Mathematics

Mathematics

Problem

In △PQR\triangle PQR, it is given that PQ=PRPQ = PR. (a) Write the relation between ∠Q\angle Q and ∠R\angle R. (b) If ∠Q=5x−7∘\angle Q = 5x - 7^\circ and ∠R=3x+17∘\angle R = 3x + 17^\circ, determine the value of xx and hence find the measure of each interior angle of the triangle. (c) State whether ∠P\angle P is acute, right, or obtuse. Give reasons for your answer.
Intuitive explanation

An isosceles triangle has a special symmetry: when two sides are equal, the angles opposite those sides must also be equal. This is the key insight that unlocks part (a). Once you use that relationship to set up an equation in part (b), you can solve for xx algebraically, then use the angle sum property (180°180°) to find all three angles. Finally, classifying ∠P\angle P just means checking whether it's less than, equal to, or greater than 90°90°.

Intuitive explanation

An isosceles triangle has a special symmetry: when two sides are equal, the angles opposite those sides must also be equal. This is the key insight that unlocks part (a). Once you use that relationship to set up an equation in part (b), you can solve for xx algebraically, then use the angle sum property (180°180°) to find all three angles. Finally, classifying ∠P\angle P just means checking whether it's less than, equal to, or greater than 90°90°.

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