Question

given: line ( d ) is parallel to ( overline{ab} ). prove: ( mangle1 + mangle2 + mangle3 = 180 ). which reason would be needed to complete this proof? a alternate exterior angle theorem b same - side interior angle theorem c alternate interior angle theorem d corresponding angle theorem

Explanation:

Step1: Recall the theorems

• Alternate Exterior Angle Theorem: When a transversal intersects two parallel lines, alternate exterior angles are congruent.
• Same - Side Interior Angle Theorem: When a transversal intersects two parallel lines, same - side interior angles are supplementary.
• Alternate Interior Angle Theorem: When a transversal intersects two parallel lines, alternate interior angles are congruent.
• Corresponding Angle Theorem: When a transversal intersects two parallel lines, corresponding angles are congruent.

Step2: Analyze the angles

In the given figure, line \(d\) is parallel to \(\overline{AB}\), and \(\angle1\) and \(\angle4\), \(\angle3\) and \(\angle5\) are pairs of angles formed by a transversal intersecting two parallel lines. These pairs of angles lie between the two parallel lines and on opposite sides of the transversal.

Answer:

C. Alternate Interior Angle Theorem