新高考數(shù)學(xué)一輪復(fù)習(xí)提升訓(xùn)練7.3 空間角（精練）（原卷版）

7.3空間角（精練）（提升版）題組一題組一線(xiàn)線(xiàn)角1．（2023·全國(guó)·高三專(zhuān)題練習(xí)）已知直三棱柱SKIPIF1<0的所有棱長(zhǎng)都相等，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則SKIPIF1<0與SKIPIF1<0所成角的正弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023·全國(guó)·高三專(zhuān)題練習(xí)（理））已知正四面體ABCD，M為BC中點(diǎn)，N為AD中點(diǎn)，則直線(xiàn)BN與直線(xiàn)DM所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·河南省杞縣高中模擬預(yù)測(cè)（理））如圖，四邊形SKIPIF1<0為圓臺(tái)SKIPIF1<0的軸截面（通過(guò)圓臺(tái)上、下底面兩個(gè)圓心的截面，其形狀為等腰梯形），SKIPIF1<0，C、D分別為OB，SKIPIF1<0的中點(diǎn)，點(diǎn)E為底面圓弧AB的中點(diǎn)，則CD與SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·浙江嘉興·模擬預(yù)測(cè)）如圖，在矩形SKIPIF1<0中，SKIPIF1<0，E，F(xiàn)，G，H分別為邊SKIPIF1<0的中點(diǎn)，將SKIPIF1<0分別沿直線(xiàn)SKIPIF1<0翻折形成四棱錐SKIPIF1<0，下列說(shuō)法正確的是（

）A．異面直線(xiàn)SKIPIF1<0所成角的取值范圍是SKIPIF1<0 B．異面直線(xiàn)SKIPIF1<0所成角的取值范圍是SKIPIF1<0C．異面直線(xiàn)SKIPIF1<0所成角的取值范圍是SKIPIF1<0 D．異面直線(xiàn)SKIPIF1<0所成角的取值范圍是SKIPIF1<05．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，四邊形SKIPIF1<0中，SKIPIF1<0．現(xiàn)將SKIPIF1<0沿SKIPIF1<0折起，當(dāng)二面角SKIPIF1<0處于SKIPIF1<0過(guò)程中，直線(xiàn)SKIPIF1<0與SKIPIF1<0所成角的余弦值取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題組二題組二線(xiàn)面角1．（2023·全國(guó)·高三專(zhuān)題練習(xí)（文））如圖，在四面體ABCD中，SKIPIF1<0平面BCD，SKIPIF1<0，P為AC的中點(diǎn)，則直線(xiàn)BP與AD所成的角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·河南省杞縣）如圖，在三棱柱SKIPIF1<0中，SKIPIF1<0平面ABC，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則異面直線(xiàn)SKIPIF1<0與SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·青海西寧·二模（理））如圖是一個(gè)正方體的平面展開(kāi)圖，則在正方體中，異面直線(xiàn)SKIPIF1<0與SKIPIF1<0所成的角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·全國(guó)·高三專(zhuān)題練習(xí)）如圖，在三棱錐SKIPIF1<0中，SKIPIF1<0，點(diǎn)O?M分別是SKIPIF1<0?SKIPIF1<0的中點(diǎn)，SKIPIF1<0底面SKIPIF1<0.(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求直線(xiàn)SKIPIF1<0與平面SKIPIF1<0所成角的大小.5．（2022·浙江湖州·模擬預(yù)測(cè)）已知四棱錐SKIPIF1<0中，底面SKIPIF1<0為等腰梯形，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是斜邊為SKIPIF1<0的等腰直角三角形．(1)若SKIPIF1<0時(shí)，求證：平面SKIPIF1<0平面SKIPIF1<0；(2)若SKIPIF1<0時(shí)，求直線(xiàn)SKIPIF1<0與平面SKIPIF1<0所成的角的正弦值．6．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，在直三棱柱SKIPIF1<0中，SKIPIF1<0，點(diǎn)SKIPIF1<0分別在棱SKIPIF1<0和棱SKIPIF1<0上，且SKIPIF1<0．(1)設(shè)SKIPIF1<0為SKIPIF1<0中點(diǎn)，求證：SKIPIF1<0平面SKIPIF1<0；(2)求直線(xiàn)SKIPIF1<0與平面SKIPIF1<0所成角的正弦值．題組三題組三二面角1．（2022·北京·景山學(xué)校模擬預(yù)測(cè)）如圖，正三棱柱SKIPIF1<0中，E，F(xiàn)分別是棱SKIPIF1<0，SKIPIF1<0上的點(diǎn)，平面SKIPIF1<0平面SKIPIF1<0，M是AB的中點(diǎn)．(1)證明：SKIPIF1<0平面BEF；(2)若SKIPIF1<0，求平面BEF與平面ABC夾角的大?。?．（2022·湖南·雅禮中學(xué)二模）如圖，在正方體SKIPIF1<0中，點(diǎn)SKIPIF1<0在線(xiàn)段SKIPIF1<0上，SKIPIF1<0，點(diǎn)SKIPIF1<0為線(xiàn)段SKIPIF1<0上的動(dòng)點(diǎn).(1)若SKIPIF1<0SKIPIF1<0平面SKIPIF1<0，求SKIPIF1<0的值；(2)當(dāng)SKIPIF1<0為SKIPIF1<0中點(diǎn)時(shí)，求二面角SKIPIF1<0的正切值.3．（2022·浙江·海寧中學(xué)模擬預(yù)測(cè)）如圖所示，在四邊形ABCD中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0現(xiàn)將SKIPIF1<0沿BD折起，使得點(diǎn)A到E的位置.(1)試在BC邊上確定一點(diǎn)F，使得SKIPIF1<0；(2)若平面SKIPIF1<0平面BCD，求二面角SKIPIF1<0所成角的正切值.4．（2022·廣東惠州·高三階段練習(xí)）如圖，在五面體SKIPIF1<0中，SKIPIF1<0為邊長(zhǎng)為2的等邊三角形，SKIPIF1<0平面SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)求證：平面SKIPIF1<0平面SKIPIF1<0；(2)若直線(xiàn)SKIPIF1<0與平面SKIPIF1<0所成角的正切值為SKIPIF1<0，求平面BDE與平面ABC所成銳二面角的余弦值.5．（2022·山東聊城·三模）已知四邊形ABCD為平行四邊形，E為CD的中點(diǎn)，AB=4，SKIPIF1<0為等邊三角形，將三角形ADE沿AE折起，使點(diǎn)D到達(dá)點(diǎn)P的位置，且平面SKIPIF1<0平面ABCE.(1)求證：SKIPIF1<0；(2)試判斷在線(xiàn)段PB上是否存在點(diǎn)F，使得平面AEF與平面AEP的夾角為45°.若存在，試確定點(diǎn)F的位置；若不存在，請(qǐng)說(shuō)明理由.6．（2022·四川省瀘縣第二中學(xué)模擬預(yù)測(cè)（理））如圖，四棱錐SKIPIF1<0中，SKIPIF1<0，底面ABCD是正方形．且平面SKIPIF1<0平面ABCD，SKIPIF1<0．(1)若SKIPIF1<0，SKIPIF1<0，F(xiàn)為AB的中點(diǎn)，N為BC的中點(diǎn)，證明四邊形MENF為梯形；(2)若點(diǎn)E為PC的中點(diǎn)，試判斷在線(xiàn)段AB上是否存在一點(diǎn)F？使得二面角SKIPIF1<0平面角為SKIPIF1<0．若存在，求出SKIPIF1<0的值．若不存在，請(qǐng)說(shuō)明理由．7．（2022·貴州·貴陽(yáng)一中高三階段練習(xí)（理））如圖，四棱錐SKIPIF1<0中，平面SKIPIF1<0平面SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．SKIPIF1<0是SKIPIF1<0中點(diǎn)，SKIPIF1<0是SKIPIF1<0上一點(diǎn)．(1)是否存在點(diǎn)SKIPIF1<0使得SKIPIF1<0平面SKIPIF1<0，若存在求SKIPIF1<0的長(zhǎng)．若不存在，請(qǐng)說(shuō)明理由；(2)二面角SKIPIF1<0的余弦值為SKIPIF1<0，求SKIPIF1<0的值．題組四空間角的綜合運(yùn)用題組四空間角的綜合運(yùn)用1．（2022·浙江·樂(lè)清市知臨中學(xué)模擬預(yù)測(cè)）如圖，正方體SKIPIF1<0的棱長(zhǎng)為a，E是棱SKIPIF1<0的動(dòng)點(diǎn)，則下列說(shuō)法正確的（

）個(gè)．①若E為SKIPIF1<0的中點(diǎn)，則直線(xiàn)SKIPIF1<0平面SKIPIF1<0②三棱錐SKIPIF1<0的體積為定值SKIPIF1<0③E為SKIPIF1<0的中點(diǎn)時(shí)，直線(xiàn)SKIPIF1<0與平面SKIPIF1<0所成的角正切值為SKIPIF1<0④過(guò)點(diǎn)SKIPIF1<0，C，E的截面的面積的范圍是SKIPIF1<0A．1 B．2 C．3 D．42．（2023·全國(guó)·高三專(zhuān)題練習(xí)（理））在矩形SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，沿對(duì)角線(xiàn)SKIPIF1<0將矩形折成一個(gè)大小為SKIPIF1<0的二面角SKIPIF1<0，若SKIPIF1<0，則下列結(jié)論中正確結(jié)論的個(gè)數(shù)為（

）①四面體SKIPIF1<0外接球的表面積為SKIPIF1<0②點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0之間的距離為SKIPIF1<0③四面體SKIPIF1<0的體積為SKIPIF1<0④異面直線(xiàn)SKIPIF1<0與SKIPIF1<0所成的角為SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·北京·首都師范大學(xué)附屬中學(xué)三模）如圖，在正方體SKIPIF1<0中，SKIPIF1<0為棱SKIPIF1<0上的動(dòng)點(diǎn)，SKIPIF1<0為棱SKIPIF1<0的中點(diǎn)，則下列選項(xiàng)正確的是（

）A．直線(xiàn)SKIPIF1<0與直線(xiàn)SKIPIF1<0相交B．當(dāng)SKIPIF1<0為棱SKIPIF1<0上的中點(diǎn)時(shí)，則點(diǎn)SKIPIF1<0在平面SKIPIF1<0的射影是點(diǎn)SKIPIF1<0C．存在點(diǎn)SKIPIF1<0，使得直線(xiàn)SKIPIF1<0與直線(xiàn)SKIPIF1<0所成角為SKIPIF1<0D．三棱錐SKIPIF1<0的體積為定值4．（2022·四川攀枝花·二模（文））如圖正方體SKIPIF1<0，中，點(diǎn)SKIPIF1<0、SKIPIF1<0分別是SKIPIF1<0、SKIPIF1<0的中點(diǎn)，SKIPIF1<0為正方形SKIPIF1<0的中心，則（

）A．直線(xiàn)SKIPIF1<0與SKIPIF1<0是異面直線(xiàn) B．直線(xiàn)SKIPIF1<0與SKIPIF1<0是相交直線(xiàn)C．直線(xiàn)SKIPIF1<0與SKIPIF1<0互相垂直 D．直線(xiàn)SKIPIF1<0與SKIPIF1<0所成角的余弦值為SKIPIF1<0.5．（2022·江蘇·如皋市第一中學(xué)）（多選）在四邊形SKIPIF1<0中(如圖1)，SKIPIF1<0，將四邊形SKIPIF1<0沿對(duì)角線(xiàn)SKIPIF1<0折成四面體SKIPIF1<0（如圖2所示），使得SKIPIF1<0，E，F(xiàn)，G分別為SKIPIF1<0的中點(diǎn)，連接SKIPIF1<0為平面SKIPIF1<0內(nèi)一點(diǎn)，則（

）A．三棱錐SKIPIF1<0的體積為SKIPIF1<0B．直線(xiàn)SKIPIF1<0與SKIPIF1<0所成的角的余弦值為SKIPIF1<0C．四面體SKIPIF1<0的外接球的表面積為SKIPIF1<0D．若SKIPIF1<0，則Q點(diǎn)的軌跡長(zhǎng)度為SKIPIF1<06．（2022·江蘇·常州市第一中學(xué)）（多選）如圖，在菱形ABCD中，AB＝2，∠BAD＝60°，將△ABD沿對(duì)角線(xiàn)BD翻折到△PBD位置，連接PC，構(gòu)成三棱錐SKIPIF1<0．設(shè)二面角SKIPIF1<0為SKIPIF1<0，直線(xiàn)SKIPIF1<0和直線(xiàn)SKIPIF1<0所成角為SKIPIF1<0，在翻折過(guò)程中，下列說(shuō)法正確的是（

）A．PC與平面BCD所成的最大角為45°B．存在某個(gè)位置，使得PB⊥CDC．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的最大值為SKIPIF1<0D．存在某個(gè)位置，使得B到平面PDC的距離為SKIPIF1<07．（2022·福建漳州）（多選）已知正