two parallel lines are intersected by a transversal at points a and b prove that bisector of co interior angles are perpendicular to each other.
Answers
To Prove:
Two parallel lines are intersected by a transversal at points a and b prove that bisector of co interior angles are perpendicular to each other.
Proof:
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.