Start with the given line segment and two angles.

Note:
The two given angles are only there to indicate the measure of the two angles.
The lines making up the given angles have random lengths that have no significance in the construction.

The first part of this construction (steps 1  4) is to copy a line segment to
form one side of the new triangle. (See Copying a Line Segment). 
1. Mark a point A that will be one vertex of the new triangle. 

2. Set the compasses' width to the length of the segment AB. 

3. With the compasses' point on A, make an arc near the future vertex B of the triangle. 

4. Mark a point B on this arc.
Then draw the line AB. This will be one side of the new triangle. 

What we do now is essentially copy the two angles. (See Copying an Angle). 
5. With the compasses at any convenient width, draw an arc across both lines of the given angle A. 

6. Without changing the compasses' width, draw an arc at point A on the new triangle.
The arc must cross AB and also cross the future side of the triangle. 

7. Set the compasses to the arc width at the given angle A.
This the distance between the points where the arc intersects the sides of the angle. 

8. Near point A draw an arc in a similar position so it crosses the arc drawn earlier.
This, in effect, 'copies' the measure of the angle at P to the angle at A. 

9. Draw a line from A through the point where the arcs intersect.
This will become the second side of the triangle. Draw it long. 

10. Repeat this process at B. Copying the angle measure from the given angle B to the new triangle at B.
The point where the lines intersect is C, the third vertex of the triangle. 

Done. The blue triangle ABC has the side and two angle measures desired. 
