Proposition 12. Reciprocal Polar Triangles

199. Theorem. If one spherical triangle is the polar triangle of another, then the second is the polar triangle of the first.

Given ABC, a spherical A, and A'B'C', its polar A.

Prove that

ABC is the polar ▲ of A'B'C'.

Proof. Since A is the pole of arc B'C',

and

C is the pole of arc A'B',

.. B' is the pole of arc AC.

§ 198

then B' is at the distance of a quadrant from A and C. § 179

$ 180

Similarly,

and

A' is the pole of arc BC,

C' is the pole of arc AB.

.. ABC is the polar ▲ of A'B'C'.

$198

The student should notice that we may just as well start with ABC as the polar triangle of A'B'C', and then prove that A'B'C' is the polar triangle of ABC.

It should also be noticed that it is not necessary that either of the triangles should be wholly within the other. For example, if we draw the figures freehand, taking AB as about 100°, AC as about 100°, and BC as about 30°, one triangle will overlap the other.

Proposition 13. Angle and Side Supplementary

200. Theorem. Any angle of either of two polar triangles is the supplement of the opposite side of the other.

Given the polar A ABC and A'B'C'.

Prove that LA and a'; ZB and b'; /C and c'; ZA' and a; LB' and b; ZC' and c are respectively supplementary.

Proof. Let arcs AB and AC produced meet arc B'C' at D and E respectively.

a'; Band b'; ZC and c' are respectively supplementary.

In a similar way, by considering the angles of AA'B'C' and producing the sides of AABC, the other relations can be proved.

Proposition 14. Sum of the Angles of a Triangle

201. Theorem. The sum of the angles of a spherical triangle is greater than 180° and less than 540°.

Given the spherical AABC.

Prove that 180° <ZA+ZB+ZC<540°.

Proof. Let A'B'C' be the polar ▲ of ABC, with the sides lettered as usual.

of both

Then

or

Now

Also,

A+a'= 180°, B+b'= 180°, ZC+c'=180°. § 200

..ZA+ZB+<C+ a' + b'+ c' = 540°,

ZA+ZB+ZC=540°- (a'+b'+c').

a'+b'+c'<360°. ..ZA+ZB+<C>180°. a'+ b'+ c'>0°.

..ZA+ZB+C<540°.

Ax. 1

Ax. 2

§ 197

202. Triangles classified as to Right Angles. Since (§ 201) a spherical triangle may have two and even three right angles, or two and even three obtuse angles, we find it convenient to speak of a spherical triangle which has two right angles as birectangular, and one which has three right angles as trirectangular.

Exercises. Spherical Polygons

1. Two sides of a spherical triangle are respectively 83° 48' and 64° 59'. What is known concerning the number of degrees in the third side?

2. Three sides of a spherical quadrilateral are respectively 87° 39', 74° 48', and 68° 56'. What is known (§ 197) concerning the number of degrees in the fourth side?

3. If two sides of a spherical triangle are quadrants, the third side measures the opposite angle.

4. In a birectangular spherical triangle the sides opposite the right angles are quadrants, and the side opposite the third angle measures that angle.

Since the angles are right angles, what two planes are perpendicular to a third plane? What two arcs must therefore pass through the pole of a third arc? Then what two arcs are quadrants? How is the third angle measured?

5. Each side of a trirectangular spherical triangle is a quadrant.

6. Three planes passed through the center of a sphere, each perpendicular to the other two, divide the spherical surface into eight congruent trirectangular triangles.

B

Find the number of degrees in the sides of a spherical triangle, given the angles of its polar triangle as follows:

7. 83°; 78°; 64°.

8. 84° 50'; 49° 38'; 104° 40'.

Find the number of degrees in the angles of a spherical triangle, given the sides of its polar triangle as follows:

9. 69° 42′ 38′′; 93° 48′ 8′′; 38° 36' 15".

10. 72° 48' 26"; 104° 38' 43"; 90°.

203. Symmetric Spherical Triangles. If through the center O of a sphere the diameters AA', BB', CC' are drawn, and the points A, B, C and also the points A', B', C' are joined by arcs of great circles, the spherical & ABC and A'B'C' are called symmetric spherical triangles.

In the same way we may form two symmetric polygons of any number of sides. We may then place the symmetric polygons thus formed in any position we choose upon the sphere.

B

204. Relation of Symmetric Triangles. Two symmetric triangles are mutually equilateral and mutually equiangular; but in general they are not congruent, since they cannot be made to coincide by superposition. Thus, in the above figure, if the AABC is made to slide on the sphere until the vertex A falls on A', it is evident that the two triangles cannot be made to coincide since, looked at from the point O, the corresponding parts of the triangles occur in reverse order.

The relation of two symmetric spherical triangles, which is similar to that of a pair of gloves, may be illustrated by cutting the triangles out of the peel of an orange.

A

خ.

205. Symmetric Isosceles Triangles. Consider, however, the case of the symmetric AABC and A'B'C' in which AB= AC, and A'B' = A'C'; that is, the two symmetric triangles are isosceles. Then because AB, AC, A'B', and A'C' are all equal, and the A and A' are equal since they were originally formed by vertical dihedral angles (§ 203), the two triangles can be made to coincide.

B

CB

If two symmetric spherical triangles are isosceles, they are superposable and hence are congruent.