Formal Geometrical Proofs

Measuring a diagram can suggest a result, but it cannot prove it. A drawing may not be to scale, and checking one particular shape does not establish a general relationship. Letters allow an argument to work without choosing particular angle sizes or lengths.

Naming angles and choosing reasons

In $\angle ABC$, the middle letter, $B$, is the vertex: the angle is between $BA$ and $BC$. The notation $AB$ denotes the length of the line segment joining $A$ and $B$. Naming the exact angle or length makes each step unambiguous.

Useful reasons include angles on a straight line summing to $180^\circ$, angles around a point summing to $360^\circ$, vertically opposite angles being equal, and the angles of a triangle summing to $180^\circ$. A quadrilateral has an angle sum of $360^\circ$. Equal sides of an isosceles triangle have equal angles opposite them.

Parallel lines provide further connections: alternate angles are equal, corresponding angles are equal, and co-interior angles sum to $180^\circ$. The parallel lines must be given or established before these facts can be used.

Circle theorems connect angles that may lie in different parts of a diagram. Look for the chord or arc linking them: angles in the same segment are equal, and the angle at the centre is twice the angle at the circumference subtended by the same arc. An angle in a semicircle is $90^\circ$. Opposite angles of a cyclic quadrilateral sum to $180^\circ$, while the alternate segment theorem connects a tangent–chord angle to an angle at the circumference.

These facts are tools for building an argument, not just rules for calculating missing numbers.

Building an angle proof

Start by separating what is given from what must be proved. Mark the given information on the diagram, then look for a triangle, parallel lines or another configuration that connects it to the target. You can plan backwards from the target, but the written proof should move forwards from established facts.

For example, let $A$, $B$ and $C$ lie on one straight line, parallel to the line through $E$ and $F$. Suppose $BE=BF$ and $\angle ABE=x^\circ$. We want to prove that $\angle EBF=(180-2x)^\circ$.

The target angle belongs to triangle $BEF$. Finding its two other angles will therefore give us a route to the result.

The proof works for every value of $x$ compatible with this configuration. No particular value of $x$ is needed.

Combining circle theorems with algebra

A circle proof often needs more than one theorem, followed by an ordinary angle sum. Suppose $ABCD$ is a cyclic quadrilateral, with tangent $PDQ$ touching the circle at $D$. Let $\angle ABC=w^\circ$, $\angle CDQ=x^\circ$ and $\angle ACD=y^\circ$. We want to prove $w=x+y$.

The target links three angles in different positions. Triangle $ADC$ brings them together: one of its angles is already $y^\circ$, and the other two can be expressed using circle theorems.

The geometry establishes the equation; the algebra puts it into the requested form.

Another useful combination occurs when $AB$ is a diameter and $C$ lies on the circumference. If the angles at $A$ and $B$ are $(5x)^\circ$ and $(10y)^\circ$, then $\angle ACB=90^\circ$ because it is an angle in a semicircle. The triangle angle sum gives $5x+10y=90$, so dividing by $5$ proves $x+2y=18$. This establishes a relationship without determining $x$ and $y$ separately.

Proving equal lengths or angles

For equality statements, look for two triangles whose corresponding parts can be matched. Congruent triangles have equal corresponding sides and angles. To establish congruence, use SSS (three sides), SAS (two sides and the included angle), ASA (two angles and a corresponding side), or RHS (a right angle, hypotenuse and another side).

For example, let $ABCD$ be a rectangle. To prove its diagonals are equal, compare triangles $ABC$ and $BAD$:

• $AB=BA$: common side.
• $BC=AD$: opposite sides of a rectangle are equal.
• $\angle ABC=\angle BAD=90^\circ$: angles of a rectangle.

The triangles are congruent by SAS, so their corresponding sides $AC$ and $BD$ are equal. These are the diagonals, which completes the proof.

Similarity can instead establish equal corresponding angles or ratios of corresponding lengths; it does not by itself establish equal lengths. Pythagoras' theorem, $a^2+b^2=c^2$, can establish relationships between lengths in a right-angled triangle, where $c$ is the hypotenuse. Choose the tool that connects the information given to the conclusion required.