Question

given: △abc with de || ac
prove: m∠a + m∠abc + m∠c = 180°
identify mistakes in the steps to prove that the sum of all the angles of a triangle is 180°
statements reasons

1. △abc with de || ac 1. given
2. ∠dba ≅ ∠b; ∠ebc ≅ ∠c 2. de || ac and alternate interior angles are congruent when lines are parallel
3. m∠dba = m∠a; m∠ebc = m∠c 3. definition of congruent 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

(1 point)
statement 2 has a mistake. it should say ∠dba ≅ ∠b, ∠ebc ≅ ∠a
statement 2 has a mistake. it should say ∠dba ≅ ∠a, ∠ebc ≅ ∠c
statement 4 has a mistake. it should say m∠a + m∠abc + m∠c = 180°
statement 4 has a mistake. it should say m∠dba + m∠abc + m∠acb = 180°

Explanation:

Step1: Analyze parallel - line angle relationships

When $\overleftrightarrow{DE}\parallel\overline{AC}$, by the alternate - interior angles theorem, $\angle DBA\cong\angle A$ and $\angle EBC\cong\angle C$. But in statement 2 of the given proof, it wrongly states $\angle DBA\cong\angle B$ and $\angle EBC\cong\angle C$.

Step2: Check other statements

Statement 1 is the given condition. Statement 3 is correct as it uses the definition of congruent angles. Statement 4 is correct as $\angle DBA+\angle ABC+\angle EBC$ forms a straight - angle which is $180^{\circ}$. Statement 5 is correct as it uses substitution.

Answer:

Statement 2 has a mistake. It should say $\angle DBA\cong\angle A;\angle EBC\cong\angle C$