Exercises and resolutions The calculator below does, step by step, the calculations with degrees, minutes and seconds for exercises 29 to 33. Try solving it first and use it to check. Enter the angles and choose the operation. It shows each step: 60″ becomes 1′ and 60′ becomes 1°. Ative o JavaScript para ver esta figura interativa. 29 Simplify the following measurements: a) $30^{\circ}70^{\prime}$ · b) $45^{\circ}150^{\prime}$ · c) $65^{\circ}39^{\prime}123^{\prime\prime}$ · d) $110^{\circ}58^{\prime}300^{\prime\prime}$ · e) $30^{\circ}56^{\prime}240^{\prime\prime}$ Ver respostaa) $31^{\circ}10^{\prime}$ · b) $47^{\circ}30^{\prime}$ · c) $65^{\circ}41^{\prime}3^{\prime\prime}$ · d) $111^{\circ}3^{\prime}$ · e) $31^{\circ}$
Why does a complete turn have 360 degrees, and not 100? The answer lies in Mesopotamian astronomy. Who discovered The names straight, acute and obtuse come from Euclid, who defines them in The Elements (definitions I.10 to I.12): straight is the angle equal to its supplementary adjacent; acute, smaller than the straight; obtuse, the biggest. The measurement in degrees has another origin, Babylonian astronomy, as explained in the curiosity “Why 360 degrees?”, below. The words minute and second come from the Latin translations of Claudius Ptolemy (2nd century AD).
Copying an angle and dividing it in half are classic constructions. Dividing it into three equal parts, however, has challenged mathematicians for more than two thousand years. Who discovered The two constructions are in The Elements of Euclid (c. 300 BC): the proposition I.9 divides an angle in half (a angle bisector) and the I.23 copies a given angle onto another straight line. The trisection of the angle with a ruler and compass — dividing any angle into three equal parts — was proven impossible in 1837 by the Frenchman Pierre Wantzel, then just over 20 years old.
Two rays with the same origin form an angle. And the first theorem about angles even has a named discoverer. Who discovered According to Proclus (5th century AD), who cited the History of Geometry by Eudemus of Rhodes, it was Thales of Miletus (c. 624–546 BC) who first realized that angles opposite the vertex are equal; Euclid only later demonstrated it “scientifically”, in proposition I.15. The definition of angle as “inclination of two lines” is in The Elements of Euclid (definition I.8).
A straight line divides the plane into two sides. A figure is convex when it has no “indentations”. The two ideas go together. Who discovered The first to carefully define a convex figure was Archimedes of Syracuse (c. 287–212 BC), in the work On the sphere and the cylinder, where he uses curves “concave to the same side” to compare perimeters. The separation of the plane by a straight line was formalized by Moritz Pasch in 1882: his axiom says that a straight line that enters a triangle on one side must exit on the other. At the end of the 19th century, Hermann Minkowski made convexity an area of mathematics.