Proof.

Let m be a given infinite straight line and A a point which is not on it. Take point B at random on the opposite side of the straight line.

Consider A as the center of a circle with radius AB. Notice intersection points C and D.

Let CD be bisected at E (proposition I.10), and draw straight lines AC, AE, AD.

Since CE is equal to DE, AE is common, and the two sides AC and AD are equal, triangles ACE and ADE are congruent (SSS proposition I.8)

In particular, the angles AEC and AED are equal and hence, per definition, these angles are right angles and the line AE is perpendicular to m.

Drag e.g. m, A, and B.