◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--1-plane-geometry / Definition: 1.09: Angle, Rectilinear, Vertex, Legs

Definition: 1.09: Angle, Rectilinear, Vertex, Legs

And when the lines containing the plane angle are straight then the angle is called rectilinear.

Modern Definition

Let \(\overline{P Q}\) and \(\overline{Q R}\) be two given segments  ¹, which meet in the point \(Q\). The inclination of \(\overline{PQ}\) and \(\overline{QR}\), is called a rectilinear angle, or simply an angle. The point \(Q\) is called the vertex of the angle and the segments are called its legs.

Please note the following important notation convention used in this adaptation of Euclid's “Elements”:

In order to avoid ambiguities in notation, given a drawing of an angle, we think of an angle as an oriented rotation of its legs around its vertex, and always assume that this rotation is counter clockwise oriented. With this in our mind, in this adaptation of Euclid's “Elements”, we always denote all angles beginning with the leg, where the thought rotation starts, and ending with the leg, where the thought rotation stops! In order to demonstrate this principle, consider the following figure:

To denote the smaller angle, we write \(\angle{PQR}\), because the leg \(\overline{PQ}\) is being rotated around the vertex \(Q\) to the leg \(\overline{QR}\). To denote the bigger angle, we write \(\angle{RQP}\), because the leg \(\overline{RQ}\) is being rotated around the vertex \(Q\) to the leg \(\overline{QP}\).

Table of Contents

1. Definition: Exterior, Interior, Alternate and Corresponding Angles
2. Definition: Sum of Angles
3. Definition: Supplemental Angles

Mentioned in:

Corollaries: 1
Definitions: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Lemmas: 17
Proofs: 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154
Propositions: 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:

@Calahan

@Casey

@Fitzpatrick

References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Footnotes