《2025年全品学练考九年级数学上册苏科版江苏专版》

例3 如图2-2-5,已知$AB是\odot O$的直径,弦$AC// OD$.
(1)求证：$\overset{\frown}{BD}= \overset{\frown}{CD}$；
(2)若$\overset{\frown}{AC}的度数为58^{\circ}$,求$∠AOD$的度数.

答案：
解：
(1)证明：如图，连接$OC$。

$\because OA = OC$，$\therefore \angle OAC = \angle ACO$。
$\because AC // OD$，
$\therefore \angle OAC = \angle BOD$，$\angle COD = \angle ACO$。
$\therefore \angle BOD = \angle COD$，$\therefore \overset{\frown}{BD} = \overset{\frown}{CD}$。
(2)由$\overset{\frown}{AC}$的度数为$58^{\circ}$，得$\angle AOC = 58^{\circ}$。
$\because \angle BOD = \angle COD$，
$\therefore \angle BOD = \frac{1}{2}\angle BOC = \frac{1}{2}(180^{\circ} - \angle AOC) = 61^{\circ}$，
$\therefore \angle AOD = 180^{\circ} - \angle BOD = 119^{\circ}$。

|总结|
|反思|
如图2-2-6,$AB$,$CD是\odot O$的两条弦.
(1)已知$AB= CD$,那么$\overset{\frown}{AB} = \overset{\frown}{CD}$,$\angle AOB = \angle COD$；
(2)已知$\overset{\frown}{AB}= \overset{\frown}{CD}$,那么$AB = CD$,$\angle AOB = \angle COD$；
(3)已知$∠AOB= ∠COD$,那么$AB = CD$,$\overset{\frown}{AB} = \overset{\frown}{CD}$.

答案：
(1)$\overset{\frown}{AB} = \overset{\frown}{CD}$ $\angle AOB = \angle COD$
(2)$AB = CD$ $\angle AOB = \angle COD$
(3)$AB = CD$ $\overset{\frown}{AB} = \overset{\frown}{CD}$