中考數(shù)學(xué)二輪復(fù)習(xí)幾何模型重點(diǎn)突破講練專題06 三角形中的雙角平分線模型（原卷版）

專題06三角形中的雙角平分線模型【模型1】雙角平分線模型如圖，已知在SKIPIF1<0中，BO，CO分別是SKIPIF1<0，SKIPIF1<0的平分線，根據(jù)角平分線的性質(zhì)和三角形內(nèi)角和定理，可得SKIPIF1<0。【模型2】一內(nèi)角一外角平分線模型如圖，已知在SKIPIF1<0中，BP，CP分別是SKIPIF1<0，SKIPIF1<0的平分線，SKIPIF1<0SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0；SKIPIF1<0；SKIPIF1<0；SKIPIF1<0【模型3】雙外角平分線模型如圖，已知在SKIPIF1<0中，BP，CP分別是SKIPIF1<0，SKIPIF1<0的平分線，根據(jù)外角定理，SKIPIF1<0，SKIPIF1<0，又SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；SKIPIF1<0SKIPIF1<0；【例1】如圖，在△ABC中，∠ABC和∠ACB的角平分線交于點(diǎn)O，延長BO與∠ACB的外角平分線交于點(diǎn)D，若∠BOC＝130°，則∠D＝_____【例2】如圖，已知△ABC，O是△ABC內(nèi)的一點(diǎn)，連接OB、OC，將∠ABO、∠ACO分別記為∠1、∠2，則∠1、∠2、∠A、∠O四個(gè)角之間的數(shù)量關(guān)系是（

）A．∠1+∠0=∠A+∠2 B．∠1+∠2+∠A+∠O=180°C．∠1+∠2+∠A+∠O=360° D．∠1+∠2+∠A=∠O【例3】（1）問題發(fā)現(xiàn)：如圖1，在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0和SKIPIF1<0的平分線交于SKIPIF1<0，則SKIPIF1<0的度數(shù)是______（2）類比探究：如圖2，在SKIPIF1<0中，SKIPIF1<0的平分線和SKIPIF1<0的外角SKIPIF1<0的角平分線交于SKIPIF1<0，則SKIPIF1<0與SKIPIF1<0的關(guān)系是______，并說明理由．（3）類比延伸：如圖3，在SKIPIF1<0中，SKIPIF1<0外角SKIPIF1<0的角平分線和SKIPIF1<0的外角SKIPIF1<0的角平分線交于SKIPIF1<0，請直接寫出SKIPIF1<0與SKIPIF1<0的關(guān)系是______．一、單選題1．如圖，在△ABC中，∠ABC和∠ACB的外角平分線交于點(diǎn)O，設(shè)∠A=m，則∠BOC=（）A． B． C． D．2．如圖：SKIPIF1<0、SKIPIF1<0是SKIPIF1<0、SKIPIF1<0的角平分線，SKIPIF1<0，SKIPIF1<0（

）A．∠BPC=70o B．∠BPC=140oC．∠BPC=110o D．∠BPC=40o3．如圖，△ABC中，∠E=18°，BE平分∠ABC，CE平分∠ACD，則∠A等于（）A．36° B．30° C．20° D．18°4．如圖，SKIPIF1<0中，SKIPIF1<0與SKIPIF1<0的平分線交于點(diǎn)SKIPIF1<0，過點(diǎn)SKIPIF1<0作SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0，交SKIPIF1<0于點(diǎn)SKIPIF1<0，那么下列結(jié)論：①SKIPIF1<0和SKIPIF1<0都是等腰三角形②SKIPIF1<0；③SKIPIF1<0；④若SKIPIF1<0，則SKIPIF1<0．其中正確的有（

）個(gè)A．1 B．2 C．3 D．45．如圖，SKIPIF1<0，SKIPIF1<0的角平分線交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的度數(shù)（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題6．如圖，在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0和SKIPIF1<0的平分線交于點(diǎn)SKIPIF1<0，得SKIPIF1<0，SKIPIF1<0和SKIPIF1<0的平分線交于點(diǎn)SKIPIF1<0，得SKIPIF1<0；SKIPIF1<0；SKIPIF1<0和SKIPIF1<0的平分線交于點(diǎn)SKIPIF1<0，則SKIPIF1<0__．（用SKIPIF1<0表示）7．如圖，在△SKIPIF1<0中，SKIPIF1<0，如果SKIPIF1<0與SKIPIF1<0的平分線交于點(diǎn)SKIPIF1<0，那么SKIPIF1<0＝_________度．8．如圖在△ABC中，BO，CO分別平分∠ABC，∠ACB，交于O，CE為外角∠ACD的平分線，交BO的延長線于點(diǎn)E，記SKIPIF1<0，SKIPIF1<0，則以下結(jié)論①SKIPIF1<0，②SKIPIF1<0，③SKIPIF1<0，④SKIPIF1<0，正確的是________．（把所有正確的結(jié)論的序號寫在橫線上）9．如圖，SKIPIF1<0的角平分線SKIPIF1<0、SKIPIF1<0相交于點(diǎn)SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0______．10．如圖，已知SKIPIF1<0，SKIPIF1<0是角平分線且SKIPIF1<0，作SKIPIF1<0的垂直平分線交SKIPIF1<0于點(diǎn)F，作SKIPIF1<0，則SKIPIF1<0周長為________．11．如圖，在△ABC中，∠A=60°，BD、CD分別平分∠ABC、∠ACB，M、N、Q分別在DB、DC、BC的延長線上，BE、CE分別平分∠MBC、∠BCN，BF、CF分別平分∠EBC、∠ECQ，則∠F=________．三、解答題12．（1）如圖所示，在SKIPIF1<0中，SKIPIF1<0分別是SKIPIF1<0和SKIPIF1<0的平分線，證明：SKIPIF1<0．（2）如圖所示，SKIPIF1<0的外角平分線SKIPIF1<0和SKIPIF1<0相交于點(diǎn)D，證明：SKIPIF1<0．（3）如圖所示，SKIPIF1<0的內(nèi)角平分線SKIPIF1<0和外角平分線SKIPIF1<0相交于點(diǎn)D，證明：SKIPIF1<0．13．如圖，在△ABC中，∠ABC、∠ACB的平分線相交于點(diǎn)O①若∠ABC=40°，∠ACB=50°，則∠BOC的度數(shù)為；②若∠A=76°，則∠BOC的度數(shù)為；③你能找出∠A與∠BOC之間的數(shù)量關(guān)系嗎？說明理由14．如圖，在△ABC中，∠ABC和∠ACB的平分線相交于點(diǎn)P．（1）若∠ABC+∠ACB＝130°，求∠BPC的度數(shù)．（2）當(dāng)∠A為多少度時(shí)，∠BPC＝3∠A？15．?dāng)?shù)學(xué)思想運(yùn)用：(1)如圖①所示，△ABC的外角平分線交于G，若∠A=80°，則∠BGC=______°，請你猜測∠BGC和∠A的數(shù)量關(guān)系：_______________．(2)如圖②所示，若△ABC的內(nèi)角平分線交于點(diǎn)I，若∠A=50°，則∠BIC=______°，請你猜測∠BIC和∠A的數(shù)量關(guān)系：__________________．(3)已知，如圖③，△ABC中，SKIPIF1<0的平分線與的平分線交于點(diǎn)，請你猜測∠D和∠A的數(shù)量關(guān)系：____________________．若，求的度數(shù)（寫出求解過程）．16．SKIPIF1<0中，SKIPIF1<0．（1）如圖①，若點(diǎn)SKIPIF1<0是SKIPIF1<0與SKIPIF1<0平分線的交點(diǎn)，求SKIPIF1<0的度數(shù)；（2）如圖②，若點(diǎn)SKIPIF1<0是SKIPIF1<0與SKIPIF1<0平分線的交點(diǎn)，求SKIPIF1<0的度數(shù)；（3）如圖③，若點(diǎn)SKIPIF1<0是SKIPIF1<0與SKIPIF1<0平分線的交點(diǎn)，求SKIPIF1<0的度數(shù)；（4）若SKIPIF1<0.請直接寫出圖①，②，③中SKIPIF1<0的度數(shù)，(用含SKIPIF1<0的代數(shù)式表示)17．【問題背景】（1）如圖1的圖形我們把它稱為“8字形”，請說明∠A+∠B=∠C+∠D；【簡單應(yīng)用】（2）如圖2，AP、CP分別平分∠BAD．∠BCD