In triangles ABC and PQR, AB/RQ = BC/QP = CA/PR, with angle A=80 degrees and angle B=60 degrees. Find angle P.
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Step-by-Step Solution
Step 1: Identify Similar Triangles
Given the ratios of corresponding sides RQAB=QPBC=PRCA, we can conclude that triangle ABC is similar to triangle RQP by the SSS (Side-Side-Side) similarity criterion. It's crucial to match the vertices correctly based on the given side ratios.
Step 2: Determine Corresponding Angles
Since the triangles △ABC and △RQP are similar, their corresponding angles must be equal. By matching the vertices from the similarity statement, we find that ∠A corresponds to ∠R, ∠B corresponds to ∠Q, and ∠C corresponds to ∠P.
Step 3: Calculate Angle C in Triangle ABC
The sum of angles in any triangle is 180∘. We are given ∠A=80∘ and ∠B=60∘. We can use this property to find the measure of ∠C in triangle ABC.
Step 4: Find Angle P
From the similarity of △ABC and △RQP, we established that ∠C corresponds to ∠P. Therefore, ∠P must be equal to ∠C, which we calculated to be 40∘.