PROPOSITION XVII. THEOREM

677. Two mutually equilateral triangles on the same sphere or on equal spheres are mutually equiangular, and are either congruent or symmetric.

Given two spherical triangles, ABC, A'B'C', on equal spheres, such that ABA'B', BC= B'C', CA = C'A'.

To prove that A=LA', ZB = LB', ZC=LC', and that AABC and A'B'C' are either congruent or symmetric.

Proof. Let O and O' be the centers of the spheres.

Pass a plane through each pair of vertices of each triangle and the center of its sphere.

Then in the trihedral angles at O and O' the face angles are equal each to its corresponding face angle.

§ 167

§ 655 Q. E. D.

.. the corresponding dihedral are respectively equal. § 499 .. the of the spherical A are respectively equal. .. the A are either congruent or symmetric, by § 676. Discussion. In the figures the parts are arranged in the same order, so that the triangles are congruent. They might be arranged as in the figures of § 676.

Discuss the proposition when the triangles are equilateral and each side is a quadrant.

Discuss the proposition when two sides of each triangle are quadrants. What is the corresponding proposition in plane geometry, and why does not the form of proof there given hold here ?

PROPOSITION XVIII. THEOREM

678. Two mutually equiangular triangles on the same sphere or on equal spheres are mutually equilateral, and are either congruent or symmetric.

Given two mutually equiangular spherical triangles T and T' on equal spheres.

To prove that T and T are mutually equilateral, and are either congruent or symmetric.

Proof. Let the AP be the polar triangle of AT, and the AP' be the polar triangle of ▲ T'.

Since the AT and T' are mutually equiangular, Given .. the polar AP and P' are mutually equilateral. § 667 .. the polar AP and P' are mutually equiangular. § 677 But the AT and T' are the polar A of AP and P'. § 666

.. the AT and T' are mutually equilateral. § 667 Therefore the AT and T' are either congruent or symmetric, by $ 677.

Q.E. D.

Discussion. The statement that mutually equiangular spherical triangles are mutually equilateral, and are either congruent or symmetric, is true only when they are on the same sphere or on equal spheres. When the spheres are unequal, the spherical triangles are unequal. In this case, however, their sides have the same arc measure, and therefore have the same ratio as the circumferences or as the radii of the spheres (§ 382).

PROPOSITION XIX. THEOREM

679. In an isosceles spherical triangle the angles opposite the equal sides are equal.

Given the spherical triangle ABC, with AB equal to AC.

Proof. Draw the arc AD of a great circle, from the vertex A

to the mid-point of the base BC.

Then A ABD and ACD are mutually equilateral.

..A ABD and ACD are mutually equiangular. § 677

1. The radius of a sphere is 4 in. From any point on the surface as a pole a circle is described upon the sphere with an opening of the compasses equal to 3 in. Find the area of this circle.

2. The edge of a regular tetrahedron is a. Find the radii r, of the inscribed and circumscribed spheres.

3. Find the diameter of the section of a sphere of diameter 10 in. made by a plane 3 in. from the center.

4. The arc of a great circle drawn from the vertex of an isosceles spherical triangle to the mid-point of the base bisects the vertical angle, is perpendicular to the base, and divides the triangle into two symmetric triangles.

PROPOSITION XX. THEOREM

680. If two angles of a spherical triangle are equal, the sides opposite these angles are equal and the triangle is isosceles.

Given the spherical triangle ABC, with angle B equal to angle C.

Proof. Let A A'B'C' be the polar triangle of A ABC.

1. To bisect a given spherical angle.

2. To construct a spherical triangle, given two sides and the included angle.

3. To construct a spherical triangle, given two angles and the included side.

4. To construct a spherical triangle, given the three sides. 5. To construct a spherical triangle, given the three angles. 6. To pass a plane tangent to a given sphere at a given point on the surface of the sphere.

7. To pass a plane tangent to a given sphere through a given straight line without the sphere.

PROPOSITION XXI. THEOREM

681. If two angles of a spherical triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater; and if two sides are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater.

Given the triangle ABC, with angle C greater than angle B.

Proof. Draw the arc CD of a great circle, making DCB equal to B. Then DB = DC.

§ 680

Now

AD+ DC > AC.

§ 663

.. AD +DB>AC, or AB > AC, by Ax. 9.

Q. E.D.

Given the triangle ABC, with AB greater than AC.

prove that ZC is greater than B.

To

Proof. The

C must be equal to, less than, or greater than

the B.

If

CB, then AB = AC;

§ 680

and if C is less than B, then AB AC, as above. But both of these conclusions are contrary to what is given.

.. ZC is greater than B.

Q.E.D.