# The labels blue means : lengths PQ = QR The label green means: lengths QS = QT. The given angle PQS = 24 (deg)

Triangles : isoceles and angles inside with other triangle
We have one main isoceles and another one inside of it. I have attached here a diagram, and we wish to find the angle in red:

The labels blue means : lengths $PQ=QR$ The label green means : lengths $QS=QT$. The given angle $PQS=24$ (deg)
We wish to find angle in red, angle RST.
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Lilliana Mason
Explanation:
Let $\stackrel{^}{QRP}=\stackrel{^}{RPQ}=x$, so $\stackrel{^}{RQP}=180-2x$, so $\stackrel{^}{RQS}=156-2x$, so the sum of the angles $\stackrel{^}{QST}$ and $\stackrel{^}{QTS}$ is $24+2x$, so they are both $x+12$, so $\stackrel{^}{RTS}=168-x$, so $\stackrel{^}{RST}=12$.
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mangicele4s
Step 1

Since PRQ and STQ are isosceles and share vertex Q, let the circle with center Q and radius QT intersect PQ at U and PR at V, and join TU. Thus, since $\frac{QU}{QP}=\frac{QT}{QR}$ then $TU\parallel RP$ and arcs so that $\mathrm{\angle }VST=\mathrm{\angle }STU$.
Step 2
But $\mathrm{\angle }SQU=2\mathrm{\angle }STU$.
Therefore $\mathrm{\angle }SQU=2\mathrm{\angle }VST$ that is $\mathrm{\angle }SQP=2\mathrm{\angle }RST$.
Thus under the given conditions, the angle we are looking for is always half the given angle.