Step 1: Construct a right triangle $ABC$ right-angled at $B$ with $\angle A=45^{\circ}$.
Step 2: By the angle sum property of a triangle,
$$\angle C=180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ}$$
Step 3: Since $\angle A=\angle C=45^{\circ}$, the triangle is isosceles and the sides opposite to these equal angles are equal.
$$BC=AB$$
Step 4: By definition of the tangent ratio for angle $A$,
$$\tan 45^{\circ}=\frac{\text{side opposite to }\angle A}{\text{side adjacent to }\angle A}=\frac{BC}{AB}$$
Step 5: Since $BC=AB$,
$$\tan 45^{\circ}=\frac{AB}{AB}=1$$
Hence proved geometrically that $\tan 45^{\circ}=1$.