Angles & Parallel Lines

Angles are measured in degrees. A right angle is 90°, a straight line is 180°, a full turn is 360°.

Two useful pairs. Complementary angles add to 90°. Supplementary angles add to 180°.

When two lines cross, the angles opposite each other are equal — these are vertical angles. The angles next to each other sit on a straight line, so they add to 180°.

When a line cuts across two parallel lines, angles in matching positions are equal (corresponding angles), and so are the ones in a Z shape (alternate angles). Angles in a C or U shape add to 180°.

The parallel postulate underwrites all of this. Given parallel lines cut by a transversal, corresponding angles are congruent; from that single fact, alternate interior angles are congruent (via vertical angles) and co-interior angles are supplementary (via the linear pair).

The relationships run both ways, which is what makes them useful for proof. If corresponding angles are congruent then the lines *are* parallel — that converse is how parallelism gets established rather than assumed.

In algebra this becomes a source of equations. Angle relationships are stated in terms of expressions such as (3x + 10)° and (5x − 20)°, and the geometric fact supplies the equation to solve. Vertical angles give 3x + 10 = 5x − 20; a linear pair gives (3x + 10) + (5x − 20) = 180. Choosing the correct relationship is the whole problem, since the algebra afterwards is routine.