Using the law of cosines and a spreadsheet, I identified many
simple triangles which share an angle.  In the list below, I
give the sides of Triangle A and Triangle B, their shared
angle, and the cosine of that angle.


Tri A	Tri B	Shared Angle	Cos(A)	Tri C
4 4 7	1 2 2	28.95502437	7/8  	2 3 4	
3 3 5	5 7 9	33.55730976	5/6  	5 8 9	
3 3 4	4 7 9	48.1896851	2/3  	7 8 9	
1 3 3	3 8 9	19.18813645	17/18 
3 5 7	3 7 8	21.7867893	13/14 
3 6 7	4 7 9	25.2087653	19/21 
3 4 6	4 8 9	26.38432975	43/48 
5 5 8	3 4 5	36.86989765	4/5  
3 5 7	5 7 8	38.2132107	11/14 
5 6 9	2 3 3	38.94244127	7/9  
2 2 3	4 5 6	41.40962211	3/4  
4 5 7	5 6 7	44.4153086	5/7  
2 3 4	6 7 8	46.56746344	11/16 
2 5 6	4 4 5	51.31781255	5/8  
3 4 5	5 5 6	53.13010235	3/5  
2 8 9	6 6 7	54.31466529	7/12 
4 5 6	8 8 9	55.77113367	9/16 
3 6 7	7 8 9	58.41186449	11/21 
3 7 8	5 7 8	60		1/2  
4 7 7	7 8 9	73.3984504	2/7  
1 2 2	6 7 8	75.52248781	1/4  
2 5 5	5 6 7	78.46304097	1/5  
2 7 7	5 7 8	81.7867893	1/7  
1 4 4	4 5 6	82.81924422	1/8  
2 9 9	3 3 4	83.62062979	1/9  
1 5 5	5 8 9	84.26082952	1/10 
1 8 8	4 6 7	86.4166783	1/16