A

Specifying three ANGLES A , B , and C does not uniquely define a TRIANGLE, but any two TRIANGLES with the same ANGLES are SIMILAR. Specifying two ANGLES of a TRIANGLE automatically gives the third since the sum of ANGLES in a TRIANGLE sums to 1808 (/p RADIANS), i.e.,

CpAB:

See also AAS THEOREM, ASA THEOREM, ASS THEOREM, SAS THEOREM, SSS THEOREM, TRIANGLE

Specifying two angles A and B and a side a uniquely determines a TRIANGLE with AREA

K a2 sin B sin C

2 sin A

a2 sin B sin(p A B)

2 sin A : (1)

The third angle is given by

CpAB; (2)

since the sum of angles of a TRIANGLE is 1808 (/p RADIANS). Solving the LAW OF SINES

a

sin A

b

sin B (3)

for b gives

ba sin B

sin A : (4)

Finally,

cb cos Aa cos Ba(sin B cot Acos B) (5)

a sin B(cot Acot B): (6)

See also AAA THEOREM, ASA THEOREM, ASS THEOREM, SAS THEOREM, SSS THEOREM, TRIANGLE

A mechanical counting device consisting of a frame holding a series of parallel rods on each of which beads are strung. Each bead represents a counting unit, and each rod a place value. The primary purpose of the abacus is not to perform actual computations, but to provide a quick means of storing numbers during a calculation. Abaci were used by the Japanese and Chinese, as well as the Romans.