Formulate criteria for parallelism of straight lines and prove one of them.

Let’s formulate three criteria for parallelism of straight lines.

First sign.

If at the intersection of two straight secant in a cross, the lying angles are equal, then the straight lines are parallel.

Let’s prove this symptom:

Let’s denote by O – the midpoint of the segment AB. Let us drop the perpendicular OH to line b and extend it to the intersection with line a.

Consider two triangles Δ OAK and Δ OBH (triangles are equal in side and two angles adjacent to it), we conclude that the angles

∠ OKA = ∠ OHB = 90 °.

Two perpendiculars to one straight line are parallel, which means a║b.

Q.E.D.

Second sign.

If at the intersection of two straight secant the corresponding angles are equal, then the straight lines are parallel.

Third sign.

If, at the intersection of two straight secant lines, the sum of one-sided angles is 180 °, then the straight lines are parallel.