Let (AB) and (CD) be 2 parallel lines by a transversal at E and F respectively.The bisector of AEF and BEF cut (CD) at M and N respectively.The bisector of CFE cuts [ME] at S, and the bisector of DFE cuts [Ne] at T.Show that: ETFS is a rectangle
The sum of angles AEF and BEF is 180 degrees. Therefore the sum of the bisectors of those angles (MEF and FEN) is 90 degrees. That sum angle (MEN or SET) is one of the angles of quadrilateral SETF. Angles BEF and EFM are equal because of the way EF intercpts parallel lines. Therefore EN and SF are parallel. SE and FT are parallel for the same reason.
Therefore EFTS is a parallelogram. Since we have shown it has one right angle, all angles are right angles, so it is a rectangle.
You will have to draw the figure, as I did, to make sense of all that.
To show that ETFS is a rectangle, we need to prove that all four angles of the quadrilateral are right angles (90 degrees).
Let's break down the problem step by step:
Step 1: Given that (AB) and (CD) are parallel lines and EF is a transversal, we can use the properties of parallel lines to find the measures of angles EFD and EFE.
Angle EFD and EFE are corresponding angles formed by a transversal (EF) intersecting parallel lines (AB and CD). Corresponding angles are congruent, which means they have the same measure. Therefore, angle EFD = angle EFE.
Step 2: The bisectors of angles AEF and BEF intersect line (CD) at points M and N, respectively.
Since the bisectors of angles divide the angles into two congruent angles, we can deduce that angle AEM = angle MEF and angle BEF = angle NEF.
Step 3: The bisector of angle CFE intersects [ME] at point S.
Since the bisector of angle CFE intersects line (ME), we can conclude that angle CFS is a right angle because the bisector of an angle is perpendicular to the opposite side of the angle.
Step 4: The bisector of angle DFE intersects [NE] at point T.
Similarly, since the bisector of angle DFE intersects line (NE), we can conclude that angle DFT is a right angle.
Step 5: Now, we have angles CFS = 90 degrees and DFT = 90 degrees.
Since adjacent angles in a quadrilateral add up to 180 degrees, we can deduce that angle EFT = 180 - (CFS + DFT) = 180 - (90 + 90) = 0 degrees.
Step 6: Angle EFT being 0 degrees means that EF is a straight line, which implies that ETFS is a rectangle.
Therefore, ETFS is proven to be a rectangle.