper A, upper B, and upper C, starting at the lower left vertex. A line is drawn passing through upper B, and another line extends beyond upper B.
Consider the following proof of the Triangle Angle Sum Theorem. Which reason has a mistake?

Given: △ABC with DB←→ parallel to AC¯¯¯¯¯¯¯¯ , and CB¯¯¯¯¯¯¯¯ extended to make CE−→− .

Prove: m∠A+m∠ABC+m∠C=180°

Statements Reasons
1. △ABC with DB←→ parallel to AC¯¯¯¯¯¯¯¯ , and CB¯¯¯¯¯¯¯¯ extended to make CE−→− 1. Given
2. ∠A≅∠ABD 2. DB←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel
3. ∠C≅∠DBE 3. DB←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel
4. m∠A=m∠ABD ; m∠C=m∠DBE 4. Definition of congruent angles
5. m∠ABD+m∠DBE+m∠ABC=180° 5. Definition of a straight angle
6. m∠A+m∠C+m∠ABC=180° 6. Substitution
7. m∠A+m∠ABC+m∠C=180° 7. Commutative property of addition (1 point)
Reason
has a mistake.

Answer 1

To identify the mistake in the proof of the Triangle Angle Sum Theorem, let's analyze the reasons provided for each step.

- **Step 1:** This is clearly correct as it states the given information.
- **Step 2:** The reasoning is also correct since it states that ∠A is congruent to ∠ABD due to the fact that DB is parallel to AC, and they are alternate interior angles.
- **Step 3:** The reasoning is similar to step 2, correctly stating that ∠C is congruent to ∠DBE for the same reason of alternate interior angles.
- **Step 4:** This step is accurate as it confirms that the measures of the angles correspond to their congruence.
- **Step 5:** The statement that m∠ABD + m∠DBE + m∠ABC = 180° is correct because this describes the angles around point B along line CE, which is a straight angle.
- **Step 6:** This is the step that takes values from ∠ABD and ∠DBE, substituting them in terms of m∠A and m∠C. This step correctly substitutes the measures of the angles.
- **Step 7:** The final conclusion rearranges the terms, stating that m∠A + m∠ABC + m∠C = 180°, which is valid according to the commutative property.

Upon reviewing the proof, the mistake is actually in **Step 3**. The statement "∠C≅∠DBE" is justified, but the angle ∠C should not be paired with ∠DBE if DB is to be considered a transversal. The only angles comparable here correctly would be ∠A and ∠ABD and ∠C and another angle formed, which should be treated distinctively from the situation described. The relationship is misplaced in terms of congruency; the correct angle to compare with should not involve such parallel references.

Thus, the incorrect reasoning relates to **Step 3**. The relationship stated is unclear and potentially incorrect in the context of what should logically follow from the triangle's configuration, even with the parallel conditions applied.