Quarry School

The pieces of an angle and how to name them

Explain it like I am five

Shine two flashlights from the same spot on the floor. Each beam starts at one place and runs on forever in one direction: that is a ray. An endpoint is a place where a path starts or stops. A ray has 1 endpoint. A line runs forever both ways, so it has 0. A line segment stops at both ends, so it has 2.

Two rays that share an endpoint make an angle. The shared spot is the vertex, the corner, like the hinge of a door. Each ray is a side of the angle. The symbol ∠ is read "angle".

Example: point O is the shared start, point A is above O, and point B is to the right of O. The rays are OA and OB. Each name starts with O, because a ray's name starts at its endpoint. The angle is ∠AOB, read "angle A O B", or ∠BOA. Both keep O in the middle, so both name the corner at O. The name ∠OAB would put the corner at A, the wrong spot.

In plain words

Picture a door viewed from above. Its hinge stays in one place while the door swings. To draw that movement, you need straight paths. A Line continues forever both ways, like a straight road with no ends. A Line segment has two ends, like a measured strip of that road. A Ray has one starting point and continues forever one way, like a flashlight beam. Two rays sharing a starting point form an Angle. That shared point is the Vertex, like the door hinge. Each ray is a Side of an angle. These drawings help you talk precisely about where a turn starts and ends. The symbol ∠ means angle; ∠AOB is read angle A O B. An endpoint is a starting or stopping point.

O is the vertexvertex
Both rays share the highlighted point, as a swinging door stays attached to its hinge.
Reminder
  • Point labels. A capital letter labels a location. O names the hinge point; it does not mean multiply by the number zero.
  • Degree notation. The symbol ° means degrees. The reading θ = 70° is theta equals seventy degrees.
Why it works. A Point labels one exact location. A line, segment and ray differ in how many ends they have, so their names must tell you which object you mean. A ray needs its endpoint first because reversing its letters changes where it starts. An angle name puts the vertex in the middle so everyone knows which corner is being measured. The letters are labels for places, like addresses on a map. Their order prevents you from measuring the wrong corner.
RuleA line has no endpoints, a line segment has two, and a ray has one. In ray OP, O is the endpoint. In ∠AOB, O is the vertex; ∠BOA names the same geometric corner.
The same idea, five ways
Say it

Say line A B, line segment A B, ray O P, and angle A O B. Say θ as theta.

Write it

A line extends both ways, a segment stops at two endpoints, a ray begins at one endpoint, and an angle consists of two rays sharing their vertex.

In math
  • line AB = line BA
  • line segment AB = line segment BA
  • ray OP: endpoint O, direction through P
  • ∠AOB: vertex O, sides OA and OB
  • ∠AOB and ∠BOA name the same geometric opening
Like

An endless road, a measured strip, and a flashlight beam give three different straight paths; two beams leaving a hinge give an angle.

See it
O is the vertexvertex
An angle has two straight sides meeting at one fixed vertex.
The same idea, other ways
By counting the ends

A road that continues both ways has no endpoints and is a line. A measured piece has two endpoints and is a line segment. A flashlight beam has one endpoint and is a ray. Count the ends before choosing the name.

OPone endpoint, forever one way
One endpoint and one continuing direction identify a ray.
By reading an address

The letters tell you where to look. Ray OP begins at O and passes through P. In ∠AOB, the middle O is the hinge, and the outer A and B mark its two sides. Naming the object tells the reader how to interpret the letters.

∠AOBvertex
The middle letter names the shared endpoint of the angle.
Read the drawn table

Read the symbol in the top row, then its name and pronunciation directly below it. For example, the column under θ pairs theta with THAY-tuh. A letter names an angle without prescribing its size.

input Greek symboloutput Name; Say itαName: alpha Say it: AL−fuhβName: beta Say it: BAY−tuhγName: gamma Say it: GAM−uhδName: delta Say it: DEL−tuhθName: theta Say it: THAY−tuhφName: phi Say it: fie
Choose a column in the top row, then read the matching information directly below it.
Greek symbolNameSay it
αalphaAL-fuh
βbetaBAY-tuh
γgammaGAM-uh
δdeltaDEL-tuh
θthetaTHAY-tuh
φphifie
.1Line

Imagine stretching a straight road forever in both directions. A Line has no ends. Points A and B mark places on it, but they do not stop it. You can name it line AB, line BA, or a lowercase letter such as l.

  • No endpoints.
  • Any two distinct points name the line.
  • The two directions continue without a last point.
  • The instructor puts a two-headed arrow above AB to name a line; the two arrows say it continues both ways. Writing line AB expresses the same meaning here.
ABgoes on forever both ways
The arrows at both ends mean the straight path keeps going.
Worked exampleA road through two marks

You are deciding which straight object has no endpoints. A straight path passes through A and B and continues beyond both. Is it a line or a line segment?

ABgoes on forever both ways
The arrows at both ends mean the straight path keeps going.
A line.
Use the object name and the order of its point labels together.
  1. It is line AB, also called line BA.Neither marked point ends the path; it continues in both directions.
Answer
A line.
Check There are zero endpoints, which is the defining feature of a line.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A line through A and B ends at A and B.
The letters mark two locations on the line. They do not create endpoints.
✓ Instead: The line continues past both A and B in both directions.
Tips and tricks
  • Two arrows mean a line. Two dots alone mark a segment's endpoints.
.2Line segment

A Line segment is like a ruler with a definite beginning and end. Segment AB starts at A and stops at B. The word segment matters because the letters AB alone can also label a line in a different drawing.

  • Two endpoints.
  • Segment AB and segment BA are the same piece.
  • Its length is the distance between its endpoints.
  • The instructor puts a bar above AB to name a segment; the bar has no extending arrowheads. Writing segment AB expresses the same meaning here.
ABstops at both ends
The two endpoint dots bound the whole piece.
Reminder
  • Distinguishing a line from a segment. A line has no endpoints, so points A and B on a line are markers rather than stops.
Worked exampleA measured piece

You are naming a straight object that stops at both ends. A straight piece begins at A and ends at B. What is it?

ABstops at both ends
The two endpoint dots bound the whole piece.
Line segment AB.
Use the object name and the order of its point labels together.
  1. Call it line segment AB.The path has two endpoints and does not continue past them.
Answer
Line segment AB.
Check Counting its ends gives two, unlike a line or a ray.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Segment AB continues forever beyond B.
A segment contains only the straight piece between its two endpoints.
✓ Instead: Segment AB stops at A and B; ray AB continues past B.
Tips and tricks
  • Count the stopping points. A segment has two.
.3Ray

A Ray is like light leaving a flashlight. It has a starting point, then one forward direction with no end. Ray OP begins at O and passes through P. P is a marker along the beam, not a stopping point.

  • One endpoint.
  • Name the endpoint first.
  • Ray OP and ray PO generally describe different rays.
  • The instructor puts a one-headed arrow above OP, directed from the endpoint O toward P, to name the ray. Writing ray OP preserves that direction here.
OPone endpoint, forever one way
One dot starts the path and one arrow shows its endless direction.
Reminder
  • Endpoint. An endpoint is a stopping or starting point. A ray has one; a line segment has two.
Worked exampleRead the ray's address

You are identifying the endpoint from the order of the ray’s letters. Which point starts ray OP?

OPone endpoint, forever one way
One dot starts the path and one arrow shows its endless direction.
O.
Use the object name and the order of its point labels together.
  1. O starts the ray.The first letter of a ray name identifies its only endpoint.
Answer
O.
Check P tells you which way to travel after leaving O.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Ray OP starts at P because P is the last letter.
The first letter names the ray's only endpoint; the second letter tells its forward direction.
✓ Instead: Ray OP starts at O, passes through P, and continues beyond P.
Tips and tricks
  • Read a ray name as an instruction: start at the first letter, travel toward the second.
.4Angle, sides and vertex

Think of the door's closed position and its open position as two flashlight beams leaving the hinge. Those rays are the sides of the Angle. Their shared Endpoint, the place where each ray begins, is the Vertex. A Point labels one exact location without giving it a length. The symbol ∠ means angle; read ∠RST as angle R S T.

  • Two sides share one vertex.
  • The vertex is the middle letter of a three-letter angle name.
  • Naming a geometric corner alone does not record extra full turns or a direction.
shared endpointvertex
The common dot stays fixed while the opening between the two rays changes.
Reminder
  • Naming a ray. Ray OA and ray OB both start at O, so their shared endpoint is O.
Worked exampleThe middle letter matters

You are finding the angle’s shared endpoint from its middle letter. Where is the vertex of ∠RST?

∠RST: vertex Svertex
The arrow records ∠RST: vertex S from the stated starting ray.
  1. The vertex is S.The middle letter identifies the shared endpoint of sides SR and ST.
Answer
S.
Check Both side names start at S, so they meet there.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: O is the vertex of ∠OAB.
The middle letter identifies the shared endpoint, and the middle letter here is A.
✓ Instead: A is the vertex of ∠OAB. For a corner at O with sides OA and OB, write ∠AOB.
Tips and tricks
  • Point to the hinge, then place its letter in the middle of the angle name.
.5Greek letters

Greek letters are angle labels, much like x can label a missing number. You do not multiply by the word theta when you see θ. You read θ as theta and use it as the name of the angle. The table gives the six labels you are asked to recognize and pronounce.

  • Greek-letter names: α (alpha), β (beta), γ (gamma), δ (delta), θ (theta), φ (phi).
  • δ is delta, θ is theta, φ is phi.
  • An angle may also be labeled with a number, such as angle 1.
α alpha, β beta, γ gamma
δ delta, θ theta, φ phi
θ = 70° means the angle named theta is 70°
These symbols name angles; learning to say them makes the formulas readable.
Reminder
  • Equality. The symbol = says both sides give the same amount. θ = 70° assigns the named angle a 70° measurement.
Worked exampleRead an angle label

You are saying the Greek letter and the measurement out loud. Read θ = 70° aloud.

θ = 70°
The arrow records θ = 70° from the stated starting ray.
  1. Say: theta equals seventy degrees.θ is the name theta, and ° is the degree symbol.
Answer
Theta equals seventy degrees.
Check Changing the label from θ to α would rename the angle without changing its size.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: θ = 70° means multiply 70° by the word theta.
Here θ is the name of the angle or its measurement, rather than an instruction to multiply.
✓ Instead: Read θ = 70° as theta equals seventy degrees.
Tips and tricks
  • Say the angle label aloud before reading its measurement.
Strategy: step by step
  1. Count the endpoints to decide whether you have a line, line segment or ray.
  2. For a ray, write the endpoint first and then any point farther along it.
  3. For an angle, identify the shared endpoint and put its letter in the middle.
  4. Use one point from each side as the other two letters. A single letter such as θ or a number such as 1 can also label the angle.
  5. In the table picture, the column under θ gives theta and THAY-tuh. Read each Greek letter by its own name before using it to label an angle.
Strategy
Identify the object, then name its points
1
Does a straight path have no endpoints and continue both ways?
YesCall it a line and use any two distinct points on it.
NoCount the endpoints.
↓
2
Does the straight path have two endpoints?
YesCall it a line segment and use its endpoints.
NoIf it has one endpoint and continues one way, call it a ray and name the endpoint first.
↓
3
Do two rays share one endpoint?
YesThey form an angle; put the shared endpoint's letter in the middle.
NoDo not name them as one angle with a common vertex.
  1. Count the endpoints and continuing directions.
  2. Use the object word to distinguish a line, segment, or ray.
  3. For a ray, write its endpoint first, then a point in its forward direction.
  4. For an angle, find the common endpoint and put its letter between one letter from each side.
  5. Read the name back and check that it sends you to the intended object.
Worked exampleName the corner and its two paths

You are identifying the shared point, the two rays and two names for their corner. Rays OA and OB start at O. Point A is above O and point B is to its right. Name the vertex, both rays and the angle in two ways.

∠AOB: vertex Overtex
The arrow records ∠AOB: vertex O from the stated starting ray.
  1. The vertex is O.Both paths start at O, the shared point that plays the role of a hinge.
  2. The rays are OA and OB, each read with O first.A ray name starts with its endpoint and then names a point in its forward direction.
  3. Name the angle ∠AOB or ∠BOA.Both names keep the vertex O in the middle and choose one point from each side.
Answer
  • Vertex: O.
  • Rays: OA and OB.
  • Angle: ∠AOB or ∠BOA.
Check In each angle name the middle letter is O. The two outer letters identify the two rays, so both names pick out the same corner.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Ray OP and ray PO are the same because they use the same letters.
The first letter identifies the endpoint. These rays begin at different points, so their directions and sets of points differ.
✓ Instead: Ray OP starts at O and passes through P; ray PO starts at P and passes through O.
✗ Not this: If the hinge is O, name its angle ∠OAB.
The middle A would tell the reader that the vertex is A, which identifies a different corner.
✓ Instead: Put O in the middle: ∠AOB or ∠BOA.
Tips and tricks
  • Write the object word next to its letters when no arrow or bar is shown: line AB, segment AB, or ray AB.
  • Point to the shared endpoint before writing a three-letter angle name. Put that letter in the middle.
Trap. Naming the angle ∠OAB. Its middle letter is A, so that name asks for a corner at A. Point to the hinge first, then put its letter in the middle.
Keep in mind
  • The vertex letter always sits in the middle: ∠PQR has its corner at Q, not at P or R.
  • A ray is named from its endpoint outward: ray OA starts at O and passes through A, while ray AO starts at A and runs the other way.
  • Count endpoints to tell the paths apart: a line has 0, a ray has 1, and a line segment has 2.
  • An angle can also go by one Greek letter: θ is theta (THAY-tuh), α is alpha (AL-fuh), and β is beta (BAY-tuh).
Memory hookA ray is a flashlight beam: one end, then forever. An angle's name is a sandwich: the vertex letter is the filling, as in ∠AOB.
Flash cards: say the answer out loud, then flip
What is a ray?
A straight path with one endpoint that runs forever in one direction, like a flashlight beam.
What is the vertex of an angle?
The shared endpoint of its two rays: the corner.
How many endpoints does each have: a line, a line segment, a ray?
  • Line: 0
  • Line segment: 2
  • Ray: 1
Rays KL and KM both start at K. Name the angle two ways.
  • ∠LKM
  • ∠MKL
Do ∠DEF and ∠EDF name the same corner?
  • No. ∠DEF has its corner at E
  • ∠EDF has its corner at D.
How do you say θ, and what does it do here?
Theta, THAY-tuh: a one-letter name for an angle.