Quarry School

A common start makes the landing place meaningful

Explain it like I am five

Picture a city map where two streets cross. The crossing is the origin, written (0, 0). The sideways street is the x-axis; the up-and-down one is the y-axis. A point's coordinates (x, y) are walking directions: x says how far right (+) or left (−), and y says how far up (+) or down (−). So (−5, 2) means 5 blocks left, then 2 up.

Standard position is the shared way to draw an angle: the vertex sits at the origin and the starting ray points right, along the positive x-axis, before the angle turns. Everyone starts alike, so a landing spot means the same thing to everyone.

The axes cut the map into four regions called quadrants, numbered I, II, III, IV counterclockwise from the upper right. Rebuild them from landmarks a quarter turn apart: 0° right, 90° up, 180° left, 270° down. Example: 235° is past 180° but short of 270°, so it ends down and to the left, in Quadrant III. An angle that ends on an axis, like 90°, is quadrantal and sits in no quadrant.

In plain words

Imagine several people giving you directions with arrows. Their arrows are hard to compare if everyone begins somewhere different and faces a different way. Standard position gives angles one shared starting picture. Put the Vertex at the Origin, the crossing of the horizontal x-axis and vertical y-axis. Point the Initial side right along the Positive x-axis. This map is the Rectangular coordinate system. The axes split it into four regions, each called a Quadrant. Now the Terminal side tells you which region the turn reaches. A ray on a dividing axis belongs to no quadrant, like a road forming the border between two neighborhoods.

140°initial sideterminal side
The highlighted rightward initial side and origin give every standard-position angle the same starting picture.
Reminder
  • Strict comparisons. 90° < 140° < 180° means 140° is greater than 90° and smaller than 180°; neither boundary is included.
  • Full turns. One additional 360° lap preserves the terminal side. For example, 415° and 55° finish on the same ray.
Why it works. A turn of 50° starting up finishes somewhere different from a turn of 50° starting right. Giving every angle the same origin and rightward initial side removes that ambiguity when comparing landing places. A full revolution is divided into 360 equal degree units by convention. Each quarter turn therefore uses 360° ÷ 4 = 90°. Those repeated quarter turns rebuild the region boundaries. You do not need to memorize a separate quadrant table.
RuleCoordinates (x, y) give the horizontal position first and vertical position second; (0, 0) records zero in both directions.
Standard position means vertex at (0, 0) and initial side on the positive x-axis. The Roman numerals I, II, III, IV mean 1, 2, 3, 4 and name the regions upper right, upper left, lower left, lower right, in counterclockwise order. A quadrantal angle ends on an axis.
The same idea, five ways
Say it

Say vertex at the origin, initial side pointing right, then name the quadrant or the axis of the terminal side.

Write it

An angle is in standard position when its vertex is (0, 0) and its initial side follows the positive x-axis.

In math
  • The four degree intervals below apply after reducing to 0° ≤ θ < 360°.
  • vertex = (0, 0); initial side = positive x-axis
  • I: 0° < θ < 90°
  • II: 90° < θ < 180°
  • III: 180° < θ < 270°
  • IV: 270° < θ < 360°
  • axis landing: θ = 90°n for an integer n
Like

Give every map arrow the same start before comparing its destination; crossing roads mark borders between four neighborhoods.

See it
Quadrant IIinitial sideterminal side
From the shared rightward start, the ray finishes between up and left in Quadrant II.
The same idea, other ways
As one shared starting mark

Two people comparing turns need the same starting location and direction. Standard position gives both: the vertex is at the crossing of the axes and the initial ray points right. An angle starting up fails the second requirement.

start right at the origininitial sideterminal side
The rightward initial ray and origin establish standard position.
As four neighborhoods and their borders

The axes are crossing roads. I is upper right, II upper left, III lower left, and IV lower right. A ray exactly on a road belongs to no neighborhood. That axis landing is a quadrantal angle when the angle is in standard position.

on the negative y-axisterminal sideinitial side
The terminal ray is on a border, so no quadrant contains it.
Read the drawn table

Read a terminal direction in the top row and the matching measurements below. In the column under Positive x-axis, right, each listed turn ends pointing right after a whole number of revolutions.

input Terminal sideoutput Degree measurements with that landingPositive x−axis, rightDegree measurements with that landing: 0°, 360°, 720°, −360°, −720°, −1080°Positive y−axis, upDegree measurements with that landing: 90°, 450°, 810°, −270°, −630°, −990°Negative x−axis, leftDegree measurements with that landing: 180°, 540°, 900°, −180°, −540°, −900°Negative y−axis, downDegree measurements with that landing: 270°, 630°, 990°, −90°, −450°, −810°
Choose a column in the top row, then read the matching information directly below it.
Terminal sideDegree measurements with that landing
Positive x-axis, right0°, 360°, 720°, −360°, −720°, −1080°
Positive y-axis, up90°, 450°, 810°, −270°, −630°, −990°
Negative x-axis, left180°, 540°, 900°, −180°, −540°, −900°
Negative y-axis, down270°, 630°, 990°, −90°, −450°, −810°
.1Standard position

Standard position is a shared start line, like runners all beginning at the same marked place. Both requirements matter: the hinge is at the origin and the initial ray points right. The direction rule from the preceding lesson still works outside this position.

  • Origin means the point (0, 0).
  • A Coordinate axis is one of the two dividing number lines.
  • The positive x-axis is the rightward ray from the origin.
50°initial sideterminal side
The initial ray points right from the crossing of the axes.
Reminder
  • Initial side. The initial side is where the movement starts, even when the terminal side finishes on an axis.
Worked exampleCheck both requirements

You are checking the vertex location and the initial direction against the standard-position definition. An angle has its vertex at the origin but starts pointing up. Is it in standard position?

initial side points upinitial sideterminal side
The arrow records initial side points up from the stated starting ray.
  1. It is not in standard position.Its initial side is on the positive y-axis, whereas standard position requires the positive x-axis.
Answer
No.
Check Its signed turn can still be positive or negative; failing the starting-position requirement does not remove the direction convention.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A vertex at the origin is enough to put an angle in standard position.
Standard position has two requirements. The initial side must also lie on the positive x-axis.
✓ Instead: Place the vertex at the origin and point the initial side right.
Tips and tricks
  • Check both the location and the starting direction before identifying the angle's standard-position landing.
.2The four quadrants

The four quadrants are four neighborhoods on your map. Start in the upper-right one and walk counterclockwise around the center: I, II, III, IV. The Roman numerals mean 1, 2, 3, 4. The axes are their borders, so the borders stay outside the regions.

  • These ranges describe the representative reduced to 0° ≤ θ < 360°, rather than every coterminal measurement.
  • I: 0° < θ < 90°.
  • II: 90° < θ < 180°.
  • III: 180° < θ < 270°.
  • IV: 270° < θ < 360°.
Quadrant IVterminal sideinitial side
The terminal ray is between down and right, inside the lower-right region.
Reminder
  • Strict inequality. 0° < θ < 90° includes angles between the two bounds but excludes both 0° and 90°.
Worked exampleRebuild instead of recite

You are finding the region between the axes that contains the terminal ray. Which quadrant contains 230° in standard position?

230°: Quadrant IIIterminal sideinitial side
The arrow records 230°: Quadrant III from the stated starting ray.
  1. 180° < 230° < 270°.Two quarter turns point left and three point down.
  2. The terminal side is in Quadrant III.The region between left and down is the third one encountered counterclockwise from the right.
Answer
Quadrant III.
Check 230° = 180° + 50°, so it turns 50° past left into the lower-left region.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 90° belongs to Quadrant I because it reaches the top of that region.
Quadrants are the open regions between axes. The 90° ray is exactly their border.
✓ Instead: 90° lies on the positive y-axis and belongs to no quadrant.
Tips and tricks
  • Reduce to one turn, then use strict inequalities between the neighboring axes.
.3Quadrantal angle

A Quadrantal angle lands on a border, like a clock hand exactly pointing right, up, left or down. It must be in standard position for that name to apply. Extra full laps keep the ray on the same border.

  • A quadrantal angle is a multiple of 90° in standard position.
  • Its terminal side lies on the x-axis or y-axis.
  • It belongs to no quadrant.
−270°terminal sideinitial side
A clockwise three-quarter turn ends up on the same upward ray as +90°.
Reminder
  • Removing complete turns. −270° + 360° = 90°. Adding a full turn preserves the terminal ray.
Worked exampleA negative angle on a border

You are following a clockwise turn to identify the axis it reaches. Where does −270° land in standard position?

−270°terminal sideinitial side
The arrow records −270° from the stated starting ray.
  1. −270° + 360° = 90°.Adding a full revolution changes the journey but not its terminal ray.
  2. It lands on the positive y-axis and is quadrantal.90° is exactly the upward boundary.
Answer
  • Axis: positive y-axis.
  • Classification: quadrantal.
Check Clockwise from right: −90° down, −180° left, −270° up.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: An angle ending on an axis is quadrantal even when its initial side points up.
The definition requires standard position as well as a terminal side on an axis.
✓ Instead: First check origin and rightward initial side, then check whether the terminal side lies on an axis.
Tips and tricks
  • For a standard-position axis landing, name the axis rather than a quadrant.
.4Degree landmarks

A Degree (°) is one of 360 equal pieces of a full turn. A Full revolution means one complete lap. A right angle is a quarter lap; a straight angle is half a lap. Those familiar pictures provide the measurements you use to rebuild all the boundaries.

  • Full revolution: 360°.
  • Right angle: 360° ÷ 4 = 90°.
  • Straight angle: 360° ÷ 2 = 180°.
  • 270° is three quarter turns.
3 of 4 wedges, each 90°
Three of the four quarter-turn wedges make 270°.
Reminder
  • Fraction of a whole. One quarter of 360° is 360° ÷ 4 = 90°. Multiply by 3 for three quarters.
Worked exampleCount quarter turns

You are totaling three equal quarter-turn sizes. How many degrees are three quarters of a full revolution?

3 of 4 wedges, each 90°
Three of four equal wedges make 270°.
  1. Each quarter is 360° ÷ 4 = 90°.Four equal quarter turns fill a complete revolution.
  2. Three quarters measure 3 × 90° = 270°.You count three equal pieces of 90° each.
Answer
270°.
Check 360° − 270° = 90°, leaving one quarter turn to finish the lap.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Three quarter turns total 180°.
Each quarter of 360° is 90°, so three quarters contain three copies of 90°.
✓ Instead: 3 × 90° = 270°; two quarter turns give 180°.
Tips and tricks
  • Draw right, up, left, down, right. Label each successive quarter turn with another 90°.
Strategy: step by step
  1. Check that the vertex is at the origin and the initial side points right. Otherwise the angle is not in standard position.
  2. Draw the landmarks: right 0°, up 90°, left 180°, down 270°, right again 360°.
  3. Locate the terminal ray between two landmarks and name the quadrant, starting I in the upper right.
  4. If the terminal ray is exactly on an axis, call the angle quadrantal and name the axis instead of a quadrant.
  5. For several turns, remove complete revolutions before comparing to the landmarks.
  6. In the table picture, read a column under Positive x-axis, right, then compare the matching information below it. Each column preserves one complete row of the reference table.
Strategy
Find a region or a boundary in standard position
1
Are both standard-position requirements satisfied?
YesCompare the terminal ray with the coordinate axes.
NoState that the angle is not in standard position; do not infer its landing from its signed measure alone.
↓
2
Is the terminal ray exactly on an axis?
YesName the axis and call the angle quadrantal.
NoName I, II, III, or IV from the open region containing the terminal ray.
  1. Check the vertex at the origin and the initial ray on the positive x-axis.
  2. Use a coterminal measurement in 0° ≤ θ < 360° when the turn has extra or negative laps.
  3. Mark the right, up, left, and down landmarks at 0°, 90°, 180°, and 270°.
  4. If the terminal ray matches a landmark, name its axis and call it quadrantal.
  5. Otherwise compare its measure with neighboring landmarks and name the region between them.
Worked exampleA region and a boundary

You are locating each finishing ray in a region or on an axis. In standard position, where do the terminal sides of 140° and 270° lie?

140°: Quadrant IIterminal sideinitial side
The arrow records 140°: Quadrant II from the stated starting ray.
270°: negative y-axisterminal sideinitial side
The arrow records 270°: negative y-axis from the stated starting ray.
  1. Mark 90° up, 180° left and 270° down.Each move from one cardinal direction to the next is a quarter turn of 90°.
  2. 90° < 140° < 180°, so 140° lies in Quadrant II.The upper-left region is between the upward and leftward rays.
  3. 270° lies on the negative y-axis and is a quadrantal angle.Three quarter turns finish exactly on the downward dividing ray, not inside either neighboring region.
Answer
  • 140°: Quadrant II.
  • 270°: negative y-axis, a quadrantal angle.
Check 140° is 50° past straight up but 40° short of left, so it lies between them. 270° is 90° short of a full revolution, so it points down.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: a small turn from right

You are locating a ray between right and up. In standard position, where does a 20° turn finish?

20°: Quadrant Iterminal sideinitial side
The arrow records 20°: Quadrant I from the stated starting ray.
  1. 0° < 20° < 90°.The turn is larger than no turn and smaller than the upward quarter-turn landmark.
  2. The terminal ray is in Quadrant I.The upper-right region is the space between right and up.
Answer
Quadrant I.
Check 90° − 20° = 70°, so the ray has not yet reached the upward axis.
Rung 2Rung 2: a region and an axis

You are locating each finishing ray in a region or on an axis. In standard position, where do the terminal sides of 140° and 270° lie?

140°: Quadrant IIterminal sideinitial side
The arrow records 140°: Quadrant II from the stated starting ray.
270°: negative y-axisterminal sideinitial side
The arrow records 270°: negative y-axis from the stated starting ray.
  1. Mark 90° up, 180° left and 270° down.Each move from one cardinal direction to the next is a quarter turn of 90°.
  2. 90° < 140° < 180°, so 140° lies in Quadrant II.The upper-left region is between the upward and leftward rays.
  3. 270° lies on the negative y-axis and is a quadrantal angle.Three quarter turns finish exactly on the downward dividing ray, not inside either neighboring region.
Answer
  • 140°: Quadrant II.
  • 270°: negative y-axis, a quadrantal angle.
Check 140° is 50° past straight up but 40° short of left, so it lies between them. 270° is 90° short of a full revolution, so it points down.
Rung 3Rung 3: remove one lap, then locate the ray

You are removing a full lap and then locating the terminal ray between axes. Reduce 405° to one lap and identify its quadrant.

405°terminal sideinitial side
The arrow records 405° from the stated starting ray.
  1. 405° − 360° = 45°.Removing a full revolution preserves the terminal side.
  2. 0° < 45° < 90°, so it is in Quadrant I.The ray lies between right and up.
Answer
  • Representative: 45°.
  • Quadrant: I.
Check 405° = 360° + 45°, a full lap followed by an upper-right turn.
Rung 4Rung 4: remove several laps, then locate the ray

You are removing complete laps and then naming the region reached. Locate 1230° in standard position.

1230°terminal sideinitial side
The arrow records 1230° from the stated starting ray.
  1. 3 × 360° = 1080°.Three complete turns fit before the remaining direction.
  2. 1230° − 1080° = 150°.Subtracting those laps preserves the terminal side and leaves a one-lap representative.
  3. 90° < 150° < 180°, so the ray is in Quadrant II.This is the upper-left region between up and left.
Answer
  • Representative: 150°.
  • Quadrant: II.
Check Removing revolutions one at a time gives 870°, then 510°, then 150°.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A vertex at the origin guarantees standard position, even if the initial side points up.
Standard position has two requirements. The origin requirement holds, but the initial side must also point along the positive x-axis.
✓ Instead: Move the initial side to the rightward axis to use standard position.
✗ Not this: 90° is in Quadrant I because it reaches the top edge of that region.
Quadrants contain the spaces between axes and exclude their borders.
✓ Instead: 90° lies on the positive y-axis and is quadrantal.
Tips and tricks
  • Write I, II, III, IV counterclockwise from upper right. The Roman numerals mean 1, 2, 3, 4.
  • Check equality before naming a quadrant. Exactly on an axis means name the axis.
Trap. Saying an angle of 90° is in Quadrant I. The upper-right region ends before its boundary. Exactly 90° is on the positive y-axis.
Keep in mind
  • Where an angle starts and which way it turns are separate facts: it always starts pointing right, and 160° then turns counterclockwise to end up and to the left, in Quadrant II.
  • Coordinate signs follow the quadrant: I is (+, +), II is (−, +), III is (−, −), and IV is (+, −), so (4, −1) lies in Quadrant IV.
  • An angle on an axis belongs to no quadrant: 90° lies on the positive y-axis, not in Quadrant I.
  • An axis angle lands on a point with a 0 coordinate, such as (0, 1) one unit straight up, and in Section 5-2 a trig fraction with that 0 on the bottom has no value, because 5 ÷ 0 would need a number that times 0 gives 5, and none exists.
Memory hookDraw a C starting at the upper right: it passes through quadrants I, II, III, IV in order. Every angle starts facing 3 o'clock.
Flash cards: say the answer out loud, then flip
What is standard position?
Vertex at the origin (0, 0) and initial side along the positive x-axis, pointing right.
What is a quadrantal angle?
  • An angle whose terminal side lies on an axis
  • it is in no quadrant.
Which quadrant holds 285°?
IV: between 270° and 360°
Which quadrant holds the point (−2, −7)?
III: 2 left and 7 down
In what order are the quadrants numbered?
I, II, III, IV, counterclockwise from the upper right
Does counterclockwise mean the angle starts pointing left?
  • No. It starts pointing right
  • counterclockwise is only the direction it turns.