265. Congruence and Symmetry of Polyhedral Angles. Polyhedral angles, like spherical polygons, are congruent when they can be made to coincide. They can be made to coincide when their corresponding parts are equal and arranged in the same order; but they are only symmetric when the parts are equal, but are arranged in opposite order.

Examples of symmetric bodies, that is, of bodies that are equal in all respects, but cannot coincide because their parts are arranged in opposite order, are numerous. Among the most familiar examples are one's hands, feet, ears, shoulders, a right and a left glove, a pair of shoes, the inside and the outside of a door, the two halves of a piano if cut from front to back, the two blades of a pair of scissors, etc.

266. Theorem VIII. (a) Two triangles on the same sphere are either congruent or symmetric if they have two sides and the included angle of one respectively equal to two sides and the included angle of the other.

(b) Two trihedral angles are either congruent or symmetric if they have two face angles and the included dihedral angle of one respectively equal to two face angles and the included dihedral angle of the other.

Superpose if the parts are arranged in like order. If not, superpose one on the triangle that is opposite to the other.

267. Theorem IX. (a) Two triangles on the same sphere are either congruent or symmetric if they have two angles and the included side of one respectively equal to two angles and the included side of the other.

(b) Two trihedral angles are either congruent or symmetric if they have two dihedral angles and the in

cluded face angle of one equal to two dihedral angles and the included face angle of the other.

247. Prove Theorem IX without superposing.

268. Theorem X. (a) In an isosceles spherical triangle, the base angles are equal.

(b) If two face angles of a trihedral angle are equal, the opposite dihedral angles are equal.

Prove as in plane geometry.

269. COR. 1. In an isosceles spherical triangle, the bisector of the vertex angle, the median to the base, the altitude, and the perpendicular bisector of the base, are all one line.

270. COR. 2. Two isosceles symmetric spherical triangles are congruent.

248. Two spherical right triangles are congruent or symmetric if they have two sides of one equal to the corresponding sides of the other.

271. Theorem XI. (a) Two triangles on the same sphere are either congruent or symmetric if they have the three sides of one equal respectively to the three sides of the other.

(b) Two trihedral angles are either congruent or symmetric if they have the three face angles of one respectively equal to the three face angles of the other.

272. Theorem XII. Two symmetric spherical triangles are equivalent.

Draw a circle through the vertices of each triangle and prove the circles equal. Draw the polar distances to the vertices of the triangles, and prove them equal. Show that

the three triangles thus formed in one of the given triangles are respectively congruent to the three triangles formed in the other triangle, -in other words, that symmetric triangles can be divided into congruent parts, and so are equivalent.

249. The median to the base of an isosceles spherical triangle divides the triangle into equivalent triangles.

250. If the opposite sides of a spherical quadrilateral are equal, either diagonal divides it into equivalent triangles; the diagonals bisect each other.

273. Polar Triangles. If for each side of a spherical triangle that one of its poles is taken that is on the same hemispherical surface as the opposite vertex (the circle of which that side is an arc being considered to form the hemispheres), and if these three points are joined by minor arcs of great circles, the resulting triangle is called the polar triangle of the original triangle. In the figure, if

A', B', C', are the poles of BC, CA, and AB respectively, since A and 4' are on the same hemispherical surface with reference to BC, B and B' are on the same hemispherical surface with reference to CA, and C and c' are on the same hemispherical surface with reference to AB, then the great circle arcs A'B', B'C', and C'A' form the polar triangle of triangle ABC.

274. Theorem XIII. If the first of two spherical triangles is the polar triangle of the second, the second is also the polar triangle of the first.

How can a point be proved the pole of an arc? No construction lines need be drawn.

Two triangles, such that each is the polar of the other, are called polar triangles.

275. Theorem XIV. In two polar triangles, each angle of one is measured by the supplement of the side opposite to it in the other.

C

B

B

It is necessary to prove that C is measured by 180° — A'B'. Since C is the pole of A'B', by what are can ≤ c be measured? Show that this arc is 180° - A'B' by using the fact that A' and B' are poles of the great circles through BC and CA respectively. They therefore have what polar distances to those circumferences?

251. The polar triangle of a right triangle has one side a quadrant. 252. If the angles of a triangle are 135°, 97°, and 88°, find the opposite sides of the polar triangle.

253. Show that if one side of a spherical polygon is more than 180°, the polygon is not convex.

254. If the sides of two polar triangles meet, in how many points can they intersect?

255. Two triangles, such that one side of each is a quadrant (quadrantal triangles), are congruent or symmetric if they have two angles of one respectively equal to two angles of the other.

276. Theorem XV. (a) Two triangles on the same sphere are either congruent or symmetric if they have the three angles of one respectively equal to the three angles of the other.

(b) Two trihedral angles are either congruent or symmetric if they have the three dihedral angles of one respectively equal to the three dihedral angles of the other.

For the sides of their polar triangles are equal (why?), so the polar triangles are congruent or symmetric. Show that the sides of the given triangles must therefore be respectively equal.

256. If two angles of a spherical triangle are equal, the opposite sides are equal. (Use the polar triangles.) State this for a trihedral angle.

277. Theorem XVI. The sum of the angles of a spherical triangle is greater than one, and less than three, straight angles.

Let the angles be A, B, C, the opposite sides be a, b, c, and the angles and sides of the polar triangle be A', B', C', and a', b', c':

then A+B+C = 540° − (a' +b+c). Why?

But a+b+c' lies between what limits?

278. Spherical Excess. § 277 proves that the sum of the angles of a spherical triangle is greater than the sum of the angles of a plane triangle. From this it follows that the sum of the angles of any spherical polygon of n sides is more than (n−2) straight angles. The amount by which the sum of the angles of a spherical polygon exceeds the sum of the angles of a plane polygon of the same number of sides is called the spherical excess of that polygon. The spherical excess of a triangle is shown in § 277 to be between zero and two straight angles.