You have seen how to use geometry and the properties of polygons to help you find unknown lengths and angles in various quadrilaterals and polygons. We will now extend this work to proving some of the properties and to solving riders. A conjecture is the mathematicians way of saying I believe that this is true, but I have no proof. The following worked examples will help make this clearer.
Given quadrilateral ABCD, with
and
, prove that
and
.
We draw the following diagram and construct the diagonals.
Given:
and
. We need to prove
and
. In the formal language of maths we say that we are required to prove (RTP)
and
.
Similarly we find that:
In parallelogram ABCD, the bisectors of the angles (AW, BX, CY and DZ) have been constructed:
You are also given that
,
,
,
,
, and
.
Prove that MNOP is a parallelogram.
Given:
,
,
,
,
, and
. RTP: MNOP is a parallelogram.
MNOP is a parallelogram (both pairs opp.
's
, and therefore both pairs opp. sides parallel too)
It is very important to note that a single counter example disproves a conjecture. Also numerous specific supporting examples do not prove a conjecture.
