Let the pupil draw a sphere and show that the vertices of one symmetrical triangle are at the ends of the diameters from the vertices of the other.]

649.

What is meant by vertical spherical polyedral angles? Draw a figure to illustrate. How are the corresponding polygons of two vertical polyedral angles related to each other?

650.

A polar triangle is formed by taking the vertices of a given spherical triangle as poles, and then describing three intersecting arcs of great circles. Thus, in the figure.

The great circles B' B", C' C", B'C' form eight spherical triangles, four of which are on the opposite side of the sphere. That one of the eight is a polar which has A' homologous to A on the same side of B C, C' and C on the same side of A B, B' and B on the same side of A C.

652.

PROPOSITION XI.

(1) In the two given As let A B C and A' B' C' be symmetrical isosceles spherical As; i. e., ABA' B', A CA' C',

BCB'C', ▲ ▲

BC B'C', and ▲ A = A', B = B', and

Can they be made to coincide?

CC.

=

[Hint.-Compare A B and A' C', A C and A' B'. [Auth.] A with A']

Compare

Can you superpose▲ A B C upon ▲ A' B' C?
Will the two As coincide? Why?

Give a complete demonstration.

(2) When the two triangles are not isosceles.

In the figure let A B C be any spherical triangle and A'B'C' its polar triangle. Draw on the surface of a sphere.

Can you prove that A B C is the polar triangle of A' B' C' ?

Sug. 1. A is the pole of what arc?
How many degrees from A to B'?
Sug. 2. C is the pole of what are?

How far from C to B'? B' is the pole of what arc?
Pupil complete proof.

Write the theorem, and call it Prop. XII.

654.

PROPOSITION XIII.

In the figure let A B C and A' B'C' be polar triangles. Can you show that each angle of one is measured by the supplement of the side opposite it in the other?

C

Sug. 1. Select any angle as A', and extend its sides, if necessary, to meet the opposite side of the other triangle at D and E. What part of a circumference is the distance from each point, D and E, to the extremities of the opposite sides? i. e., B is the pole of what arc? E is the pole of what arc? Sug. 2. DC + BE = ?

Complete the demonstration, and write Prop. XIII.

655.

PROPOSITION XIV.

D

B

In the given sphere form any spherical triangle, as A B D, and pass planes through the sides and the center of the sphere. What is the figure formed? Compare a face angle of any triedral angle with the sum of the other two. [ $449.]

Can you prove that the sum of any two sides of a spherical triangle is greater than the third side?

656.

Cor. Can you prove that any side of a spherical triangle is greater than the difference of the other two sides?

657.

PROPOSITION XV.

What is the limit of the sum of all the face angles about a polyedral angle? [534]

In the figure let A B C D be any polygon on the surface of the sphere. Pass planes through the sides and the center, C, of the sphere. What is formed by these planes?

Can you prove that the sum of the sides of any spherical polygon is less than the circumference of a great circle?

Give demonstration, and write Prop. XV.

658.

PROPOSITION XVI.

a

In the figure let A B C be any spherical triangle and A'B'C' its polar triangle. How do you measure each angle?