1
00:00:00,000 --> 00:00:03,130
假設有個∠ABC

2
00:00:03,240 --> 00:00:04,840
它看起來是這樣的

3
00:00:04,950 --> 00:00:07,320
角的頂點是B

4
00:00:07,450 --> 00:00:08,800
頂點是B

5
00:00:08,910 --> 00:00:11,500
假設點A在這裡

6
00:00:11,610 --> 00:00:14,360
點C在這裡

7
00:00:14,450 --> 00:00:18,590
再假設有一個∠DAB

8
00:00:18,690 --> 00:00:22,350
叫∠DBA吧

9
00:00:22,460 --> 00:00:24,990
因爲我想讓角的頂點在B

10
00:00:25,110 --> 00:00:26,890
假設∠DBA是這樣的

11
00:00:27,000 --> 00:00:31,610
點D是在這裡

12
00:00:31,720 --> 00:00:32,720
這就是點D

13
00:00:32,830 --> 00:00:37,110
假設我們已經知道∠DBA的度數

14
00:00:37,210 --> 00:00:40,450
假設∠DBA是40°

15
00:00:40,550 --> 00:00:41,990
這邊這個角

16
00:00:42,110 --> 00:00:44,820
它的度數是40°

17
00:00:44,930 --> 00:00:49,710
假設∠ABC的度數是

18
00:00:49,830 --> 00:00:55,050
50°

19
00:00:55,140 --> 00:00:57,530
好了 這有很多有意思的點

20
00:00:57,650 --> 00:00:59,870
第一個有趣兒的點就是

21
00:00:59,970 --> 00:01:02,900
這兩個角共用一條邊

22
00:01:03,010 --> 00:01:04,200
你可以把它們看成射線

23
00:01:04,300 --> 00:01:05,320
也可以看成直線

24
00:01:05,420 --> 00:01:06,210
還可以看成是線段

25
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但如果把它們當成射線

26
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那這兩個角共享射線BA

27
00:01:10,980 --> 00:01:13,090
如果有兩個角

28
00:01:13,190 --> 00:01:14,200
它們共用一條邊

29
00:01:14,290 --> 00:01:16,470
那這兩個角就是鄰角

30
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因爲“鄰”字面意思就是“旁邊”

31
00:01:19,540 --> 00:01:25,940
這兩個角就是鄰角

32
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你還會發現其它有意思的點兒

33
00:01:27,960 --> 00:01:28,790
這也很有意思

34
00:01:28,900 --> 00:01:32,140
我們已知∠DBA是40°

35
00:01:32,260 --> 00:01:35,010
∠ABC是50°

36
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那你就可以猜出

37
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∠DBC的度數了

38
00:01:40,830 --> 00:01:43,630
∠DBC的度數是

39
00:01:43,740 --> 00:01:45,770
如果你在這兒畫個量角器

40
00:01:45,880 --> 00:01:46,870
當然我不會畫了

41
00:01:46,970 --> 00:01:48,560
否則圖就亂七八糟了

42
00:01:48,650 --> 00:01:50,480
我還是快速畫一個

43
00:01:50,570 --> 00:01:52,330
假設這裡有個量角器

44
00:01:52,450 --> 00:01:54,590
很明顯 這個角是50°

45
00:01:54,710 --> 00:01:56,770
這個角是40°

46
00:01:56,880 --> 00:01:57,910
那你想知道

47
00:01:58,020 --> 00:02:00,300
∠DBC的度數

48
00:02:00,400 --> 00:02:02,120
它其實就是

49
00:02:02,230 --> 00:02:04,700
40°加上50°

50
00:02:04,780 --> 00:02:06,210
把這些東西都擦掉

51
00:02:06,300 --> 00:02:07,320
讓圖看得更清楚點兒

52
00:02:07,420 --> 00:02:09,780
因此∠DBC的度數

53
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就是90°

54
00:02:12,790 --> 00:02:15,490
我們知道90°的角是個特殊的角

55
00:02:15,580 --> 00:02:21,650
這個角是直角

56
00:02:21,780 --> 00:02:27,910
若兩角之和爲90°

57
00:02:28,010 --> 00:02:30,390
這兩角互爲余角

58
00:02:30,500 --> 00:02:32,010
我們也可以說

59
00:02:32,130 --> 00:02:42,460
∠DBA和∠ABC是互余

60
00:02:42,570 --> 00:02:49,840
因爲它們度數之和爲90°

61
00:02:49,980 --> 00:02:56,780
因此∠DBA加上∠ABC

62
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等於90°

63
00:02:59,990 --> 00:03:02,760
它們相加 組成了一個直角

64
00:03:02,870 --> 00:03:04,560
這又是一個

65
00:03:04,670 --> 00:03:07,140
與直角相關的術語

66
00:03:07,250 --> 00:03:10,860
當組成一個直角時

67
00:03:10,960 --> 00:03:13,320
組成直角的這兩條射線

68
00:03:13,430 --> 00:03:15,510
或者是組成直角的兩條直線

69
00:03:15,690 --> 00:03:17,430
或者是組成直角的兩條線段

70
00:03:17,540 --> 00:03:19,000
是相互垂直的

71
00:03:19,110 --> 00:03:22,850
因爲我們知道∠DBC是90°

72
00:03:22,940 --> 00:03:25,650
或者∠DBC是直角

73
00:03:25,760 --> 00:03:32,200
這就告訴了我們

74
00:03:32,290 --> 00:03:35,580
我可以說

75
00:03:35,700 --> 00:03:46,150
線段DB與BC垂直

76
00:03:46,270 --> 00:03:50,410
我們甚至可以說射線BD

77
00:03:50,510 --> 00:03:53,940
我們不用 垂直 這個詞了

78
00:03:54,050 --> 00:03:56,690
有時也可以用這個符號

79
00:03:56,770 --> 00:03:58,450
它就表示兩條直線垂直

80
00:03:58,540 --> 00:04:02,550
DB與BC垂直

81
00:04:02,660 --> 00:04:05,420
這些都是真命題

82
00:04:05,530 --> 00:04:07,920
都是從DB與BC組成的角

83
00:04:08,020 --> 00:04:10,300
推斷出來的

84
00:04:10,410 --> 00:04:13,680
這是90°的角

85
00:04:13,780 --> 00:04:15,340
當兩個角相加爲其它度數時

86
00:04:15,450 --> 00:04:18,480
我們還有其它的術語

87
00:04:18,590 --> 00:04:20,450
就比如

88
00:04:20,550 --> 00:04:24,950
這裡有個角 I

89
00:04:25,040 --> 00:04:27,380
我就現編一個

90
00:04:27,500 --> 00:04:29,760
我們叫這個角

91
00:04:29,860 --> 00:04:37,060
我們用字母 XYZ 來標記這個角

92
00:04:37,170 --> 00:04:44,640
假設∠XYZ是60°

93
00:04:44,740 --> 00:04:48,310
再假設還有一個角

94
00:04:48,420 --> 00:04:52,110
它是這樣的

95
00:04:52,200 --> 00:05:01,740
我用MNO表示這個角

96
00:05:01,830 --> 00:05:06,880
假設∠MNO是120°

97
00:05:06,990 --> 00:05:09,230
如果這兩個角相加

98
00:05:09,340 --> 00:05:10,780
我把這個寫下來

99
00:05:10,870 --> 00:05:23,500
∠MNO加∠XYZ

100
00:05:23,610 --> 00:05:25,330
等於

101
00:05:25,440 --> 00:05:29,700
等於120°加60°

102
00:05:29,800 --> 00:05:32,420
就是180°

103
00:05:32,530 --> 00:05:34,560
如果把這兩個角相加

104
00:05:34,640 --> 00:05:37,730
你就可以繞圓走半圈

105
00:05:37,840 --> 00:05:41,090
或者是繞整個半圓

106
00:05:41,180 --> 00:05:43,110
或者是半圓形量角器

107
00:05:43,220 --> 00:05:46,790
如果兩角之和爲180°

108
00:05:46,900 --> 00:05:48,730
它們就是補角

109
00:05:48,830 --> 00:05:50,930
我知道這有點難記

110
00:05:51,040 --> 00:05:52,430
90°是余角

111
00:05:52,540 --> 00:05:54,220
有兩個角互余

112
00:05:54,310 --> 00:05:56,210
如果之和是180°

113
00:05:56,320 --> 00:06:03,110
就是補角

114
00:06:03,230 --> 00:06:05,940
如果這兩個角還相鄰

115
00:06:06,050 --> 00:06:08,060
它們共用一條邊

116
00:06:08,180 --> 00:06:09,910
讓我在這兒畫

117
00:06:10,020 --> 00:06:13,460
假設有這樣一個角

118
00:06:13,580 --> 00:06:15,180
還有這樣一個角

119
00:06:15,280 --> 00:06:17,810
讓我標一些字母

120
00:06:17,920 --> 00:06:19,260
我又從新使用ABC

121
00:06:19,360 --> 00:06:23,560
這就是 ABC

122
00:06:23,660 --> 00:06:26,050
還有一個角是這樣的

123
00:06:26,160 --> 00:06:30,620
還有一個角是這樣的

124
00:06:30,730 --> 00:06:32,720
我已經用了C

125
00:06:32,830 --> 00:06:34,730
看起來是這樣的

126
00:06:34,850 --> 00:06:37,070
注意 我再說一遍

127
00:06:37,190 --> 00:06:39,330
這個角是50°

128
00:06:39,440 --> 00:06:42,370
這個角是130°

129
00:06:42,510 --> 00:06:47,270
很明顯 ∠DBA加∠ABC

130
00:06:47,490 --> 00:06:48,550
如果把它們相加

131
00:06:48,650 --> 00:06:50,630
130°加50°

132
00:06:50,740 --> 00:06:52,070
等於180°

133
00:06:52,180 --> 00:06:53,380
因此它們互補

134
00:06:53,490 --> 00:06:54,830
我把這個寫下來

135
00:06:54,950 --> 00:07:04,130
∠DBA和∠ABC互補

136
00:07:04,250 --> 00:07:07,470
因爲它們之和是180°

137
00:07:07,600 --> 00:07:10,590
而且它們還是鄰角

138
00:07:10,710 --> 00:07:13,890
它們是相鄰的

139
00:07:14,010 --> 00:07:16,630
因爲它們互補且相鄰

140
00:07:16,740 --> 00:07:18,780
如果你看這個大角

141
00:07:18,890 --> 00:07:21,530
也就是除了共用那條邊外的兩邊組成的角

142
00:07:21,650 --> 00:07:28,250
如果你看∠DBC

143
00:07:28,370 --> 00:07:30,740
它們實際上組成了一條直線

144
00:07:30,840 --> 00:07:36,980
我們可以稱它爲平角

145
00:07:37,090 --> 00:07:38,930
我給大家介紹了很多詞了

146
00:07:39,040 --> 00:07:40,870
我們已經有了很多基礎

147
00:07:40,970 --> 00:07:44,390
可以用來進行有趣的證明

148
00:07:44,500 --> 00:07:46,210
在回顧一下

149
00:07:46,310 --> 00:07:50,150
我們講了鄰角

150
00:07:50,260 --> 00:07:54,720
所有兩角之和爲90°的角都是互余

151
00:07:54,850 --> 00:07:56,280
這之和是90°

152
00:07:56,400 --> 00:07:58,130
如何它們還相鄰的話

153
00:07:58,260 --> 00:08:01,970
它們外邊的兩條邊還組成一個直角

154
00:08:02,070 --> 00:08:04,050
如果有直角了

155
00:08:04,160 --> 00:08:08,970
直角的兩條邊就相互垂直

156
00:08:09,080 --> 00:08:12,120
如果兩角之和爲180°

157
00:08:12,230 --> 00:08:14,200
它們就互補

158
00:08:14,320 --> 00:08:16,170
如果它們還相鄰

159
00:08:16,290 --> 00:08:18,230
就會構成一條直線

160
00:08:18,360 --> 00:08:19,560
換種說法就是

161
00:08:19,650 --> 00:08:20,890
如果有一個平角

162
00:08:21,020 --> 00:08:23,570
有其中一個角

163
00:08:23,680 --> 00:08:26,010
另外一個角就跟它互補

164
00:08:26,130 --> 00:08:28,080
它們之和等於180°

165
00:08:28,190 --> 00:08:28,980
今天就講到這裡