Question

Use the image to answer the 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 with arrows at both ends is drawn above the triangle.
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The line is drawn through a point above the triangle labeled as upper D, triangle vertex upper B, and another point upper E. The line is parallel to triangle segment upper A upper C.


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°


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)
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 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°
.
Statement 4 has a mistake. It should say m angle upper A plus m angle upper A upper B upper C plus m angle upper C equals 180 degrees .

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 .

Answer 1

Statement 2 has the mistake. It should read: ∠DBA ≅ ∠A ; ∠EBC ≅ ∠C.

Reason: Since DE (and thus DB and BE) is parallel to AC, ∠DBA is an alternate interior angle with ∠A, and ∠EBC is an alternate interior angle with ∠C.