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Proposition 10. Theorem.*

710. Two triangles on the same, or on equal spheres, having a side and the two adjacent angles of one equal respectively to a side and the two adjacent angles of the other, are either equal or equivalent.

Proof. One of the As, or the A symmetric with it, may be applied to the other, as in the corresponding case of plane As.

(105)

Q.E.D.

Proposition 11. Theorem.*

711. Two mutually equilateral triangles, on the same, or on equal spheres, are either equal or equivalent.

Proof. They are mutually equiangular, and equal or symmetrical.

... they are either equal or equivalent.

(706)

(708)

Q.E.D.

712. COR. 1. In an isosceles spherical triangle, the angles opposite the equal sides are equal.

For, in the ▲ ABC, let AB = AC; pass the arc AD of a great through A and the mid. pt. of BC; then the As ABD and ACD are mutually equilateral, and ... mutually equiangular. (706) B

713. COR. 2. The arc of a great circle drawn from the vertex of an isosceles spherical triangle to the middle of the base is perpendicular to the base, and bisects the vertical angle.

POLAR TRIANGLES.

714. One spherical triangle is called the polar triangle of a second spherical triangle when the sides of the first triangle have their poles at the vertices of the second. Thus, if A, B, C are the poles of the arcs of the great circles B'C', C'A', A'B', respectively, then A'B'C' is the polar triangle of ABC.

The great circles, of which B'C', C'A', A'B' are arcs, will form three

B

other spherical triangles on the hemisphere. But the one which is the polar of ABC is that whose vertex A', homologous to A, lies on the same side of BC as the vertex A; and in the same way with the other vertices.

Proposition 12. Theorem.

715. If the first of two spherical triangles is the polar triangle of the second, then the second is the polar triangle of the first.

Hyp. Let A'B'C' be the polar ▲ of

ABC.

To prove that ABC is the polar A of B

A'B'C'.

Proof. Because B is the pole of A'C',

... BA' is a quadrant.

B

(714)

(676)

Because C is the pole of A'B',

... CA' is a quadrant.

... A' is the pole of BC.

(677)

In like manner, B' is the pole of AC, and C' the pole

of AB.

Also, A and A' are on the same side of B'C', and so of the other vertices.

... ABC is the polar ▲ of A'B'C'.

Q.E. D.

Proposition 13. Theorem.

716. In two polar triangles, each angle of one is the supplement of the side opposite to it in the other.

Hyp. Let ABC, A'B'C' be a pair of polar As in which A, B, C, and A', B', C' denote the s, and a, b, c, and a', b', c' denote the sides.

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A'

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b

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Proof. Produce the sides AB, AC, until they meet B'C' at D and H.

Because A is the pole of B'C', AD and AH are quad

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Because B' is the pole of AH, arc B’H = 90°. Because C' is the pole of AD, arc C’D = 90°. ... arc B'H + arc C'D = 180°.

(676) (698)

arc B'Harc C'D =arc B'C' + arc DH. ...arc B'C' + arc DH = 180°.

But

But

arc B'C' = a',

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(Hyp.)

(Proved above)

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In the same way all the other relations may be proved.

Q.E.D.

717. SCH. Two polar triangles are also called supplemental triangles.

Proposition 14. Theorem.

718. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles.

Hyp. Let A, B, C denote the threes of the spherical ▲ ABC. To prove A + B + C > 180° and < 540°.

Proof. Let a', b', c' denote the opposite sides respectively of the polar ▲ A'B'C'. Then

B

A'

a

C

A = 180° — a′, B = 180° — b′, C = 180° — c′. (716)

Adding, A + B + C = 540° — (a' + b′ + c').

b'

Because a', b', c' are sides of a spherical ▲,

. . a' + b' + c' < 360°.

..A+B+C > 180°.

Also, because each of the ▲ is < 2 rt. LS,

..A+B+C < 6 rt. s, or <540°.

(705)

Q.E.D.

719. COR. A spherical triangle may have two, or even three, right angles; also two, or even three, obtuse angles. 720. If a spherical triangle has two right angles, it is called a bi-rectangular triangle; and if a spherical triangle has three right angles, it is called a tri-rectangular triangle.

EXERCISE.

In a bi-rectangular triangle the sides opposite the right angles are quadrants.

Proposition 15. Theorem.*

721. Two mutually equiangular triangles, on the same, or on equal spheres, are mutually equilateral, and are either equal or equivalent.

Hyp. Let the spherical As P and Q be mutually equiangular. To prove that ▲s P and Q are mutually equilateral, and either equal or equivalent.

Proof. Let P' be the polar ▲ of P, and Q' the polar ▲

of Q.

Since P and Q are mutually equiangular,

(Hyp.)

.. their polar As P' and Q' are mutually equilat

eral.

(716)

And since P' and Q' are mutually equilateral,

.. they are mutually equiangular.

(706)

But since P' and Q' are mutually equiangular,

(716)

.. As P and Q are mutually equilateral.
.. As P and Q are either equal or equivalent. (711)

Q.E.D.

NOTE.-Mutually equiangular spherical triangles are mutually equilateral, only when the triangles are on the same, or on equal spheres. When the spheres are unequal, the homologous sides of the triangles are no longer equal, but are proportional to the radii of their spheres (433); the triangles are then similar, as in the case of plane triangles.

722. COR. 1. If two angles of a spherical triangle are equal, the triangle is isosceles.

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