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My goal is to draw a figure exactly like this
The best I could do coding this was:

documentclassarticle
usepackageamsmath,amsfonts,amssymb
usepackagetikz
usepackagefloat

begindocument

begintikzpicture 

 draw (1, 0) to (1, 4); 
 draw (1, 0) to (4, 0); 
 draw (1, 4) to (4, 0);

endtikzpicture

begintikzpicture

 draw (1, 0) to (1, 3);
 draw (1, 0) to (3, 0);
 draw (1, 3) to (3, 0);

endtikzpicture 

enddocument

Which ends up looking like this. I tried putting in the values and angle points but it ended up looking much worse, so I omitted them:

So how do I get my figure to look exactly like the first figure? If it means anything, I'm using a MacBook Pro and I'm using the TexShop application.

Welcome to TeX-SE! The issue you are seeing is caused by the fact that an empty line tells TeX to start a new paragraph. So all you need to do is to remove the empty line. Here I go a slightly different route and put the second triangle in a scope that is used to move it to the right. This allows you to more easily control the distance between the triangles, and their vertical relative alignment. Please note also that it is advantageous to draw them in one stretch and close them with -- cycle because then the line joins look better.

documentclassarticle
usepackageamsmath,amsfonts,amssymb
usepackagetikz
usepackagefloat

begindocument

begintikzpicture
 draw (1, 0) node[below left]$A$ -- node[midway,left]$x+29$
 (1, 4) node[above left] $B$ 
 -- (4, 0) node[below right]$C$ -- node[midway,below]$21$
 cycle; 
beginscope[xshift=5cm,yshift=0.5cm]
 draw (1, 0) node[below left]$D$ 
 -- node[midway,left]$12$ (1, 3)
 node[above left]$E$ 
 -- (3, 0) node[below right]$F$ -- node[midway,below]$x$
 (1,0)-- cycle ;
endscope
endtikzpicture 

enddocument

an alternative, using relative coordinates tikz libraryquotes for labeling lines in triangles:

documentclassarticle
usepackageamsmath,amssymb % amsfonts is loaded by amsymb
usepackagetikz
usetikzlibraryquotes

begindocument

begintikzpicture
 draw (0, 0) node[below] A to["$x+29$"] ++ (0, 4) node[above] B
 to ++ (3,-4) node[below] C
 to["$21$"] cycle;
 draw (5,.5) node[below] D to["$12$"] ++ (0, 3) node[above] E
 to ++ (2,-3) node[below] F
 to["$x$"] cycle;
endtikzpicture

enddocument