The first thing some solvers did was to
play with the interactivity to see what things they could change
and what they couldn't.
Playing with the interactivity told us that if we change one thing,
we can't change anything else without affecting the thing we
changed first. For example, we moved point B to change the angle at
A, and this then fixed the rest of the shape (when we tried to move
C, the angle at A changed). We checked why this was the case by
constructing the shape with ruler and compasses. If we picked a
particular angle at A, there was only one possible quadrilateral
with the correct lengths. We thought about this in terms of the two
triangles ABD and BCD. This links to our work on congruent
triangles.
We then started to investigate the shapes we could make.
The smallest that angle A can be is just above $0^{\circ}$ giving a
long thin quadrilateral:
Angle C is also very close to $0^{\circ}$, and angles B and D are
both very close to $180^{\circ}$.
Then we made angle A as big as we could. If it went over
$180^{\circ}$ the quadrilateral would no longer be convex and if it
was equal to $180^{\circ}$ we would have a triangle. This picture
shows what the shape looks like when angle A is close to
$180^{\circ}$
When angle A is $180^{\circ}$, angles B and D are as small as they
can possibly be, and angle C is as large as it can possibly be. If
we tried to make B and D smaller or make C bigger, A got bigger
too.
We drew the shape as a triangle to help us to work out the minimum
values of B and D, and the maximum value of C.
BCD is a triangle with sides $7$, $6$ and $5$ units.We used the
Cosine Rule to work out angle B.
${CD}^2 = {BC}^2 +{BD}^22(BD)(CD)cosB$
$6^2=5^2+7^22 \times 5 \times 7 \times cosB$
We rearranged this to get
$$ cos B = 0.543$$
which gave $$B = 57.1^{\circ}$$
Then we worked out angles C and D in the same way.
Angle C is $78.5^{\circ}$ and angle D is $44.4^{\circ}$ (all our
numbers have been rounded to $3$ significant figures).
This shows that all the quadrilaterals
will have angle A between 0 and 180, B between 57.1 and 180, C
between 0 and 78.5 and D between 44.4 and 180. Can you find any
special quadrilaterals with angles in these ranges? Are there any
with right angles in? Any cyclic quadrilaterals? We would love to
hear about anything interesting you discover! 
