Angle Relationships in Intersecting and Parallel Lines

When two lines intersect, they form special types of angle relationships:

1. Vertically Opposite Angles
   • Definition: Opposite angles formed at the intersection are always equal.
   • Examples: $ \angle 1 = \angle 3 $ and $ \angle 2 = \angle 4 $
2. Adjacent Angles
   
   • Definition: Adjacent angles share a common side and a common vertex.
   • Key Fact: The sum of two adjacent angles can be any value.
   • Special Case: If adjacent angles lie on a straight line, their sum is $180^\circ$.
3. Supplementary Angles
   • Definition: Any two angles that sum to $180^\circ$ are supplementary.
   • Key Difference: Supplementary angles are any two angles that sum to $180^\circ$, whether they are adjacent or not.

A transversal is a straight line that crosses two or more other lines. When a transversal crosses two parallel lines, it forms special pairs of angles:

1. Corresponding Angles are Equal

   • Definition: Corresponding angles are pairs of angles that lie in the same relative position at each intersection of the transversal with the parallel lines.

   • How to Identify: They are always on the same side of the transversal (either both on the left or both on the right) and in the same relative position to the parallel lines (either both above or both below).

2. Alternate Interior Angles are Equal

   • Definition: Alternate interior angles are pairs of angles that lie between the two parallel lines and on opposite sides of the transversal.
3. Co-Interior Angles Sum to 180°

• • Definition: Co-interior angles are pairs of angles that lie between the two parallel lines and on the same side of the transversal. They always sum to $180^\circ$.