Chapter 6: Lines and Angles

This is your first real working chapter in geometry. The objects are simple , lines, rays, segments, and the angles they make , but the relationships are powerful. The chapter teaches you when two angles must be equal, when they must sum to $180^\circ$, and how a single line cutting two parallel lines suddenly determines eight angles that fit a beautiful symmetric pattern.

We will cover four foundational facts. The first is the linear pair: two adjacent angles on a straight line sum to $180^\circ$. The second is vertically opposite angles: when two lines cross, the angles diagonally opposite each other are equal. The third is the central theme , what happens when a transversal cuts two parallel lines: alternate-interior angles are equal, corresponding angles are equal, co-interior angles sum to $180^\circ$. The fourth is the angle-sum of a triangle: every triangle has interior angles summing to $180^\circ$, and the exterior angle equals the sum of the two opposite interior angles.

These facts will be used in every geometric proof in the rest of your school career. They are not optional. They are the toolkit. So we will pause on each one, prove it carefully, and practice using it.

The chapter also gives you your first taste of formal geometric writing: every claim is proved from the postulates and earlier results, every step is justified. This style is exactly the same style we set up in Chapter 5; here we put it to work.

By the end you should be able to look at a figure with two parallel lines and a transversal and read off all the angle relationships at a glance , pair by pair , and you should be able to prove the angle-sum theorem for a triangle in less than ten lines.

What's inside

• Linear pair , two adjacent angles on a straight line sum to $180^\circ$.
• Vertically opposite angles , equal when two lines cross.
• Parallel lines and transversal , corresponding, alternate-interior, alternate-exterior, and co-interior angles.
• Conditions for parallelism , when alternate (or corresponding) angles are equal, the lines are parallel.
• Angle sum of a triangle , $180^\circ$, plus the exterior-angle theorem.

Key results / Formula card

Memorise this card. Every problem in this chapter (and many in later chapters) reduces to one or two of these lines.