Proof in Geometry

Givens, conclusions and reasons

Suppose the task is to prove that the diagonals of a rectangle have equal lengths. “$ABCD$ is a rectangle” is given. “$AC=BD$” is the conclusion. You may use right angles because the definition of a rectangle supplies them; you may not begin by marking its diagonals equal.

A statement-and-reason table can organize a proof, but complete prose can be equally rigorous. The required feature is a chain in which each new statement follows from earlier facts. More lines do not automatically make an argument better.

AMSI’s treatment of congruence uses triangle comparisons as a recurring proof method. Before applying a test, identify the exact triangles and match the vertices in a consistent order.

A complete proof: rectangle diagonals are equal

Let $ABCD$ be a rectangle with vertices in order. Compare $\triangle ABC$ and $\triangle BAD$.

The side $AB$ is common. The sides $BC$ and $AD$ are equal because a rectangle is a parallelogram and opposite parallelogram sides are equal. The included angles $\angle ABC$ and $\angle BAD$ are both $90^\circ$ by the rectangle definition.

The correspondence is therefore

Corresponding third sides give $AC=BD$, proving the diagonals equal.

The proof establishes the result for every rectangle, not merely a square. It does not assume that the diagonals are perpendicular; that extra property holds only in the square case. In the diagram’s $6\times4$ rectangle, each diagonal has length $\sqrt{52}=2\sqrt{13}$, a numerical confirmation of the general argument.

Theorem and converse

The converse reverses the logical direction of a statement. “If a quadrilateral is a rectangle, its diagonals are equal” does not by itself establish “If its diagonals are equal, it is a rectangle.”

A counterexample refutes that converse. The quadrilateral with vertices $(-3,0)$, $(3,0)$, $(2,2)$ and $(-2,2)$ is a non-rectangular isosceles trapezoid. Its diagonals both have squared length $29$, yet its sides do not form four right angles.

Adding a hypothesis can make a converse true. A parallelogram with equal diagonals is a rectangle. The parallelogram condition rules out the isosceles-trapezoid counterexample.

By contrast, the converse of the Pythagorean theorem is true when its positive lengths form a triangle and the longest side is identified. The truth of a converse must be established independently, not inferred from the original theorem’s truth.

Direct reasoning and contradiction

A direct proof starts from the hypotheses and derives the conclusion. The rectangle proof above is direct.

A contradiction proof temporarily assumes the conclusion is false and derives an impossibility. To show that a triangle cannot have two right angles, suppose it did. Those two angles already total $180^\circ$, leaving $0^\circ$ for the third angle. A non-degenerate triangle has positive interior angles, so the assumption is impossible.

The contradiction depends on a valid previously established triangle angle sum. It would be circular to use “triangles have at most one right angle” as the reason that the remaining angle cannot be zero if that is precisely the claim being proved.

What a counterexample can accomplish

One valid counterexample disproves a universal statement. To disprove “Every rhombus is a square,” a single non-square rhombus is enough. For example, the vertices $(0,0)$, $(5,0)$, $(8,4)$ and $(3,4)$ form a rhombus with every side length $5$. Adjacent side vectors $(5,0)$ and $(3,4)$ have non-zero dot product $15$, so the angle is not right.

Several examples that satisfy a statement do not prove it universally. Testing ten rhombi with equal diagonals does not replace a proof that the claimed implication always holds. Examples are valuable for discovering a conjecture and exposing mistakes; a proof addresses all permitted cases.

Choosing an auxiliary construction

An extra line can expose a useful relationship. A quadrilateral diagonal creates two triangles. A radius to a tangent point creates a right angle. An altitude in an isosceles triangle can reveal a pair of right triangles. A parallel line can connect angles through a transversal.

The construction itself must be allowed. Drawing a segment is straightforward when its endpoints are known. Calling that segment an angle bisector, median or perpendicular adds a condition that must be constructed, given or proved.

After adding a line, ask what new fact it makes available. Extra lines that do not support a named relationship often complicate the drawing without advancing the proof.

Checking a proof for hidden assumptions

Read the argument without trusting the picture. Have equal sides been given or established? Does a claimed SAS angle lie between the two stated sides? Does “same segment” in a circle refer to vertices on the correct side of the chord? Does a point called a midpoint lie on the segment as well as have equal endpoint distances?

Check quantifiers and boundary cases too. “Every parallelogram” includes rectangles and rhombi. “Two sides are equal” does not say they are opposite rather than adjacent. “All angles are equal” does not make a quadrilateral equilateral.

A finished proof should make the final implication explicit. Ending with “the triangles are congruent” is insufficient if the requested result was a particular side equality. Name the corresponding parts that complete the task.

Related topics

Coordinate Proofs represents an arbitrary figure with variables to prove a general result. Vectors in Geometry uses displacement algebra for another route to geometric conclusions.