Question

maria was helping with the angles in the image below (given m∠7=25°). she was supposed to state the measure for each angle she looked. the diagram shows two parallel lines cut by a transversal.
her answer:
m∠2=25°, m∠3=25° because vertical angles are congruent. m∠6=m∠5=m∠1=m∠4=155° because ∠5 is a supplementary linear pair to ∠7 and ∠1 is a congruent alternate exterior to ∠5 and congruent to vertical angle ∠4.
is her answer complete and correct? explain.

minna: it’s partially incorrect because ∠2 and ∠3 are not vertical angles to ∠7. instead, they are 25° because ∠2 is a congruent alternate exterior angle to ∠7. all else is correct in her answer.

mona: it’s partially incorrect because supplementary
∠5, ∠6, ∠1, ∠4 are 145° not 155°. all else is correct in her answer.

minnie: it’s partially incorrect. ∠2 and ∠3 are not vertical angles to ∠7, instead they are 25° because ∠2 is a congruent alternate exterior angle to ∠7. also the other 4 angles are 145° not 155°. her reasoning was correct.

Explanation:

1. Vertical angles are opposite angles formed by intersecting lines. ∠2 and ∠3 are not vertical angles to ∠7.
2. When two parallel lines are cut by a transversal, alternate exterior angles are congruent. ∠2 and ∠7 are alternate exterior angles, so \( m\angle2 = m\angle7 = 35^\circ \), and \( m\angle3 = m\angle2 = 35^\circ \) (vertical angles).
3. For the angles related to \( 155^\circ \), if \( m\angle7 = 35^\circ \), its supplementary angle (since they form a linear pair) would be \( 180^\circ - 35^\circ = 145^\circ \), and vertical angles or corresponding angles would also be \( 145^\circ \), so the reasoning about the \( 155^\circ \) - related angles (if that was a typo or misstatement, but the core about ∠2, ∠3 and ∠7 is addressed) and the rest of the angle - related reasoning (like supplementary linear pairs, alternate exterior angles congruence) is correct except for the vertical angle misidentification for ∠2, ∠3 and ∠7.

Answer:

The first explanation (the one stating "Minnie is partially incorrect because ∠2 and ∠3 are not vertical angles to ∠7. Instead, they are 35° because ∠2 is a congruent alternate exterior angle to ∠7. All else is correct in her answer.") is correct.