From "it looks true" to "it is true" — definitions, if–then, proof, and compass & straightedge.
So far we have trusted our eyes. A figure looks symmetric; two angles look equal; a line looks straight. But eyes can be fooled — a clever drawing can make unequal things look equal — and "looks like" is not mathematics. This short lesson hands you the two tools that turn geometry from drawing into proof. The first is the language of careful statements: definitions that pin a word down, "if–then" sentences, and the one counterexample that can topple a false claim. The second is the compass and straightedge, the ancient pair that lets you build a figure exactly — with no ruler markings and no guessing. Together they are how we move from "it looks true" to "it is true."
Every piece of reasoning begins with words that mean exactly one thing. A definition pins down a word with no fuzzy edge: "a right angle is an angle of 90°." That is precise — any angle is either exactly 90° or it is not, with no room to argue. Compare it to "a big angle," which has no sharp boundary and so can never be a definition. Good definitions are the bricks; you cannot build proof on sand.
A statement (or proposition) is a sentence that is definitely true or definitely false — not both, not neither. "Every square is a rectangle" is a statement (and it is true). "Two plus two equals five" is a statement (a false one). But "Draw a nice triangle" is neither true nor false — it is a command, not a claim — so it is not a statement at all. Reasoning works only on statements, because only a statement has a truth value to track.
A definition says exactly what a word means. A statement is a sentence that is either true or false. A request, a question, or a vague phrase is not a statement.
Most useful statements in geometry have the shape "if … then …". The part after if is the hypothesis — what we are given, what we get to assume. The part after then is the conclusion — what we claim must follow. "If a triangle is equilateral, then it is isosceles" assumes equilateral and concludes isosceles. Spotting the two halves is the first move in every proof: it tells you what you may use and what you must reach.
Now swap the halves. Trade the hypothesis and the conclusion and you get the converse: "If a triangle is isosceles, then it is equilateral." Here is the trap that catches everyone at least once — a true statement can have a false converse. The original is true (three equal sides certainly give you at least two). The converse is false (a triangle can have two equal sides and a different third — isosceles but not equilateral). The two sentences are different claims, and each must earn its own verdict.
Switch between the original and its converse. Watch which half is the hypothesis, which is the conclusion — and how the verdict can change.
A statement and its converse are not the same claim. "If it rained, the grass is wet" can be true while "if the grass is wet, it rained" is false (someone may have used a hose). Always judge each direction on its own.
How do you disprove a claim? A statement like "all triangles are acute" sweeps over every triangle at once. To knock it down, you do not have to argue about all of them — you only need to exhibit one triangle that breaks it. A single right triangle, with its 90° corner, is a counterexample, and one counterexample is enough to declare the whole claim false. Forever.
The reverse does not work, and this asymmetry is the heart of the matter. To show a statement is true, one example — or a hundred — is never enough, because the next case you did not check might be the one that fails. Truth about all cases demands a proof: an argument that covers every case in one stroke. Disproving is cheap (one bad apple); proving is dear (a watertight argument).
Read each claim, then decide. If you pick "counterexample," a single case that breaks the claim appears in red.
One counterexample proves a claim false. No pile of examples proves it true — for that you need a proof.
A proof is a chain of statements that starts from what you are given and walks, one careful step at a time, to the conclusion — and every step is backed by a reason. A reason is never "it looks that way." It is a definition, a fact already proven, or a step you wrote earlier in the same proof. Lay the chain out in two columns, Statement on the left and Reason on the right, and anyone can check that each link holds.
Here is a complete short proof: vertical angles are equal (an old friend from Stage 13–14). Two lines cross at a point, making angles ∠1 and ∠2 opposite each other, with ∠3 sitting between them. We never measure; we reason.
Step up to unfold the proof in logical order. Each line names a Statement and the Reason that makes it legal.
Notice the move at the core: ∠1 and ∠3 sit on a straight line, so ∠1 + ∠3 = 180°. So do ∠2 and ∠3, so ∠2 + ∠3 = 180°. Both equal 180°, so they equal each other; subtract the shared ∠3 and you are left with ∠1 = ∠2. No ruler touched the page.
Greek geometers set themselves a game with exactly two tools, and the game has been played for 2300 years. A straightedge draws the one straight line through two points you already have. The catch: it has no markings. You may not measure a length with it, you may not slide it until something "lines up." It draws lines — nothing more.
A compass draws a circle of any radius about any center, and — this is its real power — it copies a length: open it to span two points, then swing that exact radius somewhere else. Every legal construction is built from just these two moves: lines through known points, and arcs of a set radius. No protractor, no marked ruler, no eyeballing.
Sliding a marked ruler until two marks land on two lines is not a legal construction — that is measuring, and the straightedge is not allowed to measure. The honor of the game is to build with lines and arcs alone.
From those two moves, a small kit of basic constructions follows, and every bigger figure is assembled from them: copy a segment (open the compass to its length and swing it), copy an angle, bisect an angle, build the perpendicular bisector of a segment (which also finds its midpoint — coming in 15.4), and drop a perpendicular from a point to a line (recall perpendicular lines from Stage 14). Master these and you can build the triangles that the congruence tests SSS and SAS describe — that is the bridge to 15.3.
Let us walk through one in full: bisecting an angle — splitting ∠ABC into two equal halves using only arcs and one line. Watch how the equal compass openings force the two halves to be equal; the proof is SSS congruence in disguise.
Step through the construction. Each step uses only a legal tool — a line through two points, or an arc of a fixed radius.
The first arc makes BP = BQ (same radius). The two crossing arcs make PR = QR (same radius). And BR = BR is shared. So △BPR ≅ △BQR by SSS (next lesson!), which forces ∠PBR = ∠QBR. The construction does not just look bisected — it is, and we can prove it.
A definition pins a word down exactly; a statement is a sentence that is true or false. Rewrite a statement as "if (hypothesis) then (conclusion)"; swap the halves to get the converse, which may have a different truth value. One counterexample disproves a claim; proving it true needs a proof — a chain from the given to the conclusion, each step backed by a reason. The straightedge draws lines (no measuring); the compass draws arcs and copies lengths. From those two tools come the basic constructions — copy, bisect, perpendicular — the very moves that build congruent triangles.
Write the converse of "If a triangle is equilateral, then it is isosceles," and say whether the converse is true.
Converse: "If a triangle is isosceles, then it is equilateral." This is false — a triangle can have two equal sides but a different third side (isosceles, not equilateral). A 5-5-8 triangle is a counterexample.
Give a counterexample to the claim "every rectangle is a square."
A 2-by-3 rectangle. It has four right angles (so it is a rectangle) but its sides are not all equal (so it is not a square). One counterexample is all you need.
Name the hypothesis and the conclusion of "If two angles are vertical, then they are equal."
Hypothesis (the if part): "two angles are vertical." Conclusion (the then part): "they are equal." We are given vertical; we must reach equal.
Which tool copies a length — the straightedge or the compass? Which tool may you not measure with?
The compass copies a length (open it to span two points, then swing that radius). The straightedge has no markings, so you may not measure with it — it only draws lines.
Is a single example enough to prove a statement true? Explain in one sentence.
No. One example (or a million) leaves untested cases, and the next case might fail — proving a claim true requires a proof that covers every case at once. (One counterexample, however, does suffice to prove a claim false.)
Which basic construction finds the midpoint of a segment, and what extra thing does it give you for free?
The perpendicular bisector of the segment: where it crosses the segment is the midpoint, and it also gives a line perpendicular to the segment there — two results from one construction.
Six questions to lock it in. Tap the answer you think is right.
This lesson is the hinge of Stage 15: it converts the looking that carried earlier grades into the reasoning that the rest of geometry runs on. The single most valuable habit to model aloud is making the hypothesis and conclusion explicit before touching a problem — "what am I given, what must I reach?" The two-column proof of vertical angles is worth doing by hand together; the goal is not the result (they already believe it) but the experience that each line carries a reason, never "it looks that way."
The misconception to watch is the converse error: assuming a true statement's converse is automatically true. "If equilateral then isosceles" is true; its converse is plainly false, and that one clean example inoculates students for years. A close second is treating the straightedge as a measuring tool — sliding a marked ruler until things line up. Insist on the honor of the game (lines and arcs only); it is exactly that restriction that makes a construction provable rather than merely accurate-looking.
Common Core: HS G-CO.C.9/10 (prove geometric theorems about lines, angles, and triangles), HS G-CO.D.12/13 (make formal geometric constructions with compass and straightedge — copying a segment and an angle, bisecting an angle, constructing perpendiculars), and the reasoning standard MP3 (construct viable arguments and critique the reasoning of others). These constructions return in 15.3 as the why behind SSS and SAS.