PROPOSITION 16.

If two parallel planes be cut by any plane, their common sections are parallel.

For let the two parallel planes AB, CD be cut by the plane EFGH, and let EF, GH be their common sections ; I say that EF is parallel to GH.

B E F A D K G H C
For, if not, EF, GH will, when produced, meet either in the direction of F, H or of E, G. Let them be produced, as in the direction of F, H, and let them, first, meet at K.
Now, since EFK is in the plane AB,
therefore all the points on EFK are also in the plane AB. [XI. 1]
But K is one of the points on the straight line EFK;
therefore K is in the plane AB.
For the same reason K is also in the plane CD;
therefore the planes AB, CD will meet when produced.
But they do not meet, because they are, by hypothesis, parallel;
therefore the straight lines EF, GH will not meet when produced in the direction of F, H.
Similarly we can prove that neither will the straight lines EF, GH meet when produced in the direction of E, G.
But straight lines which do not meet in either direction are parallel. [I. Def. 23]
Therefore EF is parallel to GH.
Therefore etc.
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "I.def.1" [label="I.def.1", fillcolor="#224422", tooltip="I.def.1: A point is that which has no part.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-1", target="_top"]; "I.def.2" [label="I.def.2", fillcolor="#224422", tooltip="I.def.2: A line is breadthless length.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-2", target="_top"]; "I.def.3" [label="I.def.3", fillcolor="#224422", tooltip="I.def.3: The extremities of a line are points.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-3", target="_top"]; "I.def.4" [label="I.def.4", fillcolor="#224422", tooltip="I.def.4: A straight line is a line which lies evenly with the points on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-4", target="_top"]; "I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.6" [label="I.def.6", fillcolor="#224422", tooltip="I.def.6: The extremities of a surface are lines.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-6", target="_top"]; "I.def.7" [label="I.def.7", fillcolor="#224422", tooltip="I.def.7: A plane surface is a surface which lies evenly with the straight lines on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-7", target="_top"]; "I.def.13" [label="I.def.13", fillcolor="#224422", tooltip="I.def.13: A boundary is that which is an extremity of anything.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-13", target="_top"]; "I.def.14" [label="I.def.14", fillcolor="#224422", tooltip="I.def.14: A figure is that which is contained by any boundary or boundaries.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-14", target="_top"]; "I.def.15" [label="I.def.15", fillcolor="#224422", tooltip="I.def.15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-15", target="_top"]; "I.def.16" [label="I.def.16", fillcolor="#224422", tooltip="I.def.16: And the point is called the centre of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-16", target="_top"]; "I.def.17" [label="I.def.17", fillcolor="#224422", tooltip="I.def.17: A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-17", target="_top"]; "I.def.23" [label="I.def.23", fillcolor="#224422", tooltip="I.def.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-23", target="_top"]; "V.def.1" [label="V.def.1", fillcolor="#224422", tooltip="V.def.1: A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-1", target="_top"]; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-2", target="_top"]; "VII.def.3" [label="VII.def.3", fillcolor="#224422", tooltip="VII.def.3: A number is a part of a number, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-3", target="_top"]; "XI.def.8" [label="XI.def.8", fillcolor="#224422", tooltip="XI.def.8: Parallel planes are those which do not meet.", URL="../../definitions/index.html#xi-def-8", target="_top"]; "XI.1" [label="XI.1", fillcolor="#222244", tooltip="XI.1: A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.", URL="../01/index.html", target="_top"]; "XI.16" [label="XI.16", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XI.16: If two parallel planes be cut by any plane, their common sections are parallel.", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.15" -> "I.def.1"; "I.def.16" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.17" -> "I.def.4"; "I.def.23" -> "I.def.4"; "XI.1" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "XI.1" -> "I.def.7"; "XI.16" -> "I.def.7"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "XI.1" -> "I.def.15"; "I.def.17" -> "I.def.16"; "XI.1" -> "I.def.16"; "XI.1" -> "I.def.17"; "XI.16" -> "I.def.23"; "VII.def.3" -> "V.def.1"; "VII.def.2" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "XI.1" -> "VII.def.3"; "XI.16" -> "XI.def.8"; "XI.16" -> "XI.1"; }

Required for

XI.17, XI.24, XI.25, XI.31, XI.32, XI.33, XI.34, XI.36, XI.37, XI.39, XII.3, XII.4, XII.5, XII.6, XII.7, XII.8, XII.9, XII.10, XII.11, XII.12, XII.13, XII.14, XII.15, XII.17, XII.18

See Commentary on Proposition XI.16.