Question

consider the following proof of the triangle angle - sum theorem. which statement has a mistake? how should the mistake be fixed? given: △abc with (overleftrightarrow{de}paralleloverleftrightarrow{ac}) prove: (mangle a + mangle abc + mangle c=180^{circ}) identify mistakes in the steps to prove that the sum of all the angles of a triangle is (180^{circ}) statements reasons 1. △abc with (overleftrightarrow{de}paralleloverleftrightarrow{ac}) 1. given 2. (angle dbacongangle a,angle ebccongangle c) 2. (overleftrightarrow{de}paralleloverleftrightarrow{ac}) and alternate interior angles are congruent when lines are parallel 3. (mangle dba = mangle a,mangle ebc = mangle c) 3. definition of congruent angles 4. (mangle dba + mangle abc + mangle ebc = 180^{circ}) 4. definition of a straight - angle 5. (mangle a + mangle abc + mangle c = 180^{circ}) 5. substitution

Explanation:

Step1: Analyze statement 2

The statement 2 has a mis - naming of angles. It should be $\angle DBA\cong\angle A$ and $\angle EBC\cong\angle C$ (not $\angle EDC\cong\angle C$) because of alternate interior angles for parallel lines $\overleftrightarrow{DE}\parallel\overleftrightarrow{AC}$.

Step2: Analyze the overall proof

The incorrect naming in statement 2 affects the substitution in later steps. To fix it, correct the angle - congruence statements in step 2 to $\angle DBA\cong\angle A$ and $\angle EBC\cong\angle C$ based on the property of alternate interior angles for parallel lines. Then, the correct fourth statement should be $m\angle DBA + m\angle ABC+m\angle EBC=180^{\circ}$ (since $\angle DBA$, $\angle ABC$ and $\angle EBC$ form a straight - line) and the substitution step will follow correctly.

The statement with a mistake is statement 2. The angle $\angle EDC$ should be $\angle EBC$. To fix it, change statement 2 to $\angle DBA\cong\angle A,\angle EBC\cong\angle C$ based on the property of alternate interior angles for parallel lines $\overleftrightarrow{DE}\parallel\overleftrightarrow{AC}$, and adjust the subsequent statements accordingly.

Answer:

The statement with a mistake is statement 2. It should be $\angle DBA\cong\angle A,\angle EBC\cong\angle C$ instead of $\angle DBA\cong\angle B,\angle EDC\cong\angle C$. Fix it by correcting the angle - congruence statements based on the property of alternate interior angles for parallel lines and adjusting the subsequent steps.