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It is always possible to draw a straight line perpendicular to a plane from a given point within it.

For let AB be a given plane, and C a point within it, and let it be required to draw from C a straight line to the plane AB.

From any point P without the plane let PQ be drawn I to the plane.

Through C draw CF || to PQ.

Then shall CF be 1 to the plane AB.

For if two straight lines are || and one of them is to a plane, the other is also to the same plane.

(Prop. 7)


There cannot be drawn more than one straight line perpendicular to a plane from a given point within it.


For, if it be possible, from a point C within the plane AB, let two straight lines CF, CG be drawn to the plane AB, and let the plane through CF, CG cut the plane AB in XY

Then 4s FCY, GCY are each rights, and are equal to one another; which is impossible.


(not all situated in the same plane).


Two straight lines which are each of them parallel to the same straight line, which is not in the same plane with them, are parallel to one another.

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Let AB, CF be each of them || to PQ
(AB, CF, PQ not being all in the same plane).
Then shall AB be || to CF.

In PQ take any point X,

and in the plane in which are AB, PQ draw XG 1 to PQ; also in the plane in which are CF, PQ draw XH to PQ.

Then since PQ is 1 to XG and XH;

.. PQ is to the plane GXH,

(Prop. 6)

but AB is || to PQ, .. AB is 1 to the plane


(Prop. 7)

Similarly CF is to the plane GXH;

.. AB is || to CF.

(Prop. 7)


Planes which never meet, however far they are produced in any direction, are called parallel planes.


Planes to which the same straight line is perpendicular are parallel to one another.



Let the straight line AB be 1 to each of the planes CF, GH.

Then shall CF be || GH.

For if not, if possible, let the planes CF, GH be produced to meet in XX.

In YX take any point Z; join AZ, BZ.
Then AB is to the plane CF,


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L BAZ is a

Hence in AABZ two of the 4 s are right s;

which is impossible;

(I. 13)

.. planes CF, GH will never meet however far they are


.. CF, GH are parallel.


If two parallel planes are cut by another plane, their common sections with it are parallel.



Let the two parallel planes AB, CF be cut by the plane HK.

Then shall their common sections GH, KL be parallel.

For if not, if possible, let them meet when produced in X.

Then since GH is in plane AB, .. X is in plane AB; (Prop. 1)

so X is in plane CF;

.. planes AB, CF will meet if produced;
which is impossible, since they are parallel;
.. GH, KL will not meet if produced;

.. GH, KL are parallel.

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