Definitions, Postulates & Reasoning — Print

What the practice questions cover

These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

• Bisect — Cut exactly in half.
• Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
• Midpoint — The point halfway between two others: average the x values and average the y values.
• Undefined term — Point, line and plane. Described but never defined, because any definition would use them.
• Postulate — A statement accepted without proof, such as "two points determine exactly one line."
• Segment — Two endpoints and everything between them. Written AB.
• Ray — One endpoint and everything beyond it in one direction. Written AB.
• Counterexample — A single case that makes a claim false. One is enough.
• Number line — A line with numbers increasing to the right.

Words that cannot be defined

Try defining point. "A location" — what is a location? "A spot" — what is a spot? Every attempt uses a word that needs the idea of a point.

So geometry gives up gracefully. Point, line and plane are undefined terms: described, drawn, used, but never defined.

Everything else is built from them. A segment is part of a line. An angle is two rays from a point. Each new word is defined using only words already agreed.

Notation matters

AB is a segment: it stops at both ends. AB is a ray: it starts at A and carries on past B forever.

↔AB is the whole line, going on in both directions. And plain AB with no bar means a number — the length.

So AB ≅ CD says two segments are congruent, while AB = CD says two numbers are equal. Both are correct; mixing them up is not.

Postulates: the rules of the game

Some things are so basic that proving them is impossible and assuming them is harmless. Those are postulates.

"Through any two points there is exactly one line." "Through any three points not in a line there is exactly one plane." "A line contains at least two points."

Every theorem you ever prove traces back to a handful of these. They are the floor you are standing on.

Segment and angle arithmetic

If B lies between A and C, then AB + BC = AC. That is the segment addition postulate, and it is exactly as obvious as it sounds.

Angles work the same way: if OB is inside ∠ AOC, then m∠ AOB + m∠ BOC = m∠ AOC.

Most early geometry problems are these two facts plus a little algebra.

Midpoints and bisectors

A midpoint cuts a segment into two congruent halves, so AM = MB and each is half of AB.

An angle bisector does the same for an angle, producing two congruent angles.

Both give you an equation the moment you spot them, which is why questions hide them in the wording: "M is the midpoint of AB" is a gift.