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PROPOSITION XX. THEOREM.

739. Two symmetrical spherical triangles are equivalent.

Let ABC and A'B'C' be two symmetrical spherical triangles, having A B, AC and BC equal respectively to A' B', A' C' and B' C'.

We are to prove ▲ A B C A A'B'C'.

Let P and P' be poles of small circles which pass through A, B, C and A', B', C'.

=

Now, since the arcs A B, AC and BC A'B', A'C' and B'C' respectively, the chords of the arcs A B, AC and BC chords of the arcs A' B', A' C' and B'C' respectively.

=

§ 181

.. the plane formed by the chords of these arcs are equal.

§ 108

ABC and A'B'C' which circumscribe these equal plane A are equal.

.. the six spherical distances PA, PB, P' A' etc. are equal, (being polar distances of equal on the same sphere).

.. ▲ P A B, P' A' B' are symmetrical and isosceles.

So likewise are ▲ PB C, P' B'C' and ▲ PAC, P' A' C'. ..A PAB may be applied to ▲ P' A' B' and will coincide with it.

§ 738 So likewise ▲ PBC with ▲ P'B'C' and ▲ PAC with ▲ P'A' C'.

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PROPOSITION XXI. THEOREM.

741. On the same sphere, or equal spheres, two triangles are either equal, or symmetrical and equivalent, if two sides and the included angle of the one be respectively equal to two sides and the included angle of the other.

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In the AA BC and DEF, let A= ▲ D, and the sides A B and AC equal respectively the sides DE and DF.

We are to prove A ABC and D E F equal, or symmetrical and equivalent.

I. When the parts of the two

are in the same order as in A

ABC and DEF,

AABC can be applied to ADEF, as in the corresponding case of plane A, and will coincide with it.

II. When the parts are in

§ 106

reverse order, as in AABC and D'E' F',

construct the ADEF symmetrical with respect to ▲ D'E' F'. Then ▲ D E F will have its and sides equal respectively to those of the A D'E' F'.

Now in the AABC and DE F,
ZAZ D, AB: DE and A C DF,

and these parts are arranged in the same order.

..AABC=AD EF.

$ 737

Case I.

But

A DEFA DEF,

§ 739

..A ABC≈ A D'E' F'.

Q. E. D.

PROPOSITION XXII. THEOREM.

742. Two triangles on the same sphere, or equal spheres, are either equal, or symmetrical and equivalent, if a side and two adjacent angles of the one be equal respectively to a side and two adjacent angles of the other.

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For one of the A may be applied to the other, or to its symmetrical A, as in the corresponding case of plane A.

§ 107

Q. E. D.

PROPOSITION XXIII. THEOREM.

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

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§ 492

(since they are measured by equal sides of the ▲). .. the corresponding dihedral are equal. ..the of the spherical A are respectively equal. .. the are either equal, or symmetrical and equivalent, according as their equal sides are arranged in the same, or reverse

order.

Q. E. D.

PROPOSITION XXIV. THEOREM.

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

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Let the spherical triangles A B C and D E F be mutually

equiangular.

We are to prove AABC and D E F mutually equilateral,

and equal, or symmetrical and equivalent.

Let A A'B'C' and D' E' F' be the polar ▲ of ▲ A B C and DEF respectively.

eral,

Then the AA'B'C' and D'E' F are mutually equilat

§ 731 (in two polar each side of the one is the supplement of the lying opposite to it in the other).

.. ▲ A' B'C' and D' E' F' are mutually equiangular, § 743 (two mutually equilateral ▲ on equal spheres are mutually equiangular). .. AABC and D E F are mutually equilateral;

$731

hence ABC and D E F are either equal, or symmetri

cal and equivalent,

§ 743

(two mutually equilateral ▲ on equal spheres are either equal, or symmetrical

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PROPOSITION XXV. THEOREM.

745. The angles opposite equal sides of an isosceles spherical triangle are equal.

D

In the spherical ▲ ABC, let AB A C.

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Draw arc A D of a great circle, from the vertex A to the middle of the base B C.

Then ▲ A B D and A CD are mutually equilateral.

$743

..A ABD and ACD are mutually equiangular, (two mutually equilateral ▲ on the same sphere are mutually equiangular).

..LB=LC,

(since they are homologous & of symmetrical ▲).

Q. E. D.

746. COROLLARY. The arc of a great circle drawn from the vertex of an isosceles spherical triangle to the middle of the base bisects the vertical angle, is perpendicular to the base, and divides the triangle into two symmetrical triangles.

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