Question

given ( l parallel m ), prove the sum of the angles in a triangle is ( 180^circ ).

fill in the blank with the reason that justifies the statement.

1. alternate interior angles are congruent
2. same - side interior angles are supplementary
3. corresponding angles are congruent
4. alternate interior angles are supplementary

Explanation:

Step1: Recall Parallel Line Angle Properties

When two parallel lines (\(L \parallel M\)) are cut by a transversal (the sides of the triangle), alternate interior angles are congruent. This property is used to relate the angles of the triangle to the straight - line angle (which is \(180^{\circ}\)). The other options: "Same - side interior angles are supplementary" is about angles adding to \(180^{\circ}\) (not congruence), "Same - side exterior angles are congruent" is incorrect (same - side exterior angles are supplementary in some cases but not congruent generally), and "Alternate interior angles are supplementary" is wrong (alternate interior angles are congruent).

Step2: Match with the Proof Context

In the proof of the sum of angles in a triangle, we draw a line parallel to the base of the triangle. The angles formed with the transversal (the sides of the triangle) and the parallel lines have alternate interior angles that are congruent. This allows us to re - arrange the angles of the triangle to form a straight line (which is \(180^{\circ}\)). So the correct reason is "Alternate interior angles are congruent".

Answer:

The correct option is "Alternate interior angles are congruent" (assuming the first option is labeled as such, e.g., if the options are labeled A - D: A. Alternate interior angles are congruent, then the answer is A. Alternate interior angles are congruent)