[i]Given Hilbert's axioms, show that the base angles of an isosceles triangle are congruent.[/i][i][math]\cong[/math][/i]Consider two triangles [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAOCAYAAAAmL5yKAAAAjElEQVQ4T6WS0QmAMAwFz010E0dxA1fQCXQEN9ERHMERHEEeWCilSUALhX4k965tGn6uJujvgRO4rboIcAEbMH0BDEALjEBnWXgGStV2LSyA0o+3WefFsrAAKT1d3bSoAfL0BDAtaoAy3bUoAbV016IE6OF2Z7hkN+dzkQP05zKIlqZyTUXRJEYwfgMe9d4gD5cZP28AAAAASUVORK5CYII=[/img]ABC and [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAOCAYAAAAmL5yKAAAAjElEQVQ4T6WS0QmAMAwFz010E0dxA1fQCXQEN9ERHMERHEEeWCilSUALhX4k965tGn6uJujvgRO4rboIcAEbMH0BDEALjEBnWXgGStV2LSyA0o+3WefFsrAAKT1d3bSoAfL0BDAtaoAy3bUoAbV016IE6OF2Z7hkN+dzkQP05zKIlqZyTUXRJEYwfgMe9d4gD5cZP28AAAAASUVORK5CYII=[/img]CBA. Notice that line AB [math]\cong[/math] C*B* and notice that BC [math]\cong[/math] B*A*, and angle ABC [math]\cong[/math] C*B*A*. Using Hilbert's Axiom of Congruence (III-5), it implies that angles BAC [math]\cong[/math] B*C*A*. We can further conclude that angle ABC [math]\cong[/math] ACB, and the base angles of triangle ABC are congruent!