When a claim resists a head-on proof, assume the opposite and corner yourself.
Most of our proofs so far marched straight from the givens to the goal — draw a diagonal, name a congruence, read off the conclusion. But some true statements fight a direct attack. The trick a detective uses works here too: to show something must be true, suppose for a moment that it is false, follow that supposition with honest logic, and watch it crash into something impossible. The crash proves the supposition was wrong — so the original claim stands. This is proof by contradiction, and it is the sharpest reasoning tool in geometry. We close the stage with one more invariant idea — what survives when you resize a whole figure — and that opens the door to the next room: similarity.
A direct proof walks forward: it starts at the givens and steps, rule by rule, to the conclusion. That is what we did for "opposite sides of a parallelogram are equal" — given the parallels, name the congruence, finish. An indirect proof, also called proof by contradiction or reductio ad absurdum, does the opposite. It assumes the conclusion is false, reasons from that assumption with perfectly ordinary logic, and arrives at something that cannot be true. Since correct reasoning never produces an impossibility, the only thing that could have been wrong is the assumption — so the conclusion must be true after all.
You use this reasoning every day without naming it. Suppose you claim, "It did not rain last night." How might you prove it? You reason: if it had rained, the street would be wet. But the street is bone dry. A wet-and-dry street at once is impossible — so it did not rain. You assumed the opposite of your claim, hit a contradiction, and rejected the assumption.
To prove a statement is true, it is enough to show its opposite is impossible. A claim and its negation cannot both hold; rule one out and the other must stand.
Every proof by contradiction has the same skeleton. Learn the three rungs and the exact words that signal each, and you can write one for any claim.
1. Assume. Suppose, for contradiction, that the conclusion is false — write the negation of exactly what you want to prove.
2. Reason. Argue with honest logic until you contradict a given, a known theorem, or the assumption itself. ("But this is impossible…")
3. Reject. The assumption led to nonsense, so it is false; therefore the original claim is true. (∴ …)
The signal phrases are worth memorizing: open with "Suppose, for contradiction, that …", flag the crash with "But this contradicts …", and close with "Therefore …" and the symbol ∴. The trickiest rung is the first — you must negate the conclusion, never a given. The givens stay true throughout; they are your weapons, not your target.
Step through a single claim and watch the three rungs appear. The impossible picture lights up at the contradiction, then clears once the claim is proved.
Now the real work. Each example below is a true geometric fact that is awkward to prove head-on but falls apart the instant you assume its opposite. Read each as a clean three-step argument.
Claim. Given a line ℓ and a point P not on it, there is at most one line through P parallel to ℓ.
1. Assume. Suppose, for contradiction, there are two different lines m and n, both through P, and both parallel to ℓ. 2. Reason. If m ∥ ℓ and n ∥ ℓ, then two lines parallel to the same line are parallel to each other: m ∥ n. But m and n share the point P — parallel lines never meet. 💥 3. Reject. The assumption is impossible, so there cannot be two such parallels. ∴ through P there is at most one line parallel to ℓ.
Claim. If a ⊥ ℓ and b ⊥ ℓ, then a and b never meet (so a ∥ b).
1. Assume. Suppose, for contradiction, that a and b do meet, at a point Q. 2. Reason. They cross ℓ at two different feet, say R and S, each at a right angle. Then triangle QRS has two 90° angles — at R and at S — already totalling 180° before we count ∠Q at all. But a triangle's three angles sum to exactly 180°, leaving no room for ∠Q > 0°. 💥 3. Reject. They cannot meet. ∴ two lines perpendicular to the same line are parallel.
Claim. No triangle has two right angles, and no triangle has two obtuse angles.
1. Assume. Suppose, for contradiction, a triangle has two angles each ≥ 90°. 2. Reason. Then those two alone sum to at least 180°, and the third angle is positive, so all three sum to more than 180°. But the three angles of a triangle sum to exactly 180°. 💥 3. Reject. ∴ a triangle has at most one angle that is 90° or more.
Choose one of the three theorems, then walk the three rungs. The figure shows the impossible picture the assumption forces; the contradiction line is red, the conclusion green.
Step 1 negates the conclusion, not a given. To prove "at most one parallel," you assume two parallels — you do not throw away the fact that they pass through P. Keep every given alive; the contradiction springs from the clash between your false assumption and those untouched truths.
We end the stage by stepping back. A dilation enlarges or shrinks a whole figure from a fixed center by a scale factor k: every point moves to k times its distance from the center. Photocopy a drawing at 150% and you have dilated it by k = 1.5. The remarkable part is what changes and what does not.
Under a dilation by factor k: every length is multiplied by k, but every angle is unchanged, and parallel lines stay parallel. Same shape, new size.
Because angles and parallelism survive, the image is a perfect copy of the original — just scaled. Two figures related this way are called similar: identical in shape, possibly different in size. That single idea — lengths scale, angles and parallels hold — is the foundation of the next stage, similarity. Everything we built here, the parallelogram family and the art of proof, becomes the toolkit for that room.
Slide the scale factor and watch the image grow and shrink. The marked angle never budges; the parallel pair stays parallel; only the lengths change.
Proof by contradiction proves a claim by ruling out its opposite. Three rungs: 1. Assume the conclusion is false (negate the conclusion, never a given). 2. Reason to an impossibility — a clash with a given, a theorem, or the assumption. 3. Reject the assumption; the claim is true (∴).
Three classic results yield to it: through a point off a line there is at most one parallel; two lines ⊥ to the same line are parallel; a triangle has at most one right or obtuse angle.
A dilation by factor k scales every length by k but leaves every angle unchanged and keeps parallels parallel — same shape, new size, the doorway to similarity.
List, in order, the three steps of a proof by contradiction.
1. Assume the opposite of the conclusion. 2. Reason until you reach a contradiction. 3. Reject the assumption and conclude that the original claim is true (∴).
You want to prove "through point P off line ℓ there is at most one line parallel to ℓ." What is the very first step — the assumption you write down?
Suppose, for contradiction, that there are two different lines through P, both parallel to ℓ. (Negate the conclusion "at most one" → "two or more"; keep the given that both pass through P.)
Explain, in one line, why a triangle cannot have two right angles.
Two 90° angles already total 180°, so with a positive third angle the sum would exceed 180° — but a triangle's angles sum to exactly 180°. The assumption is impossible, so at most one angle can be 90°.
A figure is enlarged by a dilation with scale factor k = 3. A side that was 4 cm long becomes how long? A 50° angle becomes how many degrees?
The side becomes 4 × 3 = 12 cm (lengths scale by k). The angle stays 50° (a dilation does not change angles).
Under a dilation, a pair of parallel sides in the original — are they still parallel in the image? Why does this matter for the next stage?
Yes — dilations send parallel lines to parallel lines. Because angles and parallels are preserved while only lengths scale, the image is the same shape as the original. That "same shape, different size" relationship is exactly similarity, the topic of the next stage.
Use proof by contradiction to argue that two distinct lines cannot cross at two different points.
Suppose, for contradiction, two distinct lines met at two different points A and B. But through two given points there is exactly one line — so both "lines" would be the same line, contradicting that they are distinct. 💥 Therefore two distinct lines meet in at most one point.
Six questions to lock it in. Tap the answer you think is right.
This lesson introduces the most powerful — and most misunderstood — form of argument in school geometry: the indirect proof, or proof by contradiction. The big idea is logical, not computational: a statement and its negation cannot both be true, so demonstrating that the negation leads to an impossibility establishes the statement. Three standard geometric results are proved this way (uniqueness of the parallel through a point, two perpendiculars to one line being parallel, and the one-right-angle limit on a triangle), each leaning on facts students already own — the parallel postulate, "parallels never meet," and the 180° angle sum.
The specific misconception to watch for: students negate the wrong thing. The first step must negate the conclusion, while every given stays in force. A learner who instead discards a given (or who negates both, or neither) will never reach a genuine contradiction. Insisting on the exact phrasing — "Suppose, for contradiction, that the conclusion is false…" while listing the givens that remain true — heads this off. A second misconception, addressed in 16.6.4, is believing a dilation changes angles; the activity makes visible that scaling multiplies lengths only, leaving angles and parallelism fixed.
Common Core alignment: HS G-CO.C.9 / G-CO.C.10 (prove theorems about lines, angles, and triangles), HS G-SRT.A.1 / G-SRT.A.2 (dilations take lines to lines and angles to congruent angles, multiplying lengths by the scale factor; define similarity via dilations), and Standard for Mathematical Practice MP3 (construct viable arguments and critique the reasoning of others).