Question

Xoy and yoz are two adjacent supplementary angles. [Ou) is the bisector ofund (Ov) is the bisector of yoz. A is a point on (Oy). M and N are respectively the feetof the perpendiculars drawn from A to (Ou) and (Ov)

1)prove that AMON is a rectangle

Answer 1

By showing that AMON has all right angles and opposite sides of equal length due to perpendicularly bisected supplementary angles and congruence of right triangles, we prove that AMON is a rectangle.

Step-by-step explanation:

To prove that AMON is a rectangle, we must show that it has four right angles and opposite sides of equal length. Since xoy and yoz are supplementary and bisected by lines (Ou) and (Ov) respectively, we know that triangle AOM and triangle AON are right-angled triangles, sharing the angle A and having the sides OM and ON as hypothenuses.

AM and AN are both perpendicular to (Ou) and (Ov) respectively, and since perpendicularly bisected supplementary angles form two 90-degree angles, angles AMO and ANO are both right angles. The angles at M and N are also right angles by definition of perpendicularly drawn lines. Thus, with all four angles being right angles and AO being common to both triangles AOM and AON, by the congruence of right triangles (hypotenuse-leg), we deduce that AM = AN and OM = ON, thereby proving AMON is a rectangle with its consecutive sides equal and all angles at 90 degrees.