diagonal of polygon?

Question

Hi,

I have 2 confusion which i always confuse

(1) suppose , i have a paralleogram . if a draw a digonal ....does it bisect its angle ?

I draw it in a piece of paper but i am not sure ....its bit illusive.........HOW do i convince myself regarding the result ?
whats the result ?


(2) suppose, i have a cyclic quadrilateral .........now , if i draw a diagonal , does it bisects its angle ?

I again draw it on a piece of paper.....but its bit illusive ....i cant take a conclusion.

plz help

Answer 1

Both questions, the answer is no.

For the first, try to make your parallelogram thinner, it will be more apparent.
Diagonals of a parallelogram bisect each other, but they are not necessarily angle bisectors.

For the second, try to place 3 of the vertices close to each other, and the fourth far from the other 3.


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Answer 2

The other well-thought out answers are OK, but you should recall your geometry. Consider a parallelogram ABCD with diagonal AC drawn. Since a parallelogram involves parellel lines, the alternate interior angles CAB and ACD are equal, as are the other two angles involved. However, there is nothing further you can say about angles DAB and BCD. If the angles WERE bisected, you could prove that you had either a kite (if one set was bisected) or a rhombus (if both sets were bisected).

Answer 3

(1) A rhombus, a square are those parallelograms, diagonals of which always bisect the angles otherwise not.
(2) A square is the only cyclic quadrilateral, diagonal of which always bisects the angles, otherwise not.

Answer 4

Wow, look at that idiot. "A square is the only cyclic quadrilateral." WRONG. ouch. Try any other rectangle. And there are plenty more too.