An angle records a turn and its direction
Move a clock hand yourself. Where the hand rests before you move it is the initial side; initial means starting. Where it stops is the terminal side; terminal means ending, like the last stop on a bus line. The angle records how far you turned and which way. A full turn is 360 degrees, written 360°.
The sign tells only the direction. A plus sign, or no sign, means counterclockwise, against the way clock hands move. A minus sign (−) means clockwise, the way clock hands move. Example: start with the hand at 3 o'clock. For 130°, turn counterclockwise, up and over, and stop pointing up and to the left. For −130°, turn clockwise, down first, and stop pointing down and to the left.
Why a sign at all: opposite turns cancel, so +130° followed by −130° brings the hand back to 3 o'clock. The sign stays the same the whole way around, so 200° is still positive after it passes 180°. The sign also ignores where the hand starts: from 12 o'clock, a 25° counterclockwise turn is +25°.
In plain wordsPicture a clock hand you can move yourself. Draw a faint copy of where it starts. That starting ray is the Initial side. Move the hand around its center, then draw where it finishes. That ending ray is the Terminal side. Terminal means ending, as in the last stop of a bus route. The center is the Vertex. The movement is a Rotation, and the angle records how much you turned and which way. Clockwise follows ordinary clock hands. Counterclockwise goes the other way. You can keep turning for several laps. The starting and ending pictures alone do not tell you how many laps you made.
- Adding signed numbers. Adding a negative subtracts its size: 90 + (−65) = 90 − 65 = 25.
- Degree landmarks. From right, a counterclockwise quarter turn reaches up at 90°, a half turn reaches left at 180°, and a full turn returns right at 360°.
Say a positive counterclockwise turn or a negative clockwise turn.
The initial side is where the rotating ray begins, the terminal side is where it ends, and the signed angle records the movement between them.
- counterclockwise turn: θ > 0
- clockwise turn: θ < 0
- no turn: θ = 0
- ending direction from right = starting direction from right + signed turn
A clock hand's starting and ending positions are photographs; the signed angle is its complete trip record.
The faint starting hand and highlighted ending hand are positions. The curved arrow between them records the signed movement. Here the start is 90° and the clockwise movement is −40°, even though the hand ends above the horizontal axis.
A bus can return to the same stop after one lap or three laps. The stop tells you where it ends, while its trip record tells you how far it traveled. Likewise, an angle records the whole turn, not only its terminal side.
+50° and −50° are opposite moves. Their sum is 0°, so doing them in succession returns the hand to its original position. Assigning opposite signs lets addition describe turning and undoing a turn.
Say counterclockwise positive before you calculate. Everyone reading your answer then knows which direction + means. A different convention could work if everyone changed together, but mixing conventions would reverse the meaning of the same signed measurement.
.1Initial side
The Initial side is the door's position before you open it. Keep that starting picture as a reference. It is a whole ray leaving the hinge, not the short curved arrow that shows the movement.
- Initial means starting.
- The ray begins at the vertex.
- It may point in any direction unless standard position is required.
- Ray and vertex. A ray begins at one endpoint. Both angle sides begin at the vertex.
You are identifying the ray where the movement begins. A hand starts pointing right and turns upward. Which ray is its initial side?
- The rightward ray is the initial side.It shows where the hand was before the movement.
- Initial means starting. Place your finger on that ray before following the turn.
.2Terminal side
The Terminal side is the door's position after you stop opening it. The door has stopped moving, but its final direction can be drawn as a ray extending from the hinge. That final straight ray is what terminal side means.
- Terminal means ending.
- The ray starts at the same vertex as the initial side.
- Several different turns can finish on this same ray.
- Initial side. The initial side shows before the movement; the terminal side shows after it.
You are identifying the ray where the movement stops. From the rightward ray, turn counterclockwise through 90° and stop. Which ray is terminal?
- The upward ray is the terminal side.A quarter turn from right finishes pointing up.
- Terminal means ending, like the last stop on a route.
.3Positive angle
For a Positive angle, move against the direction of ordinary clock hands. Starting right, a quarter turn goes upward. Starting somewhere else changes the finish, but the counterclockwise movement keeps its positive sign.
- Counterclockwise means positive.
- +90° starting from the right finishes straight up.
- There is no upper limit on the number of turns.
- Full revolution. One full revolution is 360°. One full turn followed by 60° totals 420°.
You are totaling a complete counterclockwise lap and the additional turn. Turn counterclockwise through one full revolution and another 60°. Give the signed measure.
- 360° + 60° = +420°.A full revolution is 360°, and both parts of the movement are counterclockwise.
- Starting right, a positive turn first moves upward. Read the arrow rather than the final height.
.4Negative angle
For a Negative angle, follow the way ordinary clock hands move. Starting right, a quarter turn goes downward. The minus sign describes that direction. It does not describe a negative physical length or a point below the page in every drawing.
- Clockwise means negative.
- −90° starting from the right finishes straight down.
- A negative turn can be larger than a full lap in size.
- Opposite signed turns. A +90° turn undoes a −90° turn because 90° + (−90°) = 0°.
You are giving a quarter-turn size the sign that matches its direction. From the rightward ray, turn clockwise through a quarter of a revolution.
- The angle is −90°.360° ÷ 4 = 90° gives the size, and clockwise supplies the minus sign.
- Write the minus sign as soon as you identify clockwise movement.
.5Starting somewhere else
Move the starting hand from right to up without changing your rule for positive and negative. It is like moving a game's starting square: forward and backward still mean the same movements. Standard position specifies a start; it does not define the sign. Standard position is a later name for beginning at the map’s center and pointing right. The full definition comes after coterminal angles.
- The sign depends only on direction.
- The initial direction and the signed angle are different numbers.
- For direction readings from right: ending direction = starting direction + signed turn, allowing full turns.
- Signed addition. Adding a negative amount subtracts its size: 90 + (−40) = 50.
You are recording the amount and direction of the turn and where it finishes. Start pointing up at direction 90°, then turn clockwise through 40°. State the signed angle and final direction.
- The signed angle is −40°.Clockwise supplies the negative sign wherever the initial side points.
- The final direction is 90° − 40° = 50° from the right.A clockwise movement decreases the counterclockwise direction reading.
- Signed angle: −40°.
- Final direction: 50° from the right.
- Label the starting direction, the signed movement, and the ending direction separately.
- Locate the initial side, the hand's starting position.
- Follow the curved arrow from that starting ray to the terminal side.
- Use a positive sign for counterclockwise and a negative sign for clockwise.
- Count the whole movement, including any full laps. Do not assign a sign from the ending ray's location.
Record the movement before locating its finish
- Mark the initial side and identify the curved arrow's direction.
- Give a counterclockwise movement a positive sign or a clockwise movement a negative sign.
- Include every completed lap in the measurement.
- When the question requests a final direction from right, add the signed movement to the starting direction measured from right.
- Undo the movement with an equal turn in the opposite direction to check your result.
You are recording the amount and direction of the turn and where it finishes. The initial side points straight up, at direction 90° from the right. Turn counterclockwise through 50°. What is the signed angle, and which direction does the terminal side point?
- The signed angle is +50°.The movement is counterclockwise; its starting direction does not change its sign.
- The ending direction is 90° + 50° = 140° from the right.Both direction readings are measured counterclockwise from the same rightward reference, so the turn adds to the initial direction.
- The terminal side points up and left.140° lies between the upward direction 90° and the leftward direction 180°.
- Signed angle: +50°.
- Terminal direction: 140° from the right, pointing up and left.
- Trace the curved arrow from its starting ray. Choose the sign before naming the ending region.
- A full lap still counts in the signed angle, even when the final ray matches the starting ray.
- The sign comes from the turning arrow, not from where the hand ends: 250° turns counterclockwise yet ends pointing down and to the left.
- Positive or negative depends only on the minus sign, never on being more or less than 180°: 190° is positive and −12° is negative.
- A turn can include full laps that the picture hides: 450° is one lap (360°) plus 90° more, and it ends where 90° ends.