Mathematics · Ch 1 — Angle and its Measurement

Directed Angles

1.1

Directed Angles

In elementary geometry an angle is described just by its two arms, with no sense of "which way" it opens -- an angle of $40°$ drawn turning anticlockwise looks the same size as one drawn turning clockwise, even though the two constructions are genuinely different. A directed angle removes this ambiguity by recording how the angle was generated: start with a ray $OA$ and rotate it about its end-point $O$ until it reaches the position of a ray $OB$ . The ordered pair of rays $(OA, OB)$ , together with that rotation, is the directed angle $\angle AOB$ .

$OA$ is called the initial arm, $OB$ the terminal arm, and $O$ the vertex.
If the rotation from $OA$ to $OB$ is anticlockwise, the measure of $\angle AOB$ is taken as positive.
If the rotation is clockwise, the measure is taken as negative.

Because the pair is ordered, $(OA,OB) \ne (OB,OA)$ in general, so $\angle AOB \ne \angle BOA$ even when both rotations sweep out the same amount of turning -- one is simply the reverse of the other.

Special directed angles

Zero angle: if $OA$ does not rotate at all, so the terminal arm coincides with the initial arm, the angle formed is a zero angle, $m\angle AOB = 0°$ .
One rotation angle: if $OA$ turns all the way round and comes back to coincide with itself, $m\angle AOB = 360°$ .
Straight angle: if, after rotating, $OA$ and $OB$ point in exactly opposite directions (so $A$ , $O$ , $B$ are collinear), the angle is a straight angle, $m\angle AOB = 180°$ -- exactly half of one rotation.
Right angle: one quarter of one rotation, $m\angle AOB = 90°$ -- equivalently, half of a straight angle. One full rotation therefore equals four right angles.

Angles in standard position

Place the vertex $O$ of a directed angle at the origin of a rectangular coordinate system, with the initial arm along the positive $X$ -axis. Such an angle is said to be in standard position. Only the direction of the terminal arm, together with the sign of the rotation, then distinguishes one standard-position angle from another -- an angle whose initial arm is not along the positive $X$ -axis is simply not in standard position.

Quadrant of an angle, quadrantal angles

A directed angle in standard position is said to lie in a particular quadrant according to which quadrant its terminal ray falls in -- I, II, III or IV. If the terminal ray falls exactly on one of the axes (positive/negative $X$ -axis, positive/negative $Y$ -axis) rather than strictly inside a quadrant, the angle is called a quadrantal angle (for example $0°, 90°, 180°, 270°, 360°$ and their negatives).

To find which quadrant an angle whose measure is outside $[0°,360°)$ (or negative) lies in, first reduce it modulo $360°$ to an equivalent angle between $0°$ and $360°$ (adding or subtracting whole rotations never changes the terminal ray), and read off the quadrant of that reduced angle:

Quadrant	Range of the reduced angle
I	$0°$ to $90°$
II	$90°$ to $180°$
III	$180°$ to $270°$
IV	$270°$ to $360°$

Co-terminal angles

Two or more directed angles in standard position that share the same terminal ray (though arrived at by different amounts of rotation) are called co-terminal angles. For instance $30°$ , $390°$ and $-330°$ all end on the same terminal ray, because each differs from the others by a whole number of complete rotations: $390° - 30° = 360° = 1\times 360°$ and $390° - (-330°) = 720° = 2\times 360°$ .

Key fact: two directed angles in standard position are co-terminal if and only if the difference between their measures is an integral multiple of $360°$ .

Figure 1.1Activity 1(a): angle ABC of 40°, anticlockwise

What this figure shows. A 40° angle ABC drawn by rotating the initial ray anticlockwise from AB to AC, illustrating that a directed angle's sense of rotation is part of how it is drawn.

1.1: Activity 1(a): angle ABC of 40°, anticlockwise.

Figure 1.2Activity 1(b): angle ABC of 40°, clockwise

What this figure shows. The same 40° angle ABC drawn a second time, this time by rotating the initial ray clockwise, to show that two angles of equal size can be genuinely different directed angles.

1.2: Activity 1(b): angle ABC of 40°, clockwise.

Figure 1.3Directed angle AOB formed by rotating ray OA to OB

What this figure shows. Ray OA about vertex O rotated to the position of ray OB; part (a) shows the anticlockwise rotation forming angle AOB, part (b) shows the same two rays reversed (initial arm OB, terminal arm OA) to demonstrate that the ordered pair (OA,OB) is not the same as (OB,OA).

1.3: Directed angle AOB formed by rotating ray OA to OB.

Figure 1.4Zero angle

What this figure shows. Ray OA with no rotation at all, so the terminal ray OB coincides exactly with the initial ray OA, shown as a single ray from O. This is the zero angle case in the directed-angle convention -- the two rays coincide because the terminal ray has not swept through any rotation from the initial ray.

1.4: Zero angle.

Figure 1.5One rotation angle (360°)

What this figure shows. Ray OA sweeping all the way around the vertex O and returning to coincide with its own starting position, with a curved arrow indicating the full 360° sweep.

1.5: One rotation angle (360°).

Figure 1.6Straight angle

What this figure shows. Rays OA and OB drawn in exactly opposite directions through the common vertex O, so that A, O and B lie on one straight line. This is the straight-angle (180 degree) case in the directed-angle convention -- a half-turn rotation carries the initial ray all the way around to point exactly opposite its start.

1.6: Straight angle.

Figure 1.7Right angle

What this figure shows. Ray OB positioned a quarter turn (one right angle) from ray OA, with the usual small square mark at O indicating the 90° angle. This is the right-angle (90 degree) case in the directed-angle convention, matching the quarter-turn rotation from OA to OB.

1.7: Right angle.

Figure 1.8Angles in standard position and in a quadrant

What this figure shows. Rays OP, OQ and OR drawn from the origin O, each sharing the positive X-axis OX as the common initial arm, with their terminal rays landing in the first, second and third quadrants respectively; a fourth ray OQ (paired with OP, not sharing the X-axis as initial arm) is drawn separately to show angle POQ is not in standard position.

1.8: Angles in standard position and in a quadrant.

Figure 1.9Quadrantal angles

What this figure shows. Rays OP, OQ, OR and OS drawn from the origin, each in standard position, with terminal rays lying exactly along the positive Y-axis, the negative X-axis, the negative Y-axis and back along the positive X-axis respectively, so each angle XOP, XOQ, XOR, XOS is a quadrantal angle.

1.9: Quadrantal angles.

Figure 1.10Co-terminal angles 30°, 390°, −330°

What this figure shows. Part (a) shows three directed angles, all sharing the same initial ray OA and the same terminal ray OB, drawn with a small anticlockwise arc (30°), just over one full anticlockwise loop (390°) and a large clockwise loop (−330°); part (b) relabels the same pair of rays as initial arm OA and terminal arm OB to emphasise that all three rotations end on the same terminal ray.

1.10: Co-terminal angles 30°, 390°, −330°.

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