Question

a transversal crosses two parallel lines. which statement should be used to prove that the measures of angles 1 and 5 sum to 180°?
angles 5 and 7 are congruent as vertical angles, angles 6 and 8 are congruent as vertical angles.
angles 1 and 8 are congruent as corresponding angles, angles 5 and 8 form a linear pair.
angles 1 and 3 are congruent as vertical angles, angles 7 and 8 form a linear pair.
angles 1 and 2 form a linear pair, angles 3 and 4 form a linear pair.

Explanation:

Step1: Analyze the properties of angles

We know that when a transversal crosses two parallel lines, corresponding angles are congruent. Angles 1 and 8 are corresponding angles (since the lines are parallel and cut by a transversal), so \( \angle1=\angle8 \).

Step2: Use the linear - pair property

Angles 5 and 8 form a linear pair. By the linear - pair postulate, if two angles form a linear pair, then the sum of their measures is \( 180^{\circ} \). So \( m\angle5 + m\angle8=180^{\circ} \).

Step3: Substitute using the congruent - angle relationship

Since \( \angle1=\angle8 \) (from Step 1), we can substitute \( \angle1 \) for \( \angle8 \) in the equation \( m\angle5 + m\angle8=180^{\circ} \). We get \( m\angle1 + m\angle5=180^{\circ} \).

Answer:

Angles 1 and 8 are congruent as corresponding angles; angles 5 and 8 form a linear pair.