First align the shapes then resize the triangle to fit:
Align the corner of the triangle and shape where you know they will meet:
Zoom all the way in and place the triangle corner node precisely on the shape corner node.
Zoom out, select the triangle, select the rotate tool and click right on the triangle corner node that you placed. That places the point of rotation on the corner of the triangle. Click+Drag the other triangle corner on the side next to shape until the triangle edge aligns with shape edge.
Resize triangle: this will take some trial and error. Select triangle and scale while holding Shift until the side length matches shape side length. The resizing will move the triangle away a little bit so zoom all the way in and move it back into place. Alternately to scale the triangle using % select the triangle and open Object>Transform>Scale and enter percentages. Undo and redo until size is right then move triangle back into place.