Question

given that point \\(t\\) is the incenter, \\(\angle qsr = 105^{\circ}\\), and \\(\angle rqs = 35^{\circ}\\), determine \\(\angle srt\\).

Explanation:

Find the third angle of the triangle

Using the Triangle Angle Sum Theorem in \(\triangle QSR\):

$$ \angle QSR + \angle RQS + \angle SRQ = 180^\circ $$

Substitute the given values:

$$ 105^\circ + 35^\circ + \angle SRQ = 180^\circ $$

$$ 140^\circ + \angle SRQ = 180^\circ $$

$$ \angle SRQ = 40^\circ $$

Apply the incenter property to find the target angle

Using the Incenter Theorem, the incenter \(T\) is the intersection of the angle bisectors of \(\triangle QSR\).
Therefore, the segment \(RT\) bisects \(\angle SRQ\):

$$ \angle SRT = \frac{1}{2} \angle SRQ $$

$$ \angle SRT = \frac{1}{2} (40^\circ) = 20^\circ $$

Answer:

\(20^\circ\)