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1. 下列说法中,正确的是()
A. 相等的圆心角所对的弦相等
B. 等弧所对的弦相等
C. 相等的圆心角所对的弧相等
D. 相等的弦所对的弧相等
A. 相等的圆心角所对的弦相等
B. 等弧所对的弦相等
C. 相等的圆心角所对的弧相等
D. 相等的弦所对的弧相等
答案
B
2. 如图,在$\odot O$中,$\widehat {AB}= \widehat {CD}$,则 AC 与 BD 的关系是()
A. $AC= BD$
B. $AC＜BD$
C. $AC＞BD$
D. 不能确定
A. $AC= BD$
B. $AC＜BD$
C. $AC＞BD$
D. 不能确定
答案
A
3. 如图,AB 是$\odot O$的直径,$\widehat {BC}= \widehat {CD}= \widehat {DE},∠COD= 40^{\circ }$,则$∠AEO= $ _ .
答案
$60^{\circ}$
4. (教材$P_{89}T_{3}$变式)如图,在$\odot O$中,$\widehat {AB}= \widehat {AC},∠A= 40^{\circ }$,则$∠ABC= $ _ .
答案
$70^{\circ}$
5. 如图,AB 是$\odot O$的直径,点 C,D 在$\odot O$上,$∠AOD= 110^{\circ }$.若$\widehat {BC}= \widehat {BD}$,则$∠C= $ _ .
答案
$55^{\circ}$
6. (教材$P_{89}T_{4}$变式)如图,在$\odot O$中,$AB= CD$. 求证:$AD= BC.$
答案
证明:法一:
$\because AB = CD,\therefore \widehat{AB}=\widehat{CD}$,
$\therefore \widehat{AB}-\widehat{AC}=\widehat{CD}-\widehat{AC}$,
即 $\widehat{BC}=\widehat{AD},\therefore AD = BC$;
法二:连接 $OA,OB,OC,OD$,
$\because AB = CD,\therefore ∠AOB = ∠COD$,
$\therefore ∠AOB - ∠AOC = ∠COD - ∠AOC$,
即 $∠AOD = ∠BOC,\therefore AD = BC$.
7. 如图,AB 是半圆 O 的直径,点 C,D 在半圆上,连接 CA,OD,若$AC// OD$. 求证:$\widehat {CD}= \widehat {DB}.$
答案
证明:连接 $OC$.
$\because AC// OD,\therefore ∠COD = ∠ACO$,
$∠BOD = ∠A$.
$\because OA = OC,\therefore ∠A = ∠ACO$,
$\therefore ∠COD = ∠BOD,\therefore \widehat{CD}=\widehat{DB}$.
$\because AC// OD,\therefore ∠COD = ∠ACO$,
$∠BOD = ∠A$.
$\because OA = OC,\therefore ∠A = ∠ACO$,
$\therefore ∠COD = ∠BOD,\therefore \widehat{CD}=\widehat{DB}$.
