G.56 transversal angles of parallel lines — Heath

I.29

A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.

For let the straight line / E F / fall on the parallel straight lines / A B /, / C D /;

I say that it makes the alternate angles < A G H ), < G H D ) equal, the exterior angle < E G B ) equal to the interior and opposite angle < G H D ), and the interior angles on the same side, namely < B G H ), < G H D ), equal to two right angles.

Euclid's Elements, Book I, Proposition 29 Straight line EF falling on parallel straight lines AB and CD at G and H. E A G B C H D F

For, if the angle < A G H ) is unequal to the angle < G H D ), one of them is greater.

Let the angle < A G H ) be greater.

Let the angle < B G H ) be added to each;

therefore the angles < A G H ), < B G H ) are greater than the angles < B G H ), < G H D ).

But the angles < A G H ), < B G H ) are equal to two right angles; [G.37]

therefore the angles < B G H ), < G H D ) are less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet; [G.55]

therefore / A B /, / C D /, if produced indefinitely, will meet;
but they do not meet, because they are by hypothesis parallel.
Therefore the angle < A G H ) is not unequal to the angle < G H D ),
and is therefore equal to it.

Again, the angle < A G H ) is equal to the angle < E G B ); [G.40]

[G.16]therefore the angle < E G B ) is also equal to the angle < G H D ).

Let the angle < B G H ) be added to each;

[G.36]therefore the angles < E G B ), < B G H ) are equal to the angles < B G H ), < G H D ).

But the angles < E G B ), < B G H ) are equal to two right angles; [G.37]

therefore the angles < B G H ), < G H D ) are also equal to two right angles.
Therefore etc. Q. E. D.

Dependency Graph

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"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"]; 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"G.38" [label="G.38", tooltip="G.38 (I.post.4): All right angles are equal.", fillcolor="#444422", URL="../G.38/index.html", target="_top"]; "G.40" [label="G.40", tooltip="G.40 (I.15): Intersecting lines make their vertical angles equal.", fillcolor="#222244", URL="../G.40/index.html", target="_top"]; "G.52" [label="G.52", tooltip="G.52 (I.def.23): Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.", fillcolor="#224422", URL="../G.52/index.html", target="_top"]; "G.55" [label="G.55", tooltip="G.55 (I.post.5): Two lines intersected by a transversal with interior angles on one side summing to less than two right angles meet on that side.", fillcolor="#444422", URL="../G.55/index.html", target="_top"]; "G.56" [label="G.56", tooltip="G.56 (I.29): A transversal cutting parallel lines makes alternate interior angles equal, exterior and opposite interior angles equal, and same-side interior angles supplementary.", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"]; 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"G.10" -> "G.7"; "G.22" -> "G.7"; "G.52" -> "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.40" -> "G.16"; "G.56" -> "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.40" -> "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"; 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Required for

G.57, 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.114, G.116, G.123, G.124, G.125, G.134, G.136, G.139, G.140, G.141, G.142, G.143, G.144, G.145, G.149, G.151, G.157, G.158, G.159, G.160, G.162, G.163, G.164, 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.510, G.511, G.512, G.513, G.514, G.518, G.520, G.522, G.526, 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.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