Relationship between the angles of two triangles (College level)

[Lingogetter]

I'm stuck on one specific part of a homework problem. I'll write out the whole problem, then discuss the part I'm confused on:

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Given 4 congruent polygons that meet at a vertex of a polyhedron. If the measure of the interior angles of each polygon at this vertex is alpha (or angle VUT in my image) and the measure of the dihedral angle between 2 adjacent polygonal faces is beta (or angle QRS in my image) find an equation that relates alpha and beta so you could find alpha when given beta or find beta when given alpha.

(Note: the polygons that make up the faces aren't necessarily triangles, This is just an image of the truncated vertex of the polyhedron.)

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So here's where I'm stuck. There's supposed to be a relationship between the angles of triangle QRT and angle alpha (VUT). But... I can't find it. The only thing I know is that RQ is perpendicular to UT because it's part of the dihedral angle. If I could find that relationship, I think I could find the equation that relates alpha and beta.

 

[cougartracks4]

Angle UTV

Let's capitalize on how the polygon faces were truncated into isosceles triangles. Now considering triangle UVT, angle VUT is alpha and the other two must be congruent. The angles will sum to 180 degrees, so angle UTV is (180-alpha)/2. Can we go anywhere with that?

 

[cougartracks4]

Point Y

The attachment you included looks promising. I was looking at point Y and assumed that segment could be created so that it is perpendicular to QS and is bisecting angle QRS.
Then I can represent each angle of triangle QRY. Angle QRY is beta/2. Angle RYQ is 90. Angle RQY is 90-beta/2.
I think segment QR is going to be the ticket to this problem with some sine or cosine magic... I'll let you know if I get any other promising leads.

 

[cougartracks4]

Does anyone see similar triangles?

I've got segment QR in two triangles where one involves alpha and the other involves beta.

Triangle QRT has an angle RQT of alpha/2 and a right angle at QRT. Then alpha/2 = arccos(QR/QT).

Triangle QRY has an angle QRY of beta/2 and a right angle at QYR. Then beta/2 = arcsin(QY/QR).

Solving the above equations for QR
QR = QTcos(alpha/2)
QR = QY / sin(beta/2)

So if I could find a way to relate QT to QY... Does anyone see similar triangles?
Maybe triangle QYT could be helpful?
Could constructing Q to be the midpoint of VT be helpful? Or would we want R to be the midpoint of UT?
Man, I feel so close to the answer!