PUMaC 2022 · 几何（B 组） · 第 1 题

PUMaC 2022 — Geometry (Division B) — Problem 1

A triangle △ ABC is situated on the plane and a point E is given on segment AC. Let D be a point in the plane such that lines AD and BE are parallel. Suppose that ∠ EBC = ◦ ◦ ◦ 25 , ∠ BCA = 32 , and ∠ CAB = 60 . Find the smallest possible value of ∠ DAB in degrees.

解析

A triangle △ ABC is situated on the plane and a point E is given on segment AC. Let D be a point in the plane such that lines AD and BE are parallel. Suppose that ∠ EBC = ◦ ◦ ◦ 25 , ∠ BCA = 32 , and ∠ CAB = 60 . Find the smallest possible value of ∠ DAB in degrees. Proposed by Frank Lu Answer: 63 ◦ ◦ ◦ First, using the angles that we are given, we can compute that ∠ BEC = 180 − 57 = 123 . From here, we have two cases, depending on the positioning of D. In the first case, we have that ∠ DAC = ∠ BEC (rays BE and AD point in opposite directions). In this case, we have ◦ ◦ ∠ DAC = 123 , meaning that ∠ DAB = ∠ DAC − ∠ DAB = 63 . ◦ ◦ In the second case, ∠ DAC = 180 − ∠ BEC = 57 , with rays BE, AD pointing the same way. ◦ In this case, we have that ∠ DAB = ∠ DAC + ∠ CAB = 117 . Notice that of these cases, the ◦ smallest value is 63 , which gives us our answer.