Question

what are the values of s and t?
(there is a triangle pqr with pq, pr, qr sides. angle at p is 31 degrees. pr and qr have equal marks, indicating isosceles triangle. angle at q is s, angle at r is t.)
$s = \square^\circ$
$t = \square^\circ$

Explanation:

Step1: Identify the triangle type

The triangle \( \triangle PQR \) has two sides marked as equal (the red tick marks), so it is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. The side \( PR \) and \( QR \) are equal (from the tick marks), so the angles opposite them, \( \angle Q \) (which is \( s \)) and \( \angle P \) (which is \( 31^\circ \)) are equal. So \( s = 31^\circ \).

Step2: Calculate angle \( t \)

The sum of the interior angles of a triangle is \( 180^\circ \). So we can find \( t \) (which is \( \angle R \)) by subtracting the sum of the other two angles from \( 180^\circ \). The other two angles are \( s = 31^\circ \) and \( \angle P = 31^\circ \). So \( t = 180^\circ - 31^\circ - 31^\circ \).
\[
t = 180 - 31 - 31 = 118^\circ
\]

Answer:

\( s = 31^\circ \), \( t = 118^\circ \)