PROPOSITION XXVII. THEOREM

591 The sum of the face angles of any convex polyedral angle is less than four right angles.

S

'B

HYPOTHESIS. S-ABCDE is a convex polyedral angle.

CONCLUSION. ASB + BSC + etc., is less than four rt. 4.

PROOF

Cut the edges of the polyedral angle by a plane whose section is the convex polygon ABCDE (§ 582), and join any point O within this polygon to all its vertices.

We thus have two sets of triangles, the first set having a common vertex S, the second set having a common vertex O.

Since each set has the same number of triangles, the sum of all the angles in each set is the same.

But the sum of all the base angles of the first set is greater than the sum of all the base angles of the second set, for

<SBA + SBC > < ABC, etc.

§ 590

Therefore the sum of the face angles about S is less than the

sum of the angles about O.

Ax. 6

§ 107

But the sum of the angles about O is four rt. .
Therefore the sum of the face angles about S is less than

four rt. s.

Q. E. D.

592 Two triedral angles are equal or symmetrical, if the three face angles of the one are equal respectively to the three face angles of the other.

444

HYPOTHESIS. In the triedral angles S and S', the angles ASC, ASB, and BSC are equal respectively to the angles A'S'C', A'S'B', and B'S'C'. CONCLUSION. S and S' are equal or symmetrical.

=

PROOF

=

On the edges of the triedral angles, take SA SB = SC S'A' = S'B' = S'C', and complete the ▲ ABC and ̧A'B'C'. Then ASAB = ▲ S'A'B'.

§ 162

... AB = A'B'.

§ 166

Likewise BC = B'C', and AC = A'C'.

..A ABC = A'B'C'.

§ 172

From any point D in SA draw DE in the face SAB and DF in the face SAC each 1 to SA.

Since SAB and SAC are isosceles A,

Const.

DE will meet AB in some point E, and DF will meet AC in

Draw D'E' in the face S'A'B' and D'F' in the face S'A'C'

each L to S'A', and join E'F'.

for AE = A'E', AF = A'F', and ≤ EAF = ZE'A'F'.

§ 162 Proved

But the EDF and E'D'F' measure the diedral SA and S'A' respectively.

§ 562

.. the diedral SA and S'A' are equal. Likewise the diedral SB and SC are equal respectively to the diedral S'B' and S'C'.

Thus far the demonstration applies equally to figures 1 and 2, or 1 and 3.

In figures 1 and 2 the equal parts of the triedral angles are arranged in the same order, and therefore the two triedral angles are equal.

§ 586

In figures 1 and 3 the equal parts of the triedral angles are arranged in reverse order, and therefore the two triedral angles are symmetrical.

§ 587 Q. E. D.

593 COROLLARY. If two triedral angles have the face angles of the one equal to the face angles of the other, each to each, the diedral angles of the one are respectively equal to the diedral angles of the other.

EXERCISES

1169 If two parallel lines are not perpendicular to a plane, their projections upon that plane are parallel lines.

1170 Are two lines necessarily parallel whose projections upon a plane are parallel?

1171 If a straight line intersects two parallel planes, it makes equal angles with the planes.

1172 When will the projection of a circle upon a plane be a circle ? When will the projection be a straight line?

1173 If the projections of n points upon a plane are in a straight line, the n points are all in the same plane.

1174 If a line meets a plane obliquely, with what line in the plane does it make the greatest angle?

1175 A plane can be perpendicular to only one edge of a polyedral angle.

1176 A plane can be perpendicular to only two faces of a polyedral angle.

1177 If two face angles of a triedral angle are equal, the opposite diedral angles are equal.

1178 The three planes bisecting the three diedral angles of a triedral angle intersect in the same line.

1179 If the three face angles of a triedral angle are equal, the three diedral angles are equal.

1180 The three planes passed through the edges of a triedral angle, perpendicular to the opposite faces, intersect in the same line.

1181 Find the locus of a point equidistant from two intersecting planes. 1182 Find the locus of a point equidistant from the vertices of a triangle; of a rectangle.

1183 Find the locus of a point equidistant from the edges of a triedral angle.

1184 Find the locus of a point equidistant from the faces of a triedral angle. [Let AO be the intersection of the three planes bisecting the three diedral angles of the triedral angle (Ex. 1178). Then all points in AO are equidistant from the three faces (§ 570). Any point without AO is without at least two of the angle-bisecting planes, and is therefore not equidistant from the three faces of the triedral angle. Hence AO is the locus required.]

BOOK VII

POLYEDRONS, CYLINDERS, AND CONES

POLYEDRONS

DEFINITIONS

594 A polyedron is a solid bounded by planes.
The faces of a polyedron are the polygons bounding it.
The edges of a polyedron are the intersections of its faces.
The vertices of a polyedron are the intersections of its edges.

595 A diagonal of a polyedron is a straight line joining any two of its vertices not in the same face.

596 A convex polyedron is a polyedron every section of which is a convex polygon.

All polyedrons considered in this work are convex.

597 A tetraedron is a polyedron of four faces.
A hexaedron is a polyedron of six faces.
An octaedron is a polyedron of eight faces.
A dodecaedron is a polyedron of twelve faces.
An icosaedron is a polyedron of twenty faces.