Question

【題目】如圖，∠A=90°，E為BC上一點(diǎn)，A點(diǎn)和E點(diǎn)關(guān)于BD對(duì)稱，B點(diǎn)、C點(diǎn)關(guān)于DE對(duì)稱，求∠ABC和∠C的度數(shù)．

Answer 1

【答案】解：∵A點(diǎn)和E點(diǎn)關(guān)于BD對(duì)稱，

∴∠ABD=∠EBD，即∠ABC=2∠ABD=2∠EBD．

又B點(diǎn)、C點(diǎn)關(guān)于DE對(duì)稱，

∴∠DBE=∠C，∠ABC=2∠C．

∵∠A=90°，

∴∠ABC+∠C=2∠C+∠C=3∠C=90°．

∴∠C=30°

∴∠ABC=2∠C=60°．

【解析】依據(jù)軸對(duì)稱圖形的性質(zhì)可得到∠ABD=∠EBD，然后依據(jù)B點(diǎn)、C點(diǎn)關(guān)于DE對(duì)稱可得∠DBE=∠BCD，結(jié)合上式可得：∠ABC=2∠BCD，且∠ABC+∠BCD=90°，進(jìn)而求得∠ABC、∠C的值.
【考點(diǎn)精析】關(guān)于本題考查的軸對(duì)稱的性質(zhì)，需要了解關(guān)于某條直線對(duì)稱的兩個(gè)圖形是全等形；如果兩個(gè)圖形關(guān)于某直線對(duì)稱，那么對(duì)稱軸是對(duì)應(yīng)點(diǎn)連線的垂直平分線；兩個(gè)圖形關(guān)于某直線對(duì)稱，如果它們的對(duì)應(yīng)線段或延長(zhǎng)線相交，那么交點(diǎn)在對(duì)稱軸上才能得出正確答案．