Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake? How should the mistake be fixed? Given: △ABC with DE←→∥AC¯¯¯¯¯¯¯¯ Prove: m∠A+m∠ABC+m∠C=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) Responses Statement 2 has a mistake. It should say ∠DBA≅∠A ; ∠EBC≅∠C . Statement 2 has a mistake. It should say angle upper D upper B upper A congruent to angle upper A ; angle upper E upper B upper C congruent to angle upper C . Statement 4 has a mistake. It should say m∠DBA+m∠ABC+m∠ACB=180° . Statement 4 has a mistake. It should say m angle upper D upper B upper A plus m angle upper A upper B upper C plus m angle upper A upper C upper B equals 180 degrees . Statement 2 has a mistake. It should say ∠DBA≅∠B ; ∠EBC≅∠A . Statement 2 has a mistake. It should say angle upper D upper B upper A congruent to angle upper B ; angle upper E upper B upper C congruent to angle upper A . Statement 4 has a mistake. It should say m∠A+m∠ABC+m∠C=180° .
1 answer
The mistake is in Statement 2. It currently states:
This statement incorrectly identifies the angles. The correct relationships, given that DE is parallel to AC and we are using alternate interior angles, should be:
∠DBA≅∠A and ∠EBC≅∠C.
So, the correct response would be:
Statement 2 has a mistake. It should say ∠DBA≅∠A ; ∠EBC≅∠C.
This accurately reflects the alternate interior angle theorem since DE is parallel to AC.
