## Similar Triangles with a Shared Angle: The “upside down” case

After introducing the Dilation Axiom, we studied several “Thales figures” of
similar triangles like ABC and ADE here. The triangles have a shared angle
BAC = angle DAE and sides that are proportional: AD/AD = AE/AC. This is a special
case of SAS similarity. We can think of these triangles as “nested.” Not only
are all the corresponding angles congruent and the corresponding sides proportional,
but we have learned that line DE is parallel to line BC.

However, there are other ways for similar triangles to share an angle. Imagine
reflecting triangle ADE in the angle bisector of angle BAC. Then the reflected
triangle would look like the new triangle below. In this figure, triangle ADE
(above) is *congruent* to triangle AGF (below), so triangle AGF
is *similar* to triangle ABC. We can think of these triangles as
“nested but twisted”. Line FG is never parallel to line BC, except in the isosceles
case where AB = AC.

In this figure the two triangles clearly have one angle the same. But the
correspondence of vertices adn thus the correspondence of sides is different.

What vertices in the second triangle correspond to vertices B and C in triangle
ABC? _________

What equality of ratios must be true for triangle ABC to be similar to triangle
AGF by SAS? _____

We can think of this figure as a “reflected Thales” figure.

In other examples, the figure may be a little harder to analyze because of
overlaps, but the relationship is exactly the same, so just draw a separate
picture of each triangle and figure out the corresponding vertices and what
the ratio must be for SAS.

Here are a couple of examples. In the first, side BC intersects FG. In the
second, the vertex G actually is the same point as C, so we have triangle ABC
is similar to triangle ACF.

A final pair of figures shows what happens if two triangles share a pair of
vertical angles. Again, there are two ways that the equal angles and the vertices
can correspond to give similarity by SAS. Again there are two kinds of figures,
one with parallel corresponding sides and the other not.

In each case, what is the equality of ratios that will show these triangles
are similar by SAS?

Figure on the left _______________ Figure on the right ____________________