Practice DS, Section 3, Question 24

AleksandrM wrote:If you have two angles on the same side that are 90 degrees each, then I agree, the other two angles can be a lot of things other than 90. However, the way that the statements are stated – and thus the way the figure is drawn – when the two given angles are marked as 90 each, you end up with the lower left angle and upper right angle equaling 90, which makes the other two 90 as well.

I think you're imagining that the opposite 90 degree angles are 'in line' with each other.

Do the following visualization: imagine a 90 degree angle in the bottom right corner and top left corner of a rectangle. Now rotate the top left corner angle by a few degrees, keeping it a 90 degree angle. You'll see how easy it is to make a figure which has 90 degree angles in opposite corners, and different angles in the other two corners.

AleksandrM wrote:Sorry ian, but I do not follow. I took a book. I look at it, and I see that if I make any of the 90 degree corners wider, this will change the other corners as well, making them NOT 90 degrees.

The question is: if two opposite angles in a 4-sided shape are 90 degree angles, must the shape be a rectangle?

The answer is no, certainly not. You can draw a non-rectangle with opposite angles of 90 as follows:

-draw an L in the bottom left corner of a page, so that the vertical and horizontal lines forming the L are parallel to the borders of the page;

-draw an upside-down L in the middle of the page, *but rotate it by a few degrees* so that the vertical and horizontal parts of the L are not parallel with the borders of the page;

-from the L in the bottom left, extend as far as you can the horizontal and vertical lines which form the L;

-do the same with the L in the middle of the page;

-these lines will meet, and the intersection points, along with the two corners of the two Ls, form a quadrilateral. The Ls of course are 90 degree angles, but as long as you rotated the L in the middle of the page, the other two angles will not be 90 degrees.

If you know opposite angles in a quadrilateral are both 90, all you can really say is that the remaining two angles sum to 180.