Question

if two parallel lines are intersected by a transversal then prove that the bisector of the interior angles form a rectangle

Answer 1

Draw the picture. The interior angles on the same side of the transversal (called consecutive interior angles) are supplementary (they sum to 180 degrees). Call these angles a and b.

a + b = 180 degrees.

Now divide both sides by 2:

a/2 + b/2 = 90 degrees.

a/2 and b/2 are just the halves of a and b formed by their bisectors. These bisectors intersect, forming the legs of a triangle with the transversal being the third side.

The interior angles of this triangle are a/2, b/2, and the angle formed by the intersection of the bisectors. Call this angle c.

Since the sum of the interior angles of a triangle is 180 degrees:

a/2 + b/2 + c = 180 degrees.

Since you know a/2 + b/2 = 90 degrees,

90 degrees + c = 180 degrees,

implying

c = 90 degrees.

So the bisectors are perpendicular.

Hope this helps and does not mislead or confuse you.

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