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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.



Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake?

Given: △ABC
with DE←→∥AC¯¯¯¯¯¯¯¯

Prove: m∠A+m∠ABC+m∠C=180°

Identify Total Number of 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≅∠A
; ∠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∠ACB=180°
4. Definition of a straight angle
5. m∠A+m∠ABC+m∠C=180°
5. Substitution
(1 point)
Statement
has a mistake.

Answer 1

The mistake in the proof of the Triangle Angle Sum Theorem occurs in Step 4:

4. m∠DBA + m∠ABC + m∠ACB = 180°

This statement is incorrect because it should actually reference the angles involved in the linear pair created by the transversal cut (DE) through the parallel lines (AC). The correct angle to use would be m∠ACB rather than m∠ABC, as m∠ACB is the angle at vertex C, while m∠ABC is the angle at vertex B. Consequently, the correct statement should reflect the straight angle formed, and the labeling of angles should correspond appropriately.

So the total number of mistakes in the steps to prove the Triangle Angle Sum Theorem is 1 mistake in Statement 4.