PROPOSITION XXV. THEOREM

871. If two sides of a spherical triangle are unequal, the opposite angles are unequal, and the greater angle is opposite the greater side.

angle ABC>ACB.

If ABC were equal to ACB, then AC would equal AB.

$860 If ABC were less than ACB, then AC would be less than AB. Both these conclusions are contrary to the hypothesis. Therefore

ABC ACB.

$ 869

Q. E. D.

872. COR. If two face angles of a triedral angle are unequal, the opposite diedral angles are unequal, and the greater diedral angle is opposite the greater face angle.

873. Two triangles on the same sphere are equal: I. If two sides and the included angle of one are equal respectively to two sides and the included angle of the

other.

II. If a side and the two adjacent angles of one are equal respectively to a side and the two adjacent angles of the other.

III. If the three sides of one are equal respectively to the three sides of the other.

Provided in each case that the parts given equal are arranged in the same order in both triangles.

Proof. In each case the corresponding triedral angles are equal. S$ 595, 596, 597

They can therefore be placed in coincidence.
At the same time the triangles coincide.
Therefore the two given triangles are equal.

Q. E. D.

874. Two triangles on the same sphere are symmetrical: I. If two sides and the included angle of one are equal respectively to two sides and the included angle of the

other.

II. If a side and the two adjacent angles of one are equal respectively to a side and the two adjacent angles of the

other.

III. If the three sides of one are equal respectively to the three sides of the other.

Provided in each case that the parts given equal are arranged in opposite order in the two triangles.

Proof. In each case the corresponding triedral angles at the centre are symmetrical.

$ 603 Therefore the two given triangles are symmetrical. § 851

POLAR TRIANGLES

Q. E. D.

875. Def.--If, with the vertices of a spherical triangle as poles, arcs of great circles are described, these arcs will divide the spherical surface into eight triangles. One of these is called the polar triangle of the given triangle.

The method of selecting the polar triangle from the eight is as follows: Call the given triangle ABC and the polar triangle A'B'C'. Then A' is one of the intersections of the arcs described from B and C as poles; that one which is less than a quadrant's distance from A. In a similar way B' and C' are determined.

876. If one spherical triangle is the polar triangle of another, then, reciprocally, the second spherical triangle is the polar triangle of the first.

GIVEN

that A'B'C' is the polar triangle of ABC. TO PROVE that ABC is the polar triangle of A'B'C'.

Since B is the pole of A'C', the distance A'B is a quadrant; since C is the pole of A'B', the distance A'C is a quad

rant.

819

$ 821

Similarly, B' is the pole of CA, and C' is the pole of AB. Since also the distances AA', BB', and CC' are each less than a quadrant, ABC is the polar triangle of A'B'C'.$875

Q. E. D.

877. In two polar triangles, each angle of one is measured by the supplement of the side of which its vertex is the pole in

GIVEN the polar triangles ABC and A'B'C'. Let A, B, C, and A', B', C' denote their angles, measured in degrees, and a, b, c, and a', b', c' the sides respectively opposite these angles, also measured in degrees.

TO PROVE A' + a 180°, B' + b = 180°, C' + c = 180°,

=

4+ a' = 18c°, B+ b' 180°, C+ c = 180°.

Produce A'B' and A'C' to meet BC at R and S.

Then, since B is the pole of A'S and C the pole of A'R, BS and CR are quadrants.

$819

But BC-a, and RS measures the angle A'.

$ 836 II