Question

joe tried to prove that the sum of a triangles interior angle measures is 180°

image of triangle and line

statement | reason
--- | ---
1 | construct line ℓ through c parallel to \overleftrightarrow{ab}.
2 | m∠4 = m∠2 and m∠1 = m∠5 | alternate interior angles formed by parallel lines have equal measures.
3 | m∠5 + m∠4 + m∠3 = 180° | angles that combine to form a straight angle have measures that sum to 180°.
4 | m∠1 + m∠2 + m∠3 = 180° | substitution (2,3)

what was the first mistake in joes proof?

choose 1 answer:

a) constructing a parallel line like this isnt necessarily possible.

b) angles ∠4 and ∠2 are not alternate interior angles, and neither are ∠1 and ∠5.

c) angles ∠3, ∠4, and ∠5 dont form a straight angle.

d) the substitution isnt correct.

Explanation:

Constructing a line parallel to a side through the opposite vertex is valid (Euclid's parallel postulate), so option A is wrong. Angles ∠3, ∠4, ∠5 lie on straight line ℓ, so they form a straight angle summing to 180°, making option C wrong. The substitution step replaces equal angles, so option D is wrong. ∠4 and ∠1 are alternate interior angles with ∠2 and ∠5 respectively, so Joe misidentified the alternate interior angle pairs in Step 2.

Answer:

B. Angles ∠4 and ∠2 are not alternate interior angles, and neither are ∠1 and ∠5.