PROPOSITION 14.

Planes to which the same straight line is at right angles will be parallel.

For let any straight line AB be at right angles to each of the planes CD, EF; I say that the planes are parallel.

C G A D K F B E H

For, if not, they will meet when produced. Let them meet; they will then make, as common section, a straight line. [XI. 3] Let them make GH; let a point K be taken at random on GH, and let AK, BK be joined.

Now, since AB is at right angles to the plane EF, therefore AB is also at right angles to BK which is a straight line in the plane EF produced; [XI. Def. 3]

therefore the angle ABK is right.
For the same reason the angle BAK is also right.

Thus, in the triangle ABK, the two angles ABK, BAK are equal to two right angles:

[I. 17]which is impossible.

Therefore the planes CD, EF will not meet when produced;

[XI. Def. 8]therefore the planes CD, EF are parallel.
Therefore planes to which the same straight line is at right angles are parallel.

Dependency Graph

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"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.8" [label="I.def.8", fillcolor="#224422", tooltip="I.def.8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-8", target="_top"]; "I.def.10" [label="I.def.10", fillcolor="#224422", tooltip="I.def.10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-10", target="_top"]; 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"I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", 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"]; "I.post.1" [label="I.post.1", fillcolor="#444422", tooltip="I.post.1: To draw a straight line from any point to any point.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-1", target="_top"]; "I.post.2" [label="I.post.2", fillcolor="#444422", tooltip="I.post.2: To produce a finite straight line continuously in a straight line.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-2", target="_top"]; "I.post.3" [label="I.post.3", fillcolor="#444422", tooltip="I.post.3: To describe a circle with any centre and distance.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-3", target="_top"]; "I.cn.1" [label="I.cn.1", fillcolor="#442222", tooltip="I.cn.1: Things which are equal to the same thing are also equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-1", target="_top"]; "I.cn.2" [label="I.cn.2", fillcolor="#442222", tooltip="I.cn.2: If equals be added to equals, the wholes are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-2", target="_top"]; "I.cn.3" [label="I.cn.3", fillcolor="#442222", tooltip="I.cn.3: If equals be subtracted from equals, the remainders are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-3", target="_top"]; "I.cn.4" [label="I.cn.4", fillcolor="#442222", tooltip="I.cn.4: Things which coincide with one another are equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-4", target="_top"]; 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"I.7" [label="I.7", fillcolor="#222244", tooltip="I.7: Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its…", URL="../../../../volume_I/book_i/propositions/07/index.html", target="_top"]; "I.8" [label="I.8", fillcolor="#222244", tooltip="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…", URL="../../../../volume_I/book_i/propositions/08/index.html", target="_top"]; "I.11" [label="I.11", fillcolor="#222244", tooltip="I.11: To draw a straight line at right angles to a given straight line from a given point on it.", URL="../../../../volume_I/book_i/propositions/11/index.html", target="_top"]; "I.13" [label="I.13", fillcolor="#222244", tooltip="I.13: If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.", URL="../../../../volume_I/book_i/propositions/13/index.html", target="_top"]; "I.17" [label="I.17", fillcolor="#222244", tooltip="I.17: In any triangle two angles taken together in any manner are less than two right angles.", URL="../../../../volume_I/book_i/propositions/17/index.html", target="_top"]; "XI.def.3" [label="XI.def.3", fillcolor="#224422", tooltip="XI.def.3: A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane.", URL="../../definitions/index.html#xi-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.3" [label="XI.3", fillcolor="#222244", tooltip="XI.3: If two planes cut one another, their common section is a straight line.", URL="../03/index.html", target="_top"]; "XI.14" [label="XI.14", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XI.14: Planes to which the same straight line is at right angles will be 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.post.1" -> "I.def.1"; "I.2" -> "I.def.1"; "I.7" -> "I.def.1"; "I.11" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.8" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.2" -> "I.def.3"; "I.7" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.10" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.19" -> "I.def.4"; "I.def.23" -> "I.def.4"; "I.post.1" -> "I.def.4"; "I.post.2" -> "I.def.4"; "I.1" -> "I.def.4"; "I.2" -> "I.def.4"; "I.3" -> "I.def.4"; "I.4" -> "I.def.4"; "I.5" -> "I.def.4"; "I.7" -> "I.def.4"; "I.8" -> "I.def.4"; "I.11" -> "I.def.4"; "I.13" -> "I.def.4"; "XI.def.3" -> "I.def.4"; "XI.3" -> "I.def.4"; "XI.14" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "XI.def.3" -> "I.def.7"; "XI.3" -> "I.def.7"; "XI.14" -> "I.def.7"; "I.def.10" -> "I.def.8"; "I.4" -> "I.def.8"; "I.5" -> "I.def.8"; "I.8" -> "I.def.8"; "I.13" -> "I.def.8"; "I.17" -> "I.def.8"; "I.11" -> "I.def.10"; "I.13" -> "I.def.10"; "I.17" -> "I.def.10"; "XI.def.3" -> "I.def.10"; "XI.14" -> "I.def.10"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.4" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "I.post.3" -> "I.def.16"; "I.def.20" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.17" -> "I.def.19"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "XI.14" -> "I.def.23"; "I.1" -> "I.post.1"; "I.2" -> "I.post.1"; "I.5" -> "I.post.1"; "I.7" -> "I.post.1"; "I.2" -> "I.post.2"; "I.5" -> "I.post.2"; "I.17" -> "I.post.2"; "I.1" -> "I.post.3"; "I.2" -> "I.post.3"; "I.3" -> "I.post.3"; "I.1" -> "I.cn.1"; "I.2" -> "I.cn.1"; "I.3" -> "I.cn.1"; "I.13" -> "I.cn.1"; "I.13" -> "I.cn.2"; "I.2" -> "I.cn.3"; "I.4" -> "I.cn.4"; "I.2" -> "I.1"; "I.11" -> "I.1"; "I.3" -> "I.2"; "I.5" -> "I.3"; "I.11" -> "I.3"; "I.5" -> "I.4"; "I.7" -> "I.5"; "I.8" -> "I.7"; "I.11" -> "I.8"; "I.13" -> "I.11"; "I.17" -> "I.13"; "XI.14" -> "I.17"; "XI.14" -> "XI.def.3"; "XI.14" -> "XI.def.8"; "XI.14" -> "XI.3"; }

Required for

XI.15