G.29 congruent triangles by SSS — Heath
I.8
If two
triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the
angles equal which are contained by the equal
straight lines .
A
B
C
D
G
E
F
Let < A B C > , < D E F > be two triangles having the two sides / A B / , / A C / equal to the two sides / D E / , / D F / respectively, namely / A B / to / D E / , and / A C / to / D F / ; and let them have the base / B C / equal to the base / E F / ;
I say that the angle < B A C ) is also equal to the angle < E D F ) .
For, if the triangle < A B C > be applied to the triangle < D E F > , and if the point B be placed on the point E and the straight line / B C / on / E F / ,
the point C will also coincide with F ,
because / B C / is equal to / E F / .
Then, / B C / coinciding with / E F / ,
/ B A / , / A C / will also coincide with / E D / , / D F / ;
for, if the base / B C / coincides with the base / E F / , and the sides / B A / , / A C / do not coincide with / E D / , / D F / but fall beside them as / E G / , / G F / ,
then, given two straight lines constructed on a straight line (from its
extremities ) and meeting in a point, there will have been constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and
equal to the former two respectively, namely each to that which has the same extremity with it.
But they cannot be so constructed. [G.28]
Therefore it is not possible that, if the base / B C / be applied to the base / E F / , the sides / B A / , / A C / should not coincide with / E D / , / D F / ;
they will therefore coincide ,
so that the angle < B A C ) will also coincide with the angle < E D F ) , and will be equal to it. [G.23]
If therefore etc. Q. E. D.
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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"];
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"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"];
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"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.3" -> "G.2";
"G.4" -> "G.2";
"G.6" -> "G.2";
"G.7" -> "G.2";
"G.18" -> "G.2";
"G.22" -> "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.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.17" -> "G.13";
"G.25" -> "G.13";
"G.17" -> "G.14";
"G.20" -> "G.14";
"G.25" -> "G.14";
"G.28" -> "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.20" -> "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.24" -> "G.22";
"G.25" -> "G.22";
"G.28" -> "G.22";
"G.29" -> "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";
}
Required for
G.31 G.32 G.34 G.35 G.37 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.98 G.99 G.100 G.101 G.105 G.106 G.107 G.108 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.129 G.130 G.133 G.134 G.135 G.136 G.137 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.161 G.162 G.163 G.164 G.165 G.167 G.168 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.585 G.586 G.587 G.588 G.589 G.590 G.592 G.594 G.595 G.597 G.600 G.601
