Question

the following is the proof of triangle angle sum theorem. which statement has a mistake and how should the mistake be fixed? given: $\triangle abc$ with $overleftrightarrow{de}paralleloverline{ac}$ prove: $mangle a + mangle abc + mangle c=180^{circ}$ statements reasons 1. $\triangle abc$ with $overleftrightarrow{de}paralleloverline{ac}$ 1. given 2. $angle dbacongangle b;angle ebccongangle c$ 2. $overleftrightarrow{de}paralleloverline{ac}$ and alternate - interior angles are congruent when lines are parallel. 3. $mangle dba = mangle a;mangle ebc = mangle c$ 3. definition of congruent triangles 4. $mangle dba + mangle abc + mangle ebc = 180^{circ}$ 4. definition of a straight angle

Explanation:

Step1: Analyze statement 2

The statement $\angle DBA\cong\angle B;\angle EBC\cong\angle C$ is incorrect. $\angle B$ is ambiguous as $\angle ABC$ is the angle at $B$ in $\triangle ABC$. It should be $\angle DBA\cong\angle A$ and $\angle EBC\cong\angle C$ because of alternate - interior angles when $DE\parallel AC$.

Step2: Analyze statement 3

The reason for statement 3 "Definition of congruent triangles" is wrong. It should be "Definition of congruent angles" since we are dealing with angle - congruence, not triangle - congruence.

Answer:

Statement 2 has an incorrect angle - naming ($\angle DBA\cong\angle B$ is wrong, should be $\angle DBA\cong\angle A$) and statement 3 has an incorrect reason (should be "Definition of congruent angles" instead of "Definition of congruent triangles").