Page:Spherical Trigonometry (1914).djvu/35

§36] test of equality being that one triangle should be superposable on the other, or on the antipodal triangle of the other. In this way may be proved the first three cases of the following theorem:

Two triangles on the same sphere are either congruent or symmetrically equal, and therefore have all their corresponding elements equal,

(1) When two sides and the included angle of one are respectively equal to two sides and the included angle of the other.

(2) When the three sides of one are respectively equal to the three sides of the other.

(3) When two angles and the adjacent side of one are respectively equal to two angles and the adjacent side of the other.

(4) When the three angles of one are respectively equal to the three angles of the other.

Case (4) has no analogue in plane geometry; it is derived from Case (2) by consideration of the supplemental triangles.

35. The angles at the base of an isosceles spherical triangle are equal.

For if the sides $$AB$$, $$AC$$ are equal, and if $$D$$ be the mid point of $$BC$$, the triangles $$ADB$$, $$ADC$$ have their corresponding sides equal each to each, and therefore are symmetrically equal. Hence the angles $$B$$ and $$C$$ are equal.

If $$AB$$ and $$AC$$ are quadrants, the angles at the base are right angles by Arts. 11 and 9.

36. If two angles of a spherical triangle are equal, the opposite sides are equal.

Since the primitive triangle has two equal angles, the polar triangle has two equal sides; therefore in the polar triangle the angles opposite the equal sides are equal by Art. 35. Hence in the primitive triangle the sides opposite the equal angles are equal.