8. The diagram below contains two overlapping triangles. - First, sketch the triangles separately (and label the vertices and side lengths) so that they do not overlap. - Then, justify why the two triangles are similar and write an appropriate similarity statemes - Finally, determine the value of $m$ and $n$. \[ \begin{array}{l} m \angle P=n^{2}-7^{\circ} \\ m \angle R=54^{\circ} \\ m \angle S=33^{\circ} \\ m \angle T=33^{\circ} \end{array} \]
Solution
Step 1 ：The problem provides the measures of angles P, R, S, and T in terms of variables m and n. Specifically, we have \(\angle P = n^2 - 7^\circ\), \(\angle R = 54^\circ\), \(\angle S = 33^\circ\), and \(\angle T = 33^\circ\).
Step 2 ：Since the sum of the angles in a triangle is \(180^\circ\), we can set up two equations to solve for m and n. The first equation will be \(m + 54 + 33 = 180\) and the second equation will be \(n^2 - 7 + 33 + 33 = 180\).
Step 3 ：Solving the first equation gives \(m = 180 - 54 - 33 = 93\).
Step 4 ：Solving the second equation gives \(n^2 = 180 - 59 = 121\). Taking the square root of both sides gives \(n = 11\) or \(n = -11\).
Step 5 ：However, since the measure of an angle cannot be negative, we discard \(n = -11\). Therefore, the values of m and n are \(93^\circ\) and \(11^\circ\) respectively.
Step 6 ：Final Answer: \(m = \boxed{93}\) and \(n = \boxed{11}\).
