Question

draw the figure and explain play fair axiom

Answer 1

we take the Parallel Postulate in the form known as Playfair's Axiom:

Through a given point, only one line can be drawn parallel to a given line.

This axiom, or its equivalent, seems to be necessary to prove the Pythagorean Theorem:

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

The proofs of the Pythagorean Theorem seem to divide into three main types: proofs by shearing, which depend on theorems that the areas of parallelograms (or triangles) on equal bases with equal heights are equal, proofs by similarity, which depend on calculations of proportions of sides of similar triangles, and proofs by dissection, which depend on the observation that the acute angles of a right triangle are complementary. In each case, the argument can readily be traced back to consequences of the Parallel Postulate. In this sense, in the presence of the remainder of the axioms for plane geometry, we observe that I implies II.

Many of the consequences of the Parallel Postulate, taken together with the remainder of the axioms for plane geometry, can be shown in turn to imply the Parallel Postulate. In this sense, these statements can be regarded as equivalent to the Parallel Postulate. They include such well-known theorems as

III In any triangle, the three angles sum to two right angles.
IV In any triangle, each exterior angle equals the sum of the two remote interior angles.
V If two parallel lines are cut by a transversal, the alternate interior angles are equal, and the corresponding angles are equal.
Many apparently weaker statements have also been shown equivalent to the Parallel Postulate. These include:

VI There exists some triangle whose three angles sum to two right angles
VII There exists an isosceles right triangle whose three angles sum to two right angles.
VIII There exists arbitrarily large isosceles right triangles whose angles sum to two right angles.
IX The three angles of any right triangle sum to two right angles.