Suppose you have a scalene triangle as shown in the picture below. Let's call it triangle $ABC$.
Inside that triangle you do the following things: Draw a point $F$, that lies on $\overline {AB}$, and so that $\overline {AF} =1$. Draw a point $G$, that lies on $\overline {BC}$, so that $\overline {CG}= 1$. Unite the points $F$ and $G$. Then draw the midpoints of $\overline {AC}$ and $\overline {FG}$. Let's call them $I$ and $H$ respectively. And then draw the line segment $\overline {IH}$. You will have the following figure:
But what I am really interested in is the quadrilateral, so here is a picture of it a little bit zoomed
This was a part of a problem I was solving, but I added the fact that the triangle was scalene. While I was solving the problem I thought that the quadrilaterals $AFHI$ and $HICG$ were congruent, and that there might exist some kind of relationship between their angles. I first thought $ \angle {FAI} \cong \angle {ICG}$, but because the triangle is scalene, that can´t be true. Then I thought that probably one of the angles of $AFHI$ could be congruent with one of the angles of $HICG$. So I drew in Geogebra but found that none of them were congruent. I thought that if two figures were congruent their angles would be congruent too. Or are this figures still congruent? And what relationship exists between their angles?.