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).

Figure from The School of Athens attributed to Eudemus of Rhodes; uncertain identification.
Figure from The School of Athens attributed to Eudemus of Rhodes; uncertain identification.Rafael Sanzio; clipping of reproduction by Jic and FranksValli · Wikimedia Commons · Public domain
Thales of Miletus, later artistic representation.
Thales of Miletus, later artistic representation.Luca Giordano · Wikimedia Commons · Public domain
Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.
Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

The story

What, after all, is an angle? Proclus records that the ancients discussed whether it would be a quantity (something that is measured), a quality (like a form) or a relation between two lines. The modern definition avoids the discussion: an angle is the figure formed by two rays of the same origin.

The attribution to Thales comes from authors who lived many centuries after him, and we have no text by Thales himself. Even so, it is valuable: it shows that the Greeks already saw in it the beginning of a geometry in which facts need justification.

In everyday life

  • Scissors and pliers. When opening the blades, the handles open at the same angle: they are angles opposite the apex.
  • Clock hands. At 3 h they form a right angle; at 6 h, a shallow angle.
  • Opening doors. A door with a shock absorber opens to a maximum of a certain angle so as not to hit the wall.
  • Field of view. Cell phone cameras inform the angle they can see; “wide angle” covers more.

Sources: Tales (MacTutor) · Elements, proposition I.15 · Elements, definition I.8.

Theory and constructions

DefiniçãoAngle

Angle is the meeting of two rays of the same origin, not contained in the same straight line (not collinear).

$$A\hat{O}B = \overrightarrow{OA} \cup \overrightarrow{OB}$$

The point $O$ is the vertex of the angle, and the rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are the sides. We also write $a\hat{O}b$ or simply $\widehat{ab}$, when $a$ and $b$ are the sides.

DefiniçãoIndoor and outdoor

  • The interior of $A\hat{O}B$ is the intersection of two open half-planes: $\alpha_1$, with its origin on the line $OA$ and which contains $B$, and $\beta_1$, with its origin on the line $OB$ and which contains $A$. The interior of an angle is convex.
  • The meeting of an angle with its interior is an angular sector (also called “convex angle”).
  • The exterior of $A\hat{O}B$ is the set of points that do not belong to either the angle or its interior: it is the meeting of the open half-planes opposite $\alpha_1$ and $\beta_1$. The exterior is concave.

The figure below is an angle on a protractor. Drag $B$ and switch between inside, outside and adjacent supplementary (see section IV).

Drag $B$. The protractor measures $A\hat{O}B$ in degrees and classifies the angle.

Consecutive, adjacent and vertex-opposite angles

DefiniçãoConsecutive and adjacent

  • Two angles are consecutive if, and only if, a side of one of them is also a side of the other.
  • Two consecutive angles are adjacent if and only if they have no common interior points.
$A\hat{O}B$ and $A\hat{O}C$ are consecutive with common side $\overrightarrow{OA}$, but they are not adjacent. $A\hat{O}B$ and $B\hat{O}C$ are adjacent.

DefiniçãoOpposites by the vertex (o.p.v.)

Two angles are vertex-opposed if, and only if, the sides of one of them are rays opposite the sides of the other.

Two competing lines determine two pairs of angles opposite the vertex.