Discussion
In triangle PQR, \(\angle P = 55^{\circ}\) and QR = 18 cm. In which of the following cases, \(\Delta PQR\) can be an obtuse scalene triangle ?
Answer: B
The triangle can be obtuse and scalene when \(\angle R = 15^{\circ}\) and PR \(\gt\) 18 cm.
With \(\angle P = 55^{\circ}\) and \(\angle R = 15^{\circ}\), \(\angle Q = 110^{\circ}\), an obtuse angle. The longest side is opposite the largest angle, so PR (opposite Q) is longer than QR = 18 cm, as stated, and all three angles differ, so the sides do too.
With \(\angle R = 25^{\circ}\), PQ = QR would force \(\angle R = \angle P\), a contradiction. \(\angle R = 65^{\circ}\) gives \(\angle Q = 60^{\circ}\), not obtuse. \(\angle R = 35^{\circ}\) gives a right angle at Q.
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