Question

gabriel used the following theorem: when a transversal crosses parallel lines, alternate interior angles are congruent. fill in the blanks in gabriels solution. - since ∠lmn and ∠x are a linear pair, m∠x = \\(\square\\)°. - since ∠x and ∠y are alternate interior angles, m∠y = \\(\square\\)°. - since ∠y and ∠z are a linear pair, m∠z = \\(\square\\)°.

Explanation:

Step1: Find \( m\angle x \)

A linear pair of angles sums to \( 180^\circ \). Given \( \angle LMN = 78^\circ \) (from the diagram, as the angle at \( M \) is \( 78^\circ \)), so \( m\angle x = 180 - 78 = 102^\circ \).

Step2: Find \( m\angle y \)

Alternate interior angles are congruent (given theorem). Since \( \angle x \) and \( \angle y \) are alternate interior angles, \( m\angle y = m\angle x = 102^\circ \).

Step3: Find \( m\angle z \)

A linear pair of angles sums to \( 180^\circ \). So \( m\angle z = 180 - m\angle y = 180 - 102 = 78^\circ \).

Answer:

• \( m\angle x = \boldsymbol{102}^\circ \)
• \( m\angle y = \boldsymbol{102}^\circ \)
• \( m\angle z = \boldsymbol{78}^\circ \)