G.37 adjacent angles on a line sum to two right angles — Heath
I.13
E
A
C
B
D
For let any straight line / A B / set up on the straight line / C D / make the angles < C B A ) , < A B D ) ;
I say that the angles < C B A ) , < A B D ) are either two right angles or equal to two right angles.
Now, if the angle < C B A ) is equal to the angle < A B D ) ,
[G.33] they are
two right angles .
But, if not, let / B E / be drawn from the point B at right angles to / C D / ; [G.34]
therefore the angles < C B E ) , < E B D ) are two right angles .
Then, since the angle < C B E ) is equal to the two angles < C B A ) , < A B E ) ,
let the angle < E B D ) be added to each;
[G.36] therefore the angles
< C B E ) ,
< E B D ) are equal to the three angles
< C B A ) ,
< A B E ) ,
< E B D ) .
Again, since the angle < D B A ) is equal to the two angles < D B E ) , < E B A ) ,
let the angle < A B C ) be added to each;
[G.36] therefore the angles
< D B A ) ,
< A B C ) are equal to the three angles
< D B E ) ,
< E B A ) ,
< A B C ) .
But the angles < C B E ) , < E B D ) were also proved equal to the same three angles; and things which are equal to the same thing are also equal to one another; [G.16]
therefore the angles < C B E ) , < E B D ) are also equal to the angles < D B A ) , < A B C ) .
But the angles < C B E ) , < E B D ) are two right angles ;
therefore the angles < D B A ) , < A B C ) are also equal to two right angles.
Therefore etc.
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"G.2" [label="G.2", tooltip="G.2 (I.def.2): A line [ A B ] is a projection of a relationship of two points A, B.", fillcolor="#224422", URL="../G.2/index.html", target="_top"];
"G.3" [label="G.3", tooltip="G.3 (I.def.3): The extremities of a segment / A B / are points A, B.", fillcolor="#224422", URL="../G.3/index.html", target="_top"];
"G.4" [label="G.4", tooltip="G.4 (I.def.4): A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.", fillcolor="#224422", URL="../G.4/index.html", target="_top"];
"G.5" [label="G.5", tooltip="G.5 (I.def.5): A surface is that which has length and breadth only.", fillcolor="#224422", URL="../G.5/index.html", target="_top"];
"G.6" [label="G.6", tooltip="G.6 (I.def.6): The extremities of a surface are lines or arcs.", fillcolor="#224422", URL="../G.6/index.html", target="_top"];
"G.7" [label="G.7", tooltip="G.7 (I.def.7): A plane surface lies evenly with the lines on itself.", fillcolor="#224422", URL="../G.7/index.html", target="_top"];
"G.8" [label="G.8", tooltip="G.8 (I.def.13): A boundary is the extremity of anything.", fillcolor="#224422", URL="../G.8/index.html", target="_top"];
"G.9" [label="G.9", tooltip="G.9 (I.def.14): A figure is that which is contained by any boundary or boundaries.", fillcolor="#224422", URL="../G.9/index.html", target="_top"];
"G.10" [label="G.10", tooltip="G.10 (I.def.15): A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D...", fillcolor="#224422", URL="../G.10/index.html", target="_top"];
"G.11" [label="G.11", tooltip="G.11 (I.def.16): The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.", fillcolor="#224422", URL="../G.11/index.html", target="_top"];
"G.12" [label="G.12", tooltip="G.12 (I.def.19): Rectilineal figures are contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.", fillcolor="#224422", URL="../G.12/index.html", target="_top"];
"G.13" [label="G.13", tooltip="G.13 (I.def.20): Of trilateral figures, an equilateral triangle < A B C > has three equal sides, an isosceles triangle < D E F > has two equal sides, and a scalene triangle < G H K > has three uneq...", fillcolor="#224422", URL="../G.13/index.html", target="_top"];
"G.14" [label="G.14", tooltip="G.14 (I.post.1): Construct a segment / A B / between any two points A, B.", fillcolor="#444422", URL="../G.14/index.html", target="_top"];
"G.15" [label="G.15", tooltip="G.15 (I.post.3): Construct a circle ( A B ) with center A and defining radius point B.", fillcolor="#444422", URL="../G.15/index.html", target="_top"];
"G.16" [label="G.16", tooltip="G.16 (I.cn.1): Things equal to the same thing are equal to one another.", fillcolor="#442222", URL="../G.16/index.html", target="_top"];
"G.17" [label="G.17", tooltip="G.17 (I.1): Construct an equilateral triangle on a given segment.", fillcolor="#222244", URL="../G.17/index.html", target="_top"];
"G.18" [label="G.18", tooltip="G.18 (I.post.2): Extend a segment / A B / continuously along a line [ A B ] to / B C /.", fillcolor="#444422", URL="../G.18/index.html", target="_top"];
"G.19" [label="G.19", tooltip="G.19 (I.cn.3): If equals are subtracted from equals, the remainders are equal.", fillcolor="#442222", URL="../G.19/index.html", target="_top"];
"G.20" [label="G.20", tooltip="G.20 (I.2): Construct at a given point a segment equal to a given segment.", fillcolor="#222244", URL="../G.20/index.html", target="_top"];
"G.21" [label="G.21", tooltip="G.21 (I.3): Cut off from the greater of two unequal segments a segment equal to the less.", fillcolor="#222244", URL="../G.21/index.html", target="_top"];
"G.22" [label="G.22", tooltip="G.22 (I.def.8): A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.", fillcolor="#224422", URL="../G.22/index.html", target="_top"];
"G.23" [label="G.23", tooltip="G.23 (I.cn.4): Things / A B /, / D E / that coincide with one another are equal to one another.", fillcolor="#442222", URL="../G.23/index.html", target="_top"];
"G.24" [label="G.24", tooltip="G.24 (I.4): Triangles with two sides and the included angle equal are congruent (SAS).", fillcolor="#222244", URL="../G.24/index.html", target="_top"];
"G.25" [label="G.25", tooltip="G.25 (I.5): In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.", fillcolor="#222244", URL="../G.25/index.html", target="_top"];
"G.26" [label="G.26", tooltip="G.26 (I.cn.5): The whole is greater than the part.", fillcolor="#442222", URL="../G.26/index.html", target="_top"];
"G.28" [label="G.28", tooltip="G.28 (I.7): On the same segment and on the same side, there cannot be constructed two distinct triangles with both pairs of conterminous sides equal.", fillcolor="#222244", URL="../G.28/index.html", target="_top"];
"G.29" [label="G.29", tooltip="G.29 (I.8): Triangles with all three corresponding sides equal have their corresponding angles equal (SSS).", fillcolor="#222244", URL="../G.29/index.html", target="_top"];
"G.33" [label="G.33", tooltip="G.33 (I.def.10): When a segment standing on a line makes the adjacent angles equal, each of the equal angles is a right angle, and the standing segment is perpendicular to the line.", fillcolor="#224422", URL="../G.33/index.html", target="_top"];
"G.34" [label="G.34", tooltip="G.34 (I.11): Construct a segment at right angles to a given line from a given point on it.", fillcolor="#222244", URL="../G.34/index.html", target="_top"];
"G.36" [label="G.36", tooltip="G.36 (I.cn.2): If equals are added to equals, the wholes are equal.", fillcolor="#442222", URL="../G.36/index.html", target="_top"];
"G.37" [label="G.37", tooltip="G.37 (I.13): A segment standing on a line makes either two right angles or adjacent angles summing to two right angles.", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"];
"G.2" -> "G.1";
"G.3" -> "G.1";
"G.4" -> "G.1";
"G.10" -> "G.1";
"G.11" -> "G.1";
"G.14" -> "G.1";
"G.15" -> "G.1";
"G.17" -> "G.1";
"G.20" -> "G.1";
"G.21" -> "G.1";
"G.24" -> "G.1";
"G.25" -> "G.1";
"G.28" -> "G.1";
"G.29" -> "G.1";
"G.34" -> "G.1";
"G.37" -> "G.1";
"G.3" -> "G.2";
"G.4" -> "G.2";
"G.6" -> "G.2";
"G.7" -> "G.2";
"G.18" -> "G.2";
"G.22" -> "G.2";
"G.33" -> "G.2";
"G.34" -> "G.2";
"G.37" -> "G.2";
"G.4" -> "G.3";
"G.6" -> "G.3";
"G.8" -> "G.3";
"G.20" -> "G.3";
"G.28" -> "G.3";
"G.29" -> "G.3";
"G.7" -> "G.4";
"G.10" -> "G.4";
"G.11" -> "G.4";
"G.12" -> "G.4";
"G.14" -> "G.4";
"G.17" -> "G.4";
"G.18" -> "G.4";
"G.20" -> "G.4";
"G.21" -> "G.4";
"G.22" -> "G.4";
"G.24" -> "G.4";
"G.25" -> "G.4";
"G.28" -> "G.4";
"G.29" -> "G.4";
"G.33" -> "G.4";
"G.34" -> "G.4";
"G.37" -> "G.4";
"G.6" -> "G.5";
"G.7" -> "G.5";
"G.10" -> "G.6";
"G.10" -> "G.7";
"G.22" -> "G.7";
"G.9" -> "G.8";
"G.10" -> "G.9";
"G.12" -> "G.9";
"G.11" -> "G.10";
"G.15" -> "G.10";
"G.17" -> "G.10";
"G.20" -> "G.10";
"G.21" -> "G.10";
"G.15" -> "G.11";
"G.17" -> "G.11";
"G.20" -> "G.11";
"G.21" -> "G.11";
"G.13" -> "G.12";
"G.17" -> "G.12";
"G.20" -> "G.12";
"G.24" -> "G.12";
"G.25" -> "G.12";
"G.28" -> "G.12";
"G.29" -> "G.12";
"G.34" -> "G.12";
"G.17" -> "G.13";
"G.25" -> "G.13";
"G.17" -> "G.14";
"G.20" -> "G.14";
"G.25" -> "G.14";
"G.28" -> "G.14";
"G.34" -> "G.14";
"G.17" -> "G.15";
"G.20" -> "G.15";
"G.21" -> "G.15";
"G.17" -> "G.16";
"G.20" -> "G.16";
"G.21" -> "G.16";
"G.37" -> "G.16";
"G.20" -> "G.17";
"G.34" -> "G.17";
"G.20" -> "G.18";
"G.25" -> "G.18";
"G.20" -> "G.19";
"G.25" -> "G.19";
"G.21" -> "G.20";
"G.25" -> "G.21";
"G.34" -> "G.21";
"G.24" -> "G.22";
"G.25" -> "G.22";
"G.28" -> "G.22";
"G.29" -> "G.22";
"G.33" -> "G.22";
"G.34" -> "G.22";
"G.37" -> "G.22";
"G.24" -> "G.23";
"G.29" -> "G.23";
"G.25" -> "G.24";
"G.28" -> "G.25";
"G.28" -> "G.26";
"G.29" -> "G.28";
"G.34" -> "G.29";
"G.34" -> "G.33";
"G.37" -> "G.33";
"G.37" -> "G.34";
"G.37" -> "G.36";
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Required for
G.39 G.40 G.41 G.42 G.43 G.44 G.45 G.46 G.47 G.48 G.49 G.50 G.51 G.53 G.54 G.56 G.57 G.58 G.59 G.60 G.62 G.64 G.65 G.66 G.67 G.68 G.69 G.70 G.71 G.72 G.73 G.74 G.75 G.79 G.80 G.82 G.83 G.84 G.85 G.87 G.88 G.89 G.90 G.91 G.92 G.93 G.94 G.95 G.97 G.99 G.100 G.101 G.105 G.106 G.110 G.112 G.114 G.116 G.118 G.119 G.120 G.121 G.123 G.124 G.125 G.128 G.134 G.136 G.139 G.140 G.141 G.142 G.143 G.144 G.145 G.149 G.151 G.154 G.157 G.158 G.159 G.160 G.162 G.163 G.164 G.165 G.167 G.210 G.211 G.212 G.213 G.214 G.215 G.216 G.218 G.219 G.220 G.221 G.222 G.223 G.225 G.226 G.227 G.228 G.229 G.231 G.232 G.234 G.235 G.236 G.237 G.238 G.239 G.240 G.241 G.243 G.244 G.245 G.246 G.345 G.377 G.378 G.381 G.382 G.383 G.384 G.385 G.386 G.389 G.390 G.393 G.394 G.395 G.396 G.397 G.398 G.399 G.400 G.401 G.402 G.403 G.404 G.405 G.406 G.407 G.408 G.409 G.410 G.411 G.412 G.413 G.414 G.415 G.416 G.417 G.418 G.419 G.420 G.421 G.423 G.425 G.427 G.429 G.431 G.433 G.434 G.435 G.436 G.437 G.438 G.439 G.440 G.441 G.442 G.443 G.444 G.445 G.446 G.447 G.448 G.449 G.450 G.451 G.452 G.453 G.455 G.457 G.458 G.459 G.460 G.461 G.462 G.463 G.464 G.465 G.466 G.467 G.468 G.470 G.472 G.474 G.476 G.477 G.478 G.479 G.480 G.481 G.482 G.483 G.484 G.485 G.486 G.487 G.488 G.489 G.490 G.491 G.492 G.493 G.494 G.495 G.496 G.497 G.498 G.499 G.500 G.501 G.506 G.507 G.508 G.510 G.511 G.512 G.513 G.514 G.517 G.518 G.520 G.522 G.525 G.526 G.527 G.528 G.530 G.531 G.532 G.536 G.538 G.539 G.540 G.541 G.542 G.543 G.544 G.545 G.546 G.547 G.549 G.550 G.551 G.552 G.554 G.555 G.556 G.557 G.558 G.559 G.560 G.563 G.565 G.570 G.571 G.572 G.573 G.574 G.577 G.578 G.579 G.580 G.582 G.583 G.584 G.586 G.587 G.588 G.589 G.590 G.592 G.594 G.595 G.597 G.600 G.601
