Geometry notation communicates whether a figure stops, continues, or has a measured length. SAT questions often test these small distinctions before adding algebra.
Line, segment, or ray?
A line through P and QLine PQ has arrows at both ends, contains points P and Q, and has no endpoints.
Line
A straight arrangement of infinitely many points extending without end in both directions. Line \(PQ\) and line \(QP\) name the same line.
A segment with endpoints P and QSegment PQ contains endpoints P and Q and every point between them; it has no arrows.
Segment
A finite portion of a line with two endpoints and every point between them. \(\overline{PQ}\) names the object; \(PQ\) names its numerical length.
Ray PQRay PQ begins at endpoint P, passes through Q, and continues indefinitely beyond Q.
Ray
A portion of a line with one endpoint that continues forever in one direction. In \(\overrightarrow{PQ}\), the first letter \(P\) is the endpoint.
Line, segment, and ray comparison
Object
Endpoints
Extends forever?
Direction
Typical notation
Line
0
Yes
Both directions
\(\overleftrightarrow{PQ}\)
Segment
2
No
Between endpoints
\(\overline{PQ}\)
Ray
1
Yes
One direction
\(\overrightarrow{PQ}\)
Collinear points and opposite rays
Four collinear pointsPoints A, B, C, and D lie on the same straight line.Opposite rays RP and RQPoints P, R, and Q are collinear with common endpoint R between P and Q; ray RP points left and ray RQ points right.Mini check
Check the endpoint
Points \(A,B,C\) occur in that order on a line. Which rays are opposite?
\(\overrightarrow{BA}\) and \(\overrightarrow{BC}\)
\(\overrightarrow{AB}\) and \(\overrightarrow{CB}\)
Show answer and explanation
Answer:\(\overrightarrow{BA}\) and \(\overrightarrow{BC}\).
They share endpoint \(B\), which lies between \(A\) and \(C\).
Add adjacent segment lengths
Segment Addition PostulatePoint Q lies between P and R. PQ is a units, QR is b units, and PR is a plus b units.Segment Addition Postulate
\[PQ+QR=PR\]
This equation is valid when \(Q\) lies between \(P\) and \(R\).
Worked example
Translate point order into an equation
\(Q\) lies between \(P\) and \(R\). If \(PQ=3x+2\), \(QR=x+8\), and \(PR=30\), find \(x\).
Write the relationship
\((3x+2)+(x+8)=30\).
Combine
\(4x+10=30\), so \(4x=20\).
Solve and verify
\(x=5\); the parts are \(17\) and \(13\), which total \(30\).
\(x=5\).
Midpoints divide a segment into equal halves
Midpoint M of segment PRM lies on segment PR and matching tick marks show that PM and MR are congruent.Midpoint relationships
\[PM=MR=\frac12PR\]
A midpoint must lie on the segment and create two congruent subsegments. Matching tick marks encode equality.
Worked example
Solve expressions for equal halves
\(M\) is the midpoint of \(PR\), with \(PM=5x-4\) and \(MR=3x+10\). Find \(PR\).
Set halves equal
\(5x-4=3x+10\).
Solve
\(2x=14\), so \(x=7\).
Find the whole
\(PM=MR=31\), so \(PR=62\).
\(PR=62\).
A segment bisector passes through the midpoint
A segment bisector through midpoint MLine l intersects segment PR at midpoint M. Matching ticks show PM equals MR; no right angle is implied.
A reliable segment-problem process
Read point order
Identify which point lies between the other two.
Translate the mark or word
Use addition for adjacent pieces or equality for midpoint halves.
Solve the equation
Keep every length positive.
Answer the requested quantity
If asked for the whole segment, add or double after solving.
Key takeaways
Lines, segments, and rays: key takeaways
Arrowheads reveal which directions continue forever.
Ray names begin with the endpoint.
Opposite rays share an endpoint between two collinear outer points.
For \(P-Q-R\), use \(PQ+QR=PR\).
A midpoint creates equal halves; a segment bisector passes through that midpoint.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.