Since you didn't specify which two opposite sides are parallel,
there are four solutions for x.  Only one of them, the first, x=40°
defines a convex pentagon. The other three are concave polygons.  A 
concave polygon has one or more "sunk-in places" where one or more 
of the internal angles are greater than 180°, and are called REFLEX 
angles.  Examples of pentagons defined by the four solutions are all 
drawn below.

Since two sides are parallel, one of the sides must be a transversal
between those two parallel sides.  This means that two of the interior
angles are supplementary.

The sum of the interior angles of a pentagon are

(n-2)×180° = (5-2)×180° = 3×180° = 540°

Let the 5th interior angle be y, then

x+2x+3x+4x+y = 540°
       10x+y = 540°
           y = 540°-10x

So the 5 angles are

x, 2x, 3x, 4x, 540°-10x

There are 9 cases for the 2 supplementary angles:

Case 1: x and 2x are supplementary:
x+2x=180° or 3x=180° or x=60°, so the 5 angles would be
60°, 120°, 180°, 240°, -60° not possible

Case 2: x and 3x are supplementary:
x+3x=180° or 4x=180° or x=45°, so the 5 angles would be
45°, 90°, 135°, 180°, 90°, not possible because an 
interior angle cannot be 180°, for that would have only 
4 sides.

Case 3: x and 4x are supplementary:
x+4x=180° or 5x=180° or x=36°, so the 5 angles would be 
36°, 72°, 108°, 144°, 180°, not possible because an interior
angle cannot be 180°, for that would have only 4 sides.

Case 4: x and 540°-10x are supplementary:
x+540°-10x=180° or -9x=-360° or x=40°, so the 5 angles would be
40°, 80°, 120°, 160°, 140°, which IS POSSIBLE!  

Case 5: 2x and 3x are supplementary:
x+3x=180° or 4x=180° or x=45°, so the 5 angles would be  
45°, 90°, 135°, 180°, 90°, not possible because an interior
angle cannot be 180°, for that would have only 4 sides.

Case 6: 2x and 4x are supplementary:
2x+4x=180° or 6x=180° or x=30°, so the 5 angles would be 
30°, 60°, 90°, 120°, 240°, which IS POSSIBLE!

Case 7: 2x and 540°-10x are supplementary:
2x+540°-10x=180° or -8x=-360° or x=45°, same as case 5

Case 8: 3x and 4x are supplementary:
3x+4x=180° or 7x=180° or x=°, so the 5 angles would be 
°, °,°, °, °, which IS POSSIBLE!

Case 9: 3x and 540°-10x are supplementary:
3x+540°-10x=180° or -7x=-360° or x=°, so the 5 angles 
would be °, °, °, ,°
which IS POSSIBLE!

Case 10: 4x and 540°-10x are supplementary:
4x+540°-10x=180° or -6x=-360° or x=60°, same as case 1.  NOT 
POSSIBLE!

So there are four solutions for x:

1. x=40°, 2x=80°, 3x=120°, 4x=160°, 5th angle=140°
2. x=30°, 2x=60°, 3x=90°, 4x=120°, 5th angle=240°
3. x=°, 2x=°, 3x=°, 4x=°, 5th angle=°
4. x=°, 2x=°, 3x=°, 4x=, 5th angle=°.

Here are typical ways the 5 pentagons could be.  The 
internal angles which are more than 180° which cause
the last 3 pentagons to be concave are indicated with
a red arc. 

1. 40°, 80°, 120°, 160°, 140°



2. 30°, 60°, 90°, 120°, 240°,




3. °, °,°, °, °



4. °, °, °, ,°



Edwin