Start with the given two angles A, C and nonincluded side AB.

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, extending it to the left about 50%. This will be one side of the new triangle. 

We now copy the given angle A to the end of the line AB
(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 given angle A to the triangle. 

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. 

We now copy the given angle C to be adjacent
to the previous angle A, in effect adding them.
(See Copying an Angle). 
10. With the compasses at any convenient width,
draw an arc across both lines of the given angle C. 

11. Without changing the compasses' width, draw an arc at point A on the new triangle.
The arc should be next to the previous angle and be fairly wide. 

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

13. 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 given angle C to be adjacent to angle A. 

14. Draw a line from A through the point where the new arcs intersect.
What we have done here is to essentially add the two angles A and C. The third angle at A is 180° minus that sum.
This is the measure of the triangle's angle B, which we now copy


We now copy the third angle at A to be
the triangle's angle B
(See Copying an Angle). 
15. With the compasses at any convenient width,
draw an arc across both lines of the angle EAD. 

16. Without changing the compasses' width, draw an arc at point B on the new triangle.


17. Set the compasses to the arc width at the angle EAD.
This the distance between the points where the arc intersects the sides of the angle. 

18. Near point B draw an arc in a similar position so it crosses the arc drawn earlier.
This, in effect, 'copies' the measure of the given angle EAD to be the new triangle's angle B. 

19. Draw a line from B through the point where the new arcs intersect. Continue the line far enough to complete the triangle.


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