The excess of the sum of the angles of a spherical triangle over 180° is called its spherical excess.

The excess of the sum of the angles of a spherical polygon of n sides over (n 2) 180° is called its spherical excess.

*Proposition 331 Theorem

In a bi-rectangular spherical triangle the sides opposite the right angles are quadrants, and the third side of the measure of the third angle.

Consult Prop. 320, Prop. 234, and Prop. 321.

COR. I. If two sides of a spherical triangle are quadrants, the third side is the measure of the opposite angle.

COR. II. Each side of a tri-rectangular spherical triangle is a quadrant.

COR. III. Three planes passing through the centre of a sphere mutually perpendicular to each other divide the sphere into eight congruent tri-rectangular triangles.

NOTE. On the earth the triangle formed by two meridians and the equator is an example of Prop. 331. The longitude between two places is the angle formed by the meridians passing through these places, and it is measured by the arc of the equator intercepted between the meridians.

Ex. 1177. If two sides of a spherical triangle are quadrants, the angles opposite these sides are right angles.

Ex. 1178. If three sides of a spherical triangle are quadrants, the triangle is tri-rectangular.

Ex. 1179. The polar triangle of a bi-rectangular spherical triangle is also bi-rectangular.

Ex. 1180. A tri-rectangular spherical triangle coincides with its polar triangle.

On the same sphere, or on equal spheres, two spherical triangles are either congruent or symmetrical and equivalent when a side and the two adjacent angles of one are equal respectively to a side and the two adjacent angles of the other.

CASE I. When the equal parts are arranged in the same order, the triangles are congruent.

The method of proof is the same as that of Prop. 9.

CASE II. When the equal parts are arranged in reverse order, the triangles are symmetrical and equivalent.

Hypothesis. In the AABC and A'B'C', BC = B'C', LBLB', and C = C', and the parts are arranged in reverse order.

Conclusion, AABC and A'B'C' are symmetrical and equivalent.

Proof. Let A A"B"C" be symmetrical to ▲ A'B'C'.

L

Then BC = B"C", B = B", and C = C". (?)

ZB

CONGRUENCE AND SYMMETRY OF TRIANGLES 423

Proposition 333

Theorem

On the same sphere, or on equal spheres, two spherical triangles are either congruent or symmetrical and equivalent when two sides and the included angle of one are equal respectively to two sides and the included angle of the other.

Use a method similar to that used in Prop. 332.

Proposition 334 Theorem

On the same sphere, or on equal spheres, if two spherical triangles are mutually equilateral, they are also mutually equiangular, and are either congruent or symmetrical and equivalent.

HINT. Draw radii from the center of the sphere to the vertices of the A. Consult Prop. 126, Prop. 244, and Prop. 320.

Proposition 335 Theorem

On the same sphere, or on equal spheres, if two spherical triangles are mutually equiangular, they are also mutually equilateral, and are either congruent or symmetrical and equivalent.

HINT. Construct the polar of the given A, and consult Prop. 329 and Prop. 334.

Ex. 1181. Every point in the great circle perpendicular to a great circle arc at its mid-point is equidistant from the extremities of the arc. Ex. 1182. State and prove the converse of Ex. 1181.

* Proposition 336 Theorem

In an isosceles spherical triangle, the angles opposite the equal sides are equal.

HINT. Draw great arc from the vertex of the A to the mid-point of the base, and consult Prop. 334.

COR. The great circle arc drawn from the vertex of an isosceles spherical triangle to the mid-point of base bisects the vertical angle and is perpendicular to the base.

* Proposition 337 Theorem

If two angles of a spherical triangle are equal, the sides opposite these angles are equal, and the triangle is isosceles.

HINT. Construct the polar ▲ of the given ▲, and consult Prop. 329 and Prop. 336.

* Proposition 338 Theorem

If two angles of a spherical triangle are unequal, the side opposite the greater angle is longer than the side opposite the less.

The method of proof is the same as that of Prop. 17.

* Proposition 339 Theorem

If two sides of a spherical triangle are unequal, the angle opposite the longer side is greater than the angle opposite the shorter.

Use the indirect method.

Ex. 1183. If two adjacent sides of a spherical quadrilateral are longer than the other two sides, the angle included by the two shorter sides is greater than angle included by the two longer sides.

SPHERICAL AREAS AND VOLUMES

A lune is a portion of a sphere bounded by two halves of great circles. The angle formed by the semicircles which bound a lune is called the angle of the lune. For example, ABCDA is a lune, and BAD is its angle. The lune may be designated by "lune A."

D

B

A spherical wedge, or ungula, is a solid bounded by a lune and the planes of its bounding semicircles. The lune is called the base of the wedge, and the diameter in which the planes of the semicircles intersect is called the edge of the wedge. The angle of the lune is also called the angle of the wedge.

A zone is a portion of a sphere included between two parallel planes. The sections made by the parallel planes are called the bases of the zone, and the perpendicular distance between the planes is called the altitude of the zone. A zone of one base is a zone one of whose bounding planes is tangent to the sphere.

A spherical segment is a portion of the volume of a sphere included between two parallel planes. The portions of the planes bounding the segments are called the bases of the segment, and the perpendicular distance between the planes is called the altitude of the segment.

A spherical segment of one base is a segment one of whose bounding planes is tangent to the sphere.

NOTE. The curved surface of a segment is a zone.