And when the lines containing the plane angle are straight then the angle is called

rectilinear.

Let \(\overline{P Q}\) and \(\overline{Q R}\) be two given segments ^{1}, which meet in the point \(Q\). The inclination of \(\overline{PQ}\) and \(\overline{QR}\), is called a **rectilinear angle**, or simply an **angle**. The point \(Q\) is called the **vertex** of the angle and the segments are called its **legs**.

^{1} This definition holds when we replace the term “segments” by “straight lines” or “rays”.

In order to avoid ambiguities in notation, given a drawing of an angle, we think of an angle as an *oriented rotation of its legs around its vertex*, and always assume that this rotation is **counter clockwise oriented**. With this in our mind, in this adaptation of Euclid’s “Elements”, we always denote all angles *beginning with the leg, where the thought rotation starts, and ending with the leg, where the thought rotation stops*! In order to demonstrate this principle, consider the following figure:

To denote the smaller angle, we write \(\angle{PQR}\), because the leg \(\overline{PQ}\) is being rotated around the vertex \(Q\) to the leg \(\overline{QR}\). To denote the bigger angle, we write \(\angle{RQP}\), because the leg \(\overline{RQ}\) is being rotated around the vertex \(Q\) to the leg \(\overline{QP}\).

| | | | | created: 2014-06-05 21:10:52 | modified: 2019-02-15 06:54:42 | by: *bookofproofs* | references: [626], [628], [6419]

[626] **Callahan, Daniel**: “Euclid’s ‘Elements’ Redux”, http://starrhorse.com/euclid/, 2014

[6419] **Fitzpatrick, Richard**: “Euclid’s Elements of Geometry”, http://farside.ph.utexas.edu/Books/Euclid/Euclid.html, 2007

[628] **Casey, John**: “The First Six Books of the Elements of Euclid”, http://www.gutenberg.org/ebooks/21076, 2007

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