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HMMT 二月 2014 · 几何 · 第 4 题

HMMT February 2014 — Geometry — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. In quadrilateral ABCD , \ DAC = 98 , \ DBC = 82 , \ BCD = 70 , and BC = AD . Find \ ACD . B A 82 98 C D
解析
  1. In quadrilateral ABCD , ∠ DAC = 98 , ∠ DBC = 82 , ∠ BCD = 70 , and BC = AD . Find ∠ ACD . B A ◦ 82 ◦ 98 C D ′ ′ Answer: 28 Let B be the reflection of B across CD . Note that AD = BC , and ∠ DAC + ∠ CB D = ◦ ′ ′ ′ ◦ 180 , so ACB D is a cyclic trapezoid. Thus, ACB D is an isosceles trapezoid, so ∠ ACB = 98 . Note ′ ◦ ′ ′ ◦ ◦ ◦ that ∠ DCB = ∠ BCD = 70 , so ∠ ACD = ∠ ACB − ∠ DCB = 98 − 70 = 28 .