Four one-metre rods are joined by movable hinges. The drawing shows a square. Do four equal lengths guarantee the right angles too? No: the frame can tilt while every length stays the same.
Geometry separates what is given from what must be proved. A definition fixes the meaning of a word, an axiom supplies a starting rule of the model, and a proof derives a new conclusion from permitted rules. Measurement and a convincing drawing help investigation but cannot replace that chain.
Choose the model first
A point has no size, a line has no thickness and extends without limit in both directions, and a plane has no thickness or boundary. These are ideal objects. A pencil dot and the edge of a sheet only represent them. The part of a line between two points, including its endpoints, is a segment; its length is a number with a chosen unit.
A plane is enough for a flat frame. A box requires space: length, width and height. Its image on a screen is flat again; the slope of a drawn edge does not by itself determine the actual angle in space. This course uses Euclidean models. Their conclusions cannot simply be transferred to a curved surface without checking the assumptions.
Units are part of the conditions too. 1 m and 100 cm express the same length; the number 100 without a unit does not specify the frame’s size.
Three different roles for a statement
A definition specifies which objects receive a particular name. A square is a quadrilateral with four equal sides and four right angles. A right angle measures 90°, one quarter of a full turn. Equal sides do not allow us to omit the angle requirement. Euclid’s definition of a square.
An axiom is accepted as a starting rule of the chosen system. For example, exactly one line passes through two distinct points. It is not an axiom because we have tried a ruler many times: it is an initial assumption on which further reasoning rests. The first postulate and editorial explanation.
A theorem is a statement established by proof. In “if …, then …”, the first part is the hypothesis and the second the conclusion. The proof explains why the conclusion necessarily follows from the hypothesis, definitions, axioms and earlier proved results.
What the four rods are missing
The hypothesis “this is a square” gives the conclusion “all four sides are equal”. Equal sides are therefore necessary for a square: without them, there is no square. But they are not sufficient: they may occur in a figure that is not a square.
Square
Same lengths, different angle
Every side in the right-hand diagram is also 1 m, but one angle is 60° rather than 90°. This is a counterexample to “if four sides are equal, the figure is a square”. One such example refutes a claim about all figures. A hundred square frames would not prove it.
Swapping the hypothesis and conclusion produces the converse statement. It needs its own check. A sufficient condition guarantees the conclusion: checking every requirement in the definition of a square is sufficient. Do not add a right angle to the given data merely because it “looks right” in a sketch.
A short proof without a drawing
Given: M lies between A and B on one line, AM=1 m and MB=1 m. Find AB. By the rule for adding lengths of consecutive segments:
AB = AM + MB = 1 m + 1 m = 2 m.
“Between A and B” ensures that these parts are added. The reason is the points’ arrangement and the length rule, not the pixel count between the letters. Compare and add quantities of the same kind in consistent units. Euclid’s common notions.
What a tape measure tells us
In practical work, “1.00 m” may be a rounded measurement. If a maximum error of 0.01 m is separately specified, the true length lies between 0.99 and 1.01 m. Two identical readings then do not establish exact equality. This is a given error bound, not a probability of lying in the interval.
Before drawing a geometric conclusion, ask what is given exactly, which model rules were used, and whether the drawing has silently supplied an assumption. For the frame the answer is concrete: equal rods allow different angles; a square also needs conditions on its angles.