Question

1 construct line ( \boldsymbol{ell} ) through ( c ) parallel to ( overleftrightarrow{ab} ).
2 ( mangle 4 = mangle 2 ) and ( mangle 1 = mangle 5 ) alternate interior angles formed by parallel lines have equal measures.
3 ( mangle 5 + mangle 4 + mangle 3 = 180^circ ) angles that combine to form a straight angle have measures that sum to ( 180^circ ).
4 ( mangle 1 + mangle 2 + mangle 3 = 180^circ ) substitution (2,3)
what was the first mistake in joe’s proof?
choose 1 answer:
a constructing a parallel line like this isn’t necessarily possible.
b angles ( angle 4 ) and ( angle 2 ) are not alternate interior angles, and neither are ( angle 1 ) and ( angle 5 ).
c angles ( angle 3 ), ( angle 4 ), and ( angle 5 ) don’t form a straight angle.
d the substitution isn’t correct.
do 30 problems

Explanation:

To determine the first mistake in Joe's proof, we analyze each option:

• Option A: By the Parallel Postulate, we can construct a line through a point parallel to a given line, so this is not a mistake.
• Option B: For alternate interior angles, they are formed when a transversal crosses two parallel lines. If line \( l \) is parallel to \( \overleftrightarrow{AB} \), we need to check the transversal. If the transversal is not correct, \( \angle 4 \) and \( \angle 2 \), \( \angle 1 \) and \( \angle 5 \) may not be alternate interior angles. This is a potential first mistake.
• Option C: Angles on a straight line (forming a straight angle) sum to \( 180^\circ \). If \( \angle 3 \), \( \angle 4 \), \( \angle 5 \) are on a straight line, their sum should be \( 180^\circ \), so this is not a mistake.
• Option D: The substitution step (replacing \( m\angle 4 \) with \( m\angle 2 \) and \( m\angle 5 \) with \( m\angle 1 \)) is correct if step 2 is correct, so this is not the first mistake.

The first mistake is in step 2, where the angles are misidentified as alternate interior angles.

Answer:

B. Angles \( \angle 4 \) and \( \angle 2 \) are not alternate interior angles, and neither are \( \angle 1 \) and \( \angle 5 \).