如题所述
第1个回答 2011-04-17
八年级下册重点语法和短语
Unit 1 Will people have robots?重点语法:一般将来时态的应用 do/does 的一般将来时态形式:(shall/will) do do/does 的一般将来时态的被动语态:(shall/will) be done一般将来时态的肯定句、否定句、疑问句形式:肯定句例句:People will have robots in a few years.否定句例句:People (will not/won't) have robots in a few years.一般疑问句例句:Will people have robots in a few years?特殊疑问句例句:What will people have in a few years?重点短语:won't = will not they'll = they will she'll = she will he'll = he will I'll = I will fall in love with(sb./sth.) 爱上(某人/某物) be able to do sth. 能够做某事 come true 实现 in the future 未来 hundreds of 数以百计的 thousands of 数以千计的 look for(sb./sth.) 寻找(某人/某物) will → would 情态动词 will 的原形和过去式 may → might 情态动词 may 的原形和过去式Reading Strategy(阅读方法)Look at the title and picture, and predict what you will read about. (看着标题和图片,预知你要阅读那些方面的内容。)This helps you get ready to acquire new information. (这样可以帮助你获得一些新的信息。)Unit 2 What should I do?重点语法:过去将来时态(将来时态的委婉说法) do/does 的过去将来时态形式:(should/would) do do/does 的过去将来时态的被动语态:(should/would) be done过去将来时态的肯定句、否定句、疑问句形式:肯定句例句:You should write a letter to him.否定句例句:You shouldn't write a letter to him.一般疑问句例句:Should I write a letter to him?特殊疑问句例句:What should I do?重点短语:keep sb. out 不让某人进入 What's wrong? = What's the matter? = What's the problem? 怎么了? out of style 不时髦的;过时的 call sb. up 给某人打电话 pay for sth. 为某事付款 part-time job 兼职工作 the same as = be same (to/with) 与……同样 in style 时髦的;流行的 get on [well] with sb. = get along [well] with sb. 与某人相处(好) didn't = did not couldn't = could not as ... as possible 尽可能……(eg/ as soon as possible 尽快) all kinds of 各种;许多 on the one hand 一方面 on the other hand 另一方面 ask sb. for sth. = ask sb. to do sth. 请求某人做某事 ask sb. not to do sth. 请求某人不要做某事 spend (money) on sth. = spend (money) [in] doing sth. 花钱做某事 sth. cost sb. (money) 某人花钱为了某事 take sb. sometime to do sth. 花某人时间做某事 find out 查明 find sb. doing sth. 发现某人做某事 be angry with sb. 生某人的气 be angry at sth. 生某事的气 the same age as = as old as 与某人年龄一样 have fight with sb. 与某人打架 learn to do sth. 学会做某事 not ... until ... 直到……才…… compare sth.(A) with sth.(B) 把某事(A)与某事(B)作比较 it's time for sth. = it's time to do sth. 到该做某事的时间了 maybe adv. 或许 may be (情态动词 + 动词原形)可能是 shall → should 情态动词 shall 的原形和过去式 pay → paid → paid 动词 pay 的原形、过去式和过去分词Reading Strategy(阅读方法)You will learn to use new words better if you use a learner's dictionary. (时刻学着应用新单词来学习比时刻使用字典这种途径方法更好。)A bilingual dictionary sometimes gives the wrong meaning for the situation you want. (在某些你需要的场合下,一本双语字典有时会给你错误的解释。)Unit 3 What were you doing when the UFO arrived?重点语法:过去进行时态 do/does 的过去进行时态形式:(was/were) doing do/does 的过去进行时态的被动语态:(was/were) being done过去将来时态的肯定句、否定句和疑问句形式:肯定句例句:I was walking down the street when a UFO landed.否定句例句:I wasn't walking down the street when a UFO landed.一般疑问句例句:Were you walking down the street when a UFO landed?特殊疑问句例句:What were you doing when a UFO landed?动词 when 和 while 的选择:when 后加瞬间动词,while 后加延续性动词。例句:The boy was walking down the street when the UFO landed. =While the boy was walking down the street, the UFO landed.感叹句结构:(1) How + adj. + the + 主语 + 谓语动词 =(2) What + (a/an) + [adj.] + n. + 主语 + 谓语动词例句:What a beautiful flower [it is]! =How beautiful the flower is! What beautiful flowers [they are]! =How beautiful the flowers are!重点短语:get out 出去;离开 take off 起飞 run away 逃跑;跑掉 come in 进来 hear about = hear of 听说 take place 发生 as ... as 像……一样(eg/ as old as him 像他一样老) anywhere = everywhere = here and there 任何地方 think about 考虑 think of 认为 get up = get out of the bed 起床 at the doctor's 在诊所 every day 每一天 everyday adj. 日常的 most adj. 大部分 the most 最多的 in space 在太空中 national hero 民族英雄 all over the world = in the world 全世界Reading Strategy(阅读方法)The title can be helpful for you to understand a text. (一篇文章的标题可以帮助你理解整篇文章。)It's also a good idea to read the first sentence of each paragraph before you read. (在阅读整篇文章之前,阅读每段的第一句话也是一个很有效的方法。)Unit 4 He said I was hard-working.重点语法:宾语从句结构:主语 + 谓语动词 + 宾语从句(主语 + 谓语动词 + 宾语/表语)例句:----I'm good at English. He says. (改为加宾语从句的复合句) ----He says I'm good at English.注意:①主句是一般现在时态,宾语从句的时态不受其影响。 例句:He says I'm good at English now. He says I was good at mathematics when I was young. ②主句是过去时态,宾语从句也要用过去时态。 例句:He said I was good at mathematics when I was young yesterday. He said I was good at English now yesterday. ③宾语从句是客观真理时永远用一般现在时态。 例句:Our teacher says 24 hours make a day. Our teacher said the sun gives us so many energy yesterday. ④动词原形不能作主语,必须用其 -ing 形式。 例句:She said helping others changed her life.重点短语:direct speech 直接引语 reported speech = indirect speech 间接引语 first of all = at first 首先 pass on 传递 be supposed to do sth. 应该做某事 be good at = do well in 在某方面做得好 in good health 身体健康 get over 克服 open up 打开 care for = take care of = look after 照料;照顾 not any more = not any longer = no longer 不再 have a cold 感冒 end-of-year exam 年终考试 get nervous 变得紧张 forget to do sth. 忘记做某事(该事未做) forget doing sth. 忘记做某事(该事已做) it's + adj. + [for sb.] + to do sth. 做某事[对某人来说]……(加形容词) context 上下文Reading Strategy(阅读方法)First read for meaning, not for detail. (首先理解文段的大致意思,不在于文段的细节部分。)You can understand the meaning of a word you don't know from the context. (至于不懂的单词,你可以通过上下文来寻找它的正确释义。)Unit 5 If you go to the party, you'll have a great time!重点语法:if 引导的条件状语从句结构:主句 + if + 条件状语从句 if + 条件状语从句 + [(comma)] + 主句注意:在 if 引导的条件状语从句中,主句应用将来时态,状语从句用一般现在时态。例句:You'll have a great time if you go to the party. =If you go to the party, you'll have a great time.重点短语:take away 拿走 around the world = all over the world 在世界各地 make a living 谋生 all the time = always 一直 What's the problem? = What's the matter? = What's wrong? 怎么了? in order to do sth. 为了做某事 make sb. do sth. 使得某人做某事(to 省略,该结构是一个不带 to 的不定式。) make sb. adj. 使得某人……(加形容词) make sb. done 使得某人被做 be famous for 为……而出名 be famous as 作为……而出名 in class 在课堂上 spend ...(time/money) on sth. = spend ...(time/money) in doing sth. 花……(时间/钱)用于做某事 see sb. do sth. 看见某人做某事(强调整个过程) see sb. doing sth. 看见某人做某事(强调偶然性) say → said → said 动词 say 的原形、过去式和过去分词 tell → told → told 动词 tell 的原形、过去式和过去分词 eat → ate → eaten 动词 eat 的原形、过去式和过去分词 speak → spoke → spoken 动词 speak 的原形、过去式和过去分词Unit 6 How long have you been collecting shells?重点语法:现在完成进行时态 do/does 的现在完成进行时态形式:have/has been doing do/does 的现在完成进行时态的被动语态:have/has been being done现在完成进行时态所应用的场合:①某事从过去发生一直持续到现在都在做②过去发生的动作对现在造成影响例:我已上了三年初中。 I have been in Junior School for 3 years. 自从那次他与我谈过心后,我天天都在进步。 I have been making progress since he talked with me that time.现在完成进行时态的肯定句、否定句和疑问句形式:肯定句例句:I have been skating for five hours.否定句例句:I haven't been skating for five hours.一般疑问句例句:Have you been skating for five hours?特殊疑问句例句:How long have you been skating?注意:瞬间动词不能和一段时间连用。例句:你借这本书已经多长时间了? How long have you been keeping this book?重点短语:run out of 用完;用尽 by the way 顺便说说 be interested in doing sth. 对某事感兴趣 more than 比……多 far away 在远处 would like to do sth. = want to do sth. = feel like doing sth. 想要做某事 send sb. sth. = send sth. to sb. 把某物赠送给某人 in fact 实际上 room 房间(用于可数名词);空间(用于不可数名词) common → more common → the most common 形容词 common 的原级、比较级和最高级Reading Strategy(阅读方法)Let your eyes "scan" the text quickly to find details that you're looking for. (在阅读文章之前,用眼睛“横扫”整篇文章,快速寻找你需要的文章要点。)You can find information quickly without reading the whole text. (这样你就不用细读整篇文章,就能寻找到你需要的一些信息。)Unit 7 Would you mind turning down the music?重点语法:mind [one's] doing sth. 介意(某人)做某事重点短语:not at all 一点也不 turn down 调节使音量变小 right away = in a minute = at once 立刻;马上 wait in line 排队等候 cut in line 插队等候 hasn't = has not keep ... down 压低声音;使缓和 at first = first of all 首先 take care 当心;小心 take care of = care about = look after 关心;照顾 break the rule 违规 obey the rule 遵守规定 put out 熄灭 pick sth. up 捡起某物 wait for sb. 等候某人 depend on 依赖;依靠 get back = return 要回 mean → meant → meant 动词 mean 的原形、过去式和过去分词Reading Strategy(阅读方法)As we read, we need to find "topic sentences".(在我们阅读的时候,我们需要寻找“主题语句”,也就是和文章中心最相关的语句。) These sentences usually gives us a "summary", or overall meaning of each paragraph and help us understand what the paragraph is about.(这些语句通常会给我们一些文章的“概要”,或者每个文段的全部意思,来帮助我们理解段落大意。) After the
topic sentence comes more detail and explanation.(当“主题语句”出现后,该段的一些解释和细节也就会随之出现。)Unit 8 Why don't you get her a scarf?重点语法:询问别人为什么要做或者不做某事 why don't you do sth. = why not do sth. 例句:Why don't you get her a camera? = Why not get her a camera? what about = how about 例句:How about some tennis balls? = What about some tennis balls?重点短语:fall asleep 入睡 give away 赠送;分发 hear of = hear about 听说 take an interest in = be interested in 对……感兴趣 make friends with 与……交友 make progress 取得进步 keep → kept → kept 动词 keep 的原形、过去式和过去分词 feed → fed → fed 动词 feed 的原形、过去式和过去分词 fall → fell → fallen 动词 fall 的原形、过去式和过去分词 hear → heard → heard 动词 hear 的原形、过去式和过去分词Reading Strategy(阅读方法)To understand the important ideas from the text, we must "summarize".(为了了解文段最主要的意图,我们必须要进行总结。) Do this by
Unit 1 Will people have robots?重点语法:一般将来时态的应用 do/does 的一般将来时态形式:(shall/will) do do/does 的一般将来时态的被动语态:(shall/will) be done一般将来时态的肯定句、否定句、疑问句形式:肯定句例句:People will have robots in a few years.否定句例句:People (will not/won't) have robots in a few years.一般疑问句例句:Will people have robots in a few years?特殊疑问句例句:What will people have in a few years?重点短语:won't = will not they'll = they will she'll = she will he'll = he will I'll = I will fall in love with(sb./sth.) 爱上(某人/某物) be able to do sth. 能够做某事 come true 实现 in the future 未来 hundreds of 数以百计的 thousands of 数以千计的 look for(sb./sth.) 寻找(某人/某物) will → would 情态动词 will 的原形和过去式 may → might 情态动词 may 的原形和过去式Reading Strategy(阅读方法)Look at the title and picture, and predict what you will read about. (看着标题和图片,预知你要阅读那些方面的内容。)This helps you get ready to acquire new information. (这样可以帮助你获得一些新的信息。)Unit 2 What should I do?重点语法:过去将来时态(将来时态的委婉说法) do/does 的过去将来时态形式:(should/would) do do/does 的过去将来时态的被动语态:(should/would) be done过去将来时态的肯定句、否定句、疑问句形式:肯定句例句:You should write a letter to him.否定句例句:You shouldn't write a letter to him.一般疑问句例句:Should I write a letter to him?特殊疑问句例句:What should I do?重点短语:keep sb. out 不让某人进入 What's wrong? = What's the matter? = What's the problem? 怎么了? out of style 不时髦的;过时的 call sb. up 给某人打电话 pay for sth. 为某事付款 part-time job 兼职工作 the same as = be same (to/with) 与……同样 in style 时髦的;流行的 get on [well] with sb. = get along [well] with sb. 与某人相处(好) didn't = did not couldn't = could not as ... as possible 尽可能……(eg/ as soon as possible 尽快) all kinds of 各种;许多 on the one hand 一方面 on the other hand 另一方面 ask sb. for sth. = ask sb. to do sth. 请求某人做某事 ask sb. not to do sth. 请求某人不要做某事 spend (money) on sth. = spend (money) [in] doing sth. 花钱做某事 sth. cost sb. (money) 某人花钱为了某事 take sb. sometime to do sth. 花某人时间做某事 find out 查明 find sb. doing sth. 发现某人做某事 be angry with sb. 生某人的气 be angry at sth. 生某事的气 the same age as = as old as 与某人年龄一样 have fight with sb. 与某人打架 learn to do sth. 学会做某事 not ... until ... 直到……才…… compare sth.(A) with sth.(B) 把某事(A)与某事(B)作比较 it's time for sth. = it's time to do sth. 到该做某事的时间了 maybe adv. 或许 may be (情态动词 + 动词原形)可能是 shall → should 情态动词 shall 的原形和过去式 pay → paid → paid 动词 pay 的原形、过去式和过去分词Reading Strategy(阅读方法)You will learn to use new words better if you use a learner's dictionary. (时刻学着应用新单词来学习比时刻使用字典这种途径方法更好。)A bilingual dictionary sometimes gives the wrong meaning for the situation you want. (在某些你需要的场合下,一本双语字典有时会给你错误的解释。)Unit 3 What were you doing when the UFO arrived?重点语法:过去进行时态 do/does 的过去进行时态形式:(was/were) doing do/does 的过去进行时态的被动语态:(was/were) being done过去将来时态的肯定句、否定句和疑问句形式:肯定句例句:I was walking down the street when a UFO landed.否定句例句:I wasn't walking down the street when a UFO landed.一般疑问句例句:Were you walking down the street when a UFO landed?特殊疑问句例句:What were you doing when a UFO landed?动词 when 和 while 的选择:when 后加瞬间动词,while 后加延续性动词。例句:The boy was walking down the street when the UFO landed. =While the boy was walking down the street, the UFO landed.感叹句结构:(1) How + adj. + the + 主语 + 谓语动词 =(2) What + (a/an) + [adj.] + n. + 主语 + 谓语动词例句:What a beautiful flower [it is]! =How beautiful the flower is! What beautiful flowers [they are]! =How beautiful the flowers are!重点短语:get out 出去;离开 take off 起飞 run away 逃跑;跑掉 come in 进来 hear about = hear of 听说 take place 发生 as ... as 像……一样(eg/ as old as him 像他一样老) anywhere = everywhere = here and there 任何地方 think about 考虑 think of 认为 get up = get out of the bed 起床 at the doctor's 在诊所 every day 每一天 everyday adj. 日常的 most adj. 大部分 the most 最多的 in space 在太空中 national hero 民族英雄 all over the world = in the world 全世界Reading Strategy(阅读方法)The title can be helpful for you to understand a text. (一篇文章的标题可以帮助你理解整篇文章。)It's also a good idea to read the first sentence of each paragraph before you read. (在阅读整篇文章之前,阅读每段的第一句话也是一个很有效的方法。)Unit 4 He said I was hard-working.重点语法:宾语从句结构:主语 + 谓语动词 + 宾语从句(主语 + 谓语动词 + 宾语/表语)例句:----I'm good at English. He says. (改为加宾语从句的复合句) ----He says I'm good at English.注意:①主句是一般现在时态,宾语从句的时态不受其影响。 例句:He says I'm good at English now. He says I was good at mathematics when I was young. ②主句是过去时态,宾语从句也要用过去时态。 例句:He said I was good at mathematics when I was young yesterday. He said I was good at English now yesterday. ③宾语从句是客观真理时永远用一般现在时态。 例句:Our teacher says 24 hours make a day. Our teacher said the sun gives us so many energy yesterday. ④动词原形不能作主语,必须用其 -ing 形式。 例句:She said helping others changed her life.重点短语:direct speech 直接引语 reported speech = indirect speech 间接引语 first of all = at first 首先 pass on 传递 be supposed to do sth. 应该做某事 be good at = do well in 在某方面做得好 in good health 身体健康 get over 克服 open up 打开 care for = take care of = look after 照料;照顾 not any more = not any longer = no longer 不再 have a cold 感冒 end-of-year exam 年终考试 get nervous 变得紧张 forget to do sth. 忘记做某事(该事未做) forget doing sth. 忘记做某事(该事已做) it's + adj. + [for sb.] + to do sth. 做某事[对某人来说]……(加形容词) context 上下文Reading Strategy(阅读方法)First read for meaning, not for detail. (首先理解文段的大致意思,不在于文段的细节部分。)You can understand the meaning of a word you don't know from the context. (至于不懂的单词,你可以通过上下文来寻找它的正确释义。)Unit 5 If you go to the party, you'll have a great time!重点语法:if 引导的条件状语从句结构:主句 + if + 条件状语从句 if + 条件状语从句 + [(comma)] + 主句注意:在 if 引导的条件状语从句中,主句应用将来时态,状语从句用一般现在时态。例句:You'll have a great time if you go to the party. =If you go to the party, you'll have a great time.重点短语:take away 拿走 around the world = all over the world 在世界各地 make a living 谋生 all the time = always 一直 What's the problem? = What's the matter? = What's wrong? 怎么了? in order to do sth. 为了做某事 make sb. do sth. 使得某人做某事(to 省略,该结构是一个不带 to 的不定式。) make sb. adj. 使得某人……(加形容词) make sb. done 使得某人被做 be famous for 为……而出名 be famous as 作为……而出名 in class 在课堂上 spend ...(time/money) on sth. = spend ...(time/money) in doing sth. 花……(时间/钱)用于做某事 see sb. do sth. 看见某人做某事(强调整个过程) see sb. doing sth. 看见某人做某事(强调偶然性) say → said → said 动词 say 的原形、过去式和过去分词 tell → told → told 动词 tell 的原形、过去式和过去分词 eat → ate → eaten 动词 eat 的原形、过去式和过去分词 speak → spoke → spoken 动词 speak 的原形、过去式和过去分词Unit 6 How long have you been collecting shells?重点语法:现在完成进行时态 do/does 的现在完成进行时态形式:have/has been doing do/does 的现在完成进行时态的被动语态:have/has been being done现在完成进行时态所应用的场合:①某事从过去发生一直持续到现在都在做②过去发生的动作对现在造成影响例:我已上了三年初中。 I have been in Junior School for 3 years. 自从那次他与我谈过心后,我天天都在进步。 I have been making progress since he talked with me that time.现在完成进行时态的肯定句、否定句和疑问句形式:肯定句例句:I have been skating for five hours.否定句例句:I haven't been skating for five hours.一般疑问句例句:Have you been skating for five hours?特殊疑问句例句:How long have you been skating?注意:瞬间动词不能和一段时间连用。例句:你借这本书已经多长时间了? How long have you been keeping this book?重点短语:run out of 用完;用尽 by the way 顺便说说 be interested in doing sth. 对某事感兴趣 more than 比……多 far away 在远处 would like to do sth. = want to do sth. = feel like doing sth. 想要做某事 send sb. sth. = send sth. to sb. 把某物赠送给某人 in fact 实际上 room 房间(用于可数名词);空间(用于不可数名词) common → more common → the most common 形容词 common 的原级、比较级和最高级Reading Strategy(阅读方法)Let your eyes "scan" the text quickly to find details that you're looking for. (在阅读文章之前,用眼睛“横扫”整篇文章,快速寻找你需要的文章要点。)You can find information quickly without reading the whole text. (这样你就不用细读整篇文章,就能寻找到你需要的一些信息。)Unit 7 Would you mind turning down the music?重点语法:mind [one's] doing sth. 介意(某人)做某事重点短语:not at all 一点也不 turn down 调节使音量变小 right away = in a minute = at once 立刻;马上 wait in line 排队等候 cut in line 插队等候 hasn't = has not keep ... down 压低声音;使缓和 at first = first of all 首先 take care 当心;小心 take care of = care about = look after 关心;照顾 break the rule 违规 obey the rule 遵守规定 put out 熄灭 pick sth. up 捡起某物 wait for sb. 等候某人 depend on 依赖;依靠 get back = return 要回 mean → meant → meant 动词 mean 的原形、过去式和过去分词Reading Strategy(阅读方法)As we read, we need to find "topic sentences".(在我们阅读的时候,我们需要寻找“主题语句”,也就是和文章中心最相关的语句。) These sentences usually gives us a "summary", or overall meaning of each paragraph and help us understand what the paragraph is about.(这些语句通常会给我们一些文章的“概要”,或者每个文段的全部意思,来帮助我们理解段落大意。) After the
topic sentence comes more detail and explanation.(当“主题语句”出现后,该段的一些解释和细节也就会随之出现。)Unit 8 Why don't you get her a scarf?重点语法:询问别人为什么要做或者不做某事 why don't you do sth. = why not do sth. 例句:Why don't you get her a camera? = Why not get her a camera? what about = how about 例句:How about some tennis balls? = What about some tennis balls?重点短语:fall asleep 入睡 give away 赠送;分发 hear of = hear about 听说 take an interest in = be interested in 对……感兴趣 make friends with 与……交友 make progress 取得进步 keep → kept → kept 动词 keep 的原形、过去式和过去分词 feed → fed → fed 动词 feed 的原形、过去式和过去分词 fall → fell → fallen 动词 fall 的原形、过去式和过去分词 hear → heard → heard 动词 hear 的原形、过去式和过去分词Reading Strategy(阅读方法)To understand the important ideas from the text, we must "summarize".(为了了解文段最主要的意图,我们必须要进行总结。) Do this by
第2个回答 2012-04-18
八年级下册数学 1 过两点有且只有一条直线 2 两点之间线段最短 3 同角或等角的补角相等 4 同角或等角的余角相等 5 过一点有且只有一条直线和已知直线垂直 6 直线外一点与直线上各点连接的所有线段中,垂线段最短 7 平行公理 经过直线外一点,有且只有一条直线与这条直线平行 8 如果两条直线都和第三条直线平行,这两条直线也互相平行 9 同位角相等,两直线平行 10 内错角相等,两直线平行 11 同旁内角互补,两直线平行 12两直线平行,同位角相等 13 两直线平行,内错角相等 14 两直线平行,同旁内角互补 15 定理 三角形两边的和大于第三边 16 推论 三角形两边的差小于第三边 17 三角形内角和定理 三角形三个内角的和等于180° 18 推论1 直角三角形的两个锐角互余 19 推论2 三角形的一个外角等于和它不相邻的两个内角的和 20 推论3 三角形的一个外角大于任何一个和它不相邻的内角 21 全等三角形的对应边、对应角相等 22边角边公理(SAS) 有两边和它们的夹角对应相等的两个三角形全等 23 角边角公理( ASA)有两角和它们的夹边对应相等的两个三角形全等 24 推论(AAS) 有两角和其中一角的对边对应相等的两个三角形全等 25 边边边公理(SSS) 有三边对应相等的两个三角形全等 26 斜边、直角边公理(HL) 有斜边和一条直角边对应相等的两个直角三角形全等 27 定理1 在角的平分线上的点到这个角的两边的距离相等 28 定理2 到一个角的两边的距离相同的点,在这个角的平分线上 29 角的平分线是到角的两边距离相等的所有点的集合 30 等腰三角形的性质定理 等腰三角形的两个底角相等 (即等边对等角) 31 推论1 等腰三角形顶角的平分线平分底边并且垂直于底边 32 等腰三角形的顶角平分线、底边上的中线和底边上的高互相重合 33 推论3 等边三角形的各角都相等,并且每一个角都等于60° 34 等腰三角形的判定定理 如果一个三角形有两个角相等,那么这两个角所对的边也相等(等角对等边) 35 推论1 三个角都相等的三角形是等边三角形 36 推论 2 有一个角等于60°的等腰三角形是等边三角形 37 在直角三角形中,如果一个锐角等于30°那么它所对的直角边等于斜边的一半 38 直角三角形斜边上的中线等于斜边上的一半 39 定理 线段垂直平分线上的点和这条线段两个端点的距离相等 40 逆定理 和一条线段两个端点距离相等的点,在这条线段的垂直平分线上 41 线段的垂直平分线可看作和线段两端点距离相等的所有点的集合 42 定理1 关于某条直线对称的两个图形是全等形 43 定理 2 如果两个图形关于某直线对称,那么对称轴是对应点连线的垂直平分线 44定理3 两个图形关于某直线对称,如果它们的对应线段或延长线相交,那么交点在对称轴上 45逆定理 如果两个图形的对应点连线被同一条直线垂直平分,那么这两个图形关于这条直线对称 46勾股定理 直角三角形两直角边a、b的平方和、等于斜边c的平方,即a^2+b^2=c^2 47勾股定理的逆定理 如果三角形的三边长a、b、c有关系a^2+b^2=c^2 ,那么这个三角形是直角三角形 48定理 四边形的内角和等于360° 49四边形的外角和等于360° 50多边形内角和定理 n边形的内角的和等于(n-2)×180° 51推论 任意多边的外角和等于360° 52平行四边形性质定理1 平行四边形的对角相等 53平行四边形性质定理2 平行四边形的对边相等 54推论 夹在两条平行线间的平行线段相等 55平行四边形性质定理3 平行四边形的对角线互相平分 56平行四边形判定定理1 两组对角分别相等的四边形是平行四边形 57平行四边形判定定理2 两组对边分别相等的四边形是平行四边形 58平行四边形判定定理3 对角线互相平分的四边形是平行四边形 59平行四边形判定定理4 一组对边平行相等的四边形是平行四边形 60矩形性质定理1 矩形的四个角都是直角 61矩形性质定理2 矩形的对角线相等 62矩形判定定理1 有三个角是直角的四边形是矩形 63矩形判定定理2 对角线相等的平行四边形是矩形 64菱形性质定理1 菱形的四条边都相等 65菱形性质定理2 菱形的对角线互相垂直,并且每一条对角线平分一组对角 66菱形面积=对角线乘积的一半,即S=(a×b)÷2 67菱形判定定理1 四边都相等的四边形是菱形 68菱形判定定理2 对角线互相垂直的平行四边形是菱形 69正方形性质定理1 正方形的四个角都是直角,四条边都相等 70正方形性质定理2正方形的两条对角线相等,并且互相垂直平分,每条对角线平分一组对角 71定理1 关于中心对称的两个图形是全等的 72定理2 关于中心对称的两个图形,对称点连线都经过对称中心,并且被对称中心平分 73逆定理 如果两个图形的对应点连线都经过某一点,并且被这一 点平分,那么这两个图形关于这一点对称 74等腰梯形性质定理 等腰梯形在同一底上的两个角相等 75等腰梯形的两条对角线相等 76等腰梯形判定定理 在同一底上的两个角相等的梯形是等腰梯形 77对角线相等的梯形是等腰梯形 78平行线等分线段定理 如果一组平行线在一条直线上截得的线段 相等,那么在其他直线上截得的线段也相等 79 推论1 经过梯形一腰的中点与底平行的直线,必平分另一腰 80 推论2 经过三角形一边的中点与另一边平行的直线,必平分第 三边 81 三角形中位线定理 三角形的中位线平行于第三边,并且等于它 的一半 82 梯形中位线定理 梯形的中位线平行于两底,并且等于两底和的 一半 L=(a+b)÷2 S=L×h 83 (1)比例的基本性质 如果a:b=c:d,那么ad=bc 如果ad=bc,那么a:b=c:d 84 (2)合比性质 如果a/b=c/d,那么(a±b)/b=(c±d)/d 85 (3)等比性质 如果a/b=c/d=…=m/n(b+d+…+n≠0),那么 (a+c+…+m)/(b+d+…+n)=a/b 86 平行线分线段成比例定理 三条平行线截两条直线,所得的对应 线段成比例 87 推论 平行于三角形一边的直线截其他两边(或两边的延长线),所得的对应线段成比例 88 定理 如果一条直线截三角形的两边(或两边的延长线)所得的对应线段成比例,那么这条直线平行于三角形的第三边 89 平行于三角形的一边,并且和其他两边相交的直线,所截得的三角形的三边与原三角形三边对应成比例 90 定理 平行于三角形一边的直线和其他两边(或两边的延长线)相交,所构成的三角形与原三角形相似 91 相似三角形判定定理1 两角对应相等,两三角形相似(ASA) 92 直角三角形被斜边上的高分成的两个直角三角形和原三角形相似 93 判定定理2 两边对应成比例且夹角相等,两三角形相似(SAS) 94 判定定理3 三边对应成比例,两三角形相似(SSS) 95 定理 如果一个直角三角形的斜边和一条直角边与另一个直角三 角形的斜边和一条直角边对应成比例,那么这两个直角三角形相似 96 性质定理1 相似三角形对应高的比,对应中线的比与对应角平 分线的比都等于相似比 97 性质定理2 相似三角形周长的比等于相似比 98 性质定理3 相似三角形面积的比等于相似比的平方 99 任意锐角的正弦值等于它的余角的余弦值,任意锐角的余弦值等 于它的余角的正弦值 100任意锐角的正切值等于它的余角的余切值,任意锐角的余切值等 于它的余角的正切值 101圆是定点的距离等于定长的点的集合 102圆的内部可以看作是圆心的距离小于半径的点的集合 103圆的外部可以看作是圆心的距离大于半径的点的集合 104同圆或等圆的半径相等 105到定点的距离等于定长的点的轨迹,是以定点为圆心,定长为半 径的圆 106和已知线段两个端点的距离相等的点的轨迹,是着条线段的垂直 平分线 107到已知角的两边距离相等的点的轨迹,是这个角的平分线 108到两条平行线距离相等的点的轨迹,是和这两条平行线平行且距 离相等的一条直线 109定理 不在同一直线上的三点确定一个圆。 110垂径定理 垂直于弦的直径平分这条弦并且平分弦所对的两条弧 111推论1 ①平分弦(不是直径)的直径垂直于弦,并且平分弦所对的两条弧 ②弦的垂直平分线经过圆心,并且平分弦所对的两条弧 ③平分弦所对的一条弧的直径,垂直平分弦,并且平分弦所对的另一条弧 112推论2 圆的两条平行弦所夹的弧相等 113圆是以圆心为对称中心的中心对称图形 114定理 在同圆或等圆中,相等的圆心角所对的弧相等,所对的弦 相等,所对的弦的弦心距相等 115推论 在同圆或等圆中,如果两个圆心角、两条弧、两条弦或两 弦的弦心距中有一组量相等那么它们所对应的其余各组量都相等 116定理 一条弧所对的圆周角等于它所对的圆心角的一半 117推论1 同弧或等弧所对的圆周角相等;同圆或等圆中,相等的圆周角所对的弧也相等 118推论2 半圆(或直径)所对的圆周角是直角;90°的圆周角所 对的弦是直径 119推论3 如果三角形一边上的中线等于这边的一半,那么这个三角形是直角三角形 120定理 圆的内接四边形的对角互补,并且任何一个外角都等于它 的内对角 121①直线L和⊙O相交 d<r ②直线L和⊙O相切 d=r ③直线L和⊙O相离 d>r 122切线的判定定理 经过半径的外端并且垂直于这条半径的直线是圆的切线 123切线的性质定理 圆的切线垂直于经过切点的半径 124推论1 经过圆心且垂直于切线的直线必经过切点 125推论2 经过切点且垂直于切线的直线必经过圆心 126切线长定理 从圆外一点引圆的两条切线,它们的切线长相等, 圆心和这一点的连线平分两条切线的夹角 127圆的外切四边形的两组对边的和相等 128弦切角定理 弦切角等于它所夹的弧对的圆周角 129推论 如果两个弦切角所夹的弧相等,那么这两个弦切角也相等 130相交弦定理 圆内的两条相交弦,被交点分成的两条线段长的积 相等 131推论 如果弦与直径垂直相交,那么弦的一半是它分直径所成的 两条线段的比例中项 132切割线定理 从圆外一点引圆的切线和割线,切线长是这点到割 线与圆交点的两条线段长的比例中项 133推论 从圆外一点引圆的两条割线,这一点到每条割线与圆的交点的两条线段长的积相等 134如果两个圆相切,那么切点一定在连心线上 135①两圆外离 d>R+r ②两圆外切 d=R+r ③两圆相交 R-r<dr) ④两圆内切 d=R-r(R>r) ⑤两圆内含dr) 136定理 相交两圆的连心线垂直平分两圆的公共弦 137定理 把圆分成n(n≥3): ⑴依次连结各分点所得的多边形是这个圆的内接正n边形 ⑵经过各分点作圆的切线,以相邻切线的交点为顶点的多边形是这个圆的外切正n边形 138定理 任何正多边形都有一个外接圆和一个内切圆,这两个圆是同心圆 139正n边形的每个内角都等于(n-2)×180°/n 140定理 正n边形的半径和边心距把正n边形分成2n个全等的直角三角形 141正n边形的面积Sn=pnrn/2 p表示正n边形的周长 142正三角形面积√3a/4 a表示边长 143如果在一个顶点周围有k个正n边形的角,由于这些角的和应为 360°,因此k×(n-2)180°/n=360°化为(n-2)(k-2)=4 144弧长计算公式:L=n兀R/180 145扇形面积公式:S扇形=n兀R^2/360=LR/2 146内公切线长= d-(R-r) 外公切线长= d-(R+r) (还有一些,大家帮补充吧) 实用工具:常用数学公式 公式分类 公式表达式 乘法与因式分 a2-b2=(a+b)(a-b) a3+b3=(a+b)(a2-ab+b2) a3-b3=(a-b(a2+ab+b2) 三角不等式 |a+b|≤|a|+|b| |a-b|≤|a|+|b| |a|≤b<=>-b≤a≤b |a-b|≥|a|-|b| -|a|≤a≤|a| 一元二次方程的解 -b+√(b2-4ac)/2a -b-√(b2-4ac)/2a 根与系数的关系 X1+X2=-b/a X1*X2=c/a 注:韦达定理 判别式 b2-4ac=0 注:方程有两个相等的实根 b2-4ac>0 注:方程有两个不等的实根 b2-4ac<0 注:方程没有实根,有共轭复数根 三角函数公式 两角和公式 sin(A+B)=sinAcosB+cosAsinB sin(A-B)=sinAcosB-sinBcosA cos(A+B)=cosAcosB-sinAsinB cos(A-B)=cosAcosB+sinAsinB tan(A+B)=(tanA+tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB) ctg(A+B)=(ctgActgB-1)/(ctgB+ctgA) ctg(A-B)=(ctgActgB+1)/(ctgB-ctgA) 倍角公式 tan2A=2tanA/(1-tan2A) ctg2A=(ctg2A-1)/2ctga cos2a=cos2a-sin2a=2cos2a-1=1-2sin2a 半角公式 sin(A/2)=√((1-cosA)/2) sin(A/2)=-√((1-cosA)/2) cos(A/2)=√((1+cosA)/2) cos(A/2)=-√((1+cosA)/2) tan(A/2)=√((1-cosA)/((1+cosA)) tan(A/2)=-√((1-cosA)/((1+cosA)) ctg(A/2)=√((1+cosA)/((1-cosA)) ctg(A/2)=-√((1+cosA)/((1-cosA)) 和差化积 2sinAcosB=sin(A+B)+sin(A-B) 2cosAsinB=sin(A+B)-sin(A-B) 2cosAcosB=cos(A+B)-sin(A-B) -2sinAsinB=cos(A+B)-cos(A-B) sinA+sinB=2sin((A+B)/2)cos((A-B)/2 cosA+cosB=2cos((A+B)/2)sin((A-B)/2) tanA+tanB=sin(A+B)/cosAcosB tanA-tanB=sin(A-B)/cosAcosB ctgA+ctgBsin(A+B)/sinAsinB -ctgA+ctgBsin(A+B)/sinAsinB 某些数列前n项和 1+2+3+4+5+6+7+8+9+…+n=n(n+1)/2 1+3+5+7+9+11+13+15+…+(2n-1)=n2 2+4+6+8+10+12+14+…+(2n)=n(n+1) 12+22+32+42+52+62+72+82+…+n2=n(n+1)(2n+1)/6 13+23+33+43+53+63+…n3=n2(n+1)2/4 1*2+2*3+3*4+4*5+5*6+6*7+…+n(n+1)=n(n+1)(n+2)/3 正弦定理 a/sinA=b/sinB=c/sinC=2R 注: 其中 R 表示三角形的外接圆半径 余弦定理 b2=a2+c2-2accosB 注:角B是边a和边c的夹角 圆的标准方程 (x-a)2+(y-b)2=r2 注:(a,b)是圆心坐标 圆的一般方程 x2+y2+Dx+Ey+F=0 注:D2+E2-4F>0 抛物线标准方程 y2=2px y2=-2px x2=2py x2=-2py 直棱柱侧面积 S=c*h 斜棱柱侧面积 S=c'*h 正棱锥侧面积 S=1/2c*h' 正棱台侧面积 S=1/2(c+c')h' 圆台侧面积 S=1/2(c+c')l=pi(R+r)l 球的表面积 S=4pi*r2 圆柱侧面积 S=c*h=2pi*h 圆锥侧面积 S=1/2*c*l=pi*r*l 弧长公式 l=a*r a是圆心角的弧度数r >0 扇形面积公式 s=1/2*l*r 锥体体积公式 V=1/3*S*H 圆锥体体积公式 V=1/3*pi*r2h 斜棱柱体积 V=S'L 注:其中,S'是直截面面积, L是侧棱长 柱体体积公式 V=s*h 圆柱体 V=pi*r2h