Segments, Rays, Lines and Angles | Grade 7 Math | LevelKite

In geometry, a point marks one exact location. In these figures, each point is drawn as a dot and labelled with a capital letter, such as $A$ . In this lesson you will tell apart three straight figures built from points. Then you will use one of them, the ray, to name and sort angles.

No figure here is drawn to scale. You read every fact from the labels, arrowheads, marks and stated measures, and never from how big something looks.

Three straight figures look alike, but their ends are different:

A line segment is a straight path between two points called its endpoints. It is drawn as a stroke between two endpoint dots with no arrowheads, so it stops at both ends. Name it by its endpoints in either order: segment $AB$ or segment $BA$ .
A ray starts at one endpoint and goes on forever in one direction. It is drawn from an endpoint dot, through a second named point, and it ends in one arrowhead. Name it with the endpoint first: ray $CD$ starts at $C$ and passes through $D$ .
A line goes on forever in both directions. It is drawn through two named points with an arrowhead at each end. Name it by those two points in either order: line $EF$ or line $FE$ .

In the figure below, segment $AB$ has no arrowheads. Ray $CD$ has one arrowhead past $D$ . Line $EF$ has an arrowhead at each end.

Segment

AB

stops at two dots, ray

CD

starts at

C

and has one arrowhead, and line

EF

has an arrowhead at each end.

One segment, one line and two rays that start at the same dot

E

.

Worked example

Read the ends, then name

In the figure above, identify each straight figure as a segment, a ray or a line, and name it by its letters.

The stroke from $A$ to $B$ has a dot at each end and no arrowheads, so it stops at both ends. Two endpoints make a segment. Both ends are endpoints, so either order of letters names it.
$\text{segment } AB \text{ or segment } BA$
The stroke through $C$ and $D$ has an arrowhead at each end, so it goes on forever both ways. That makes it a line. A line has no endpoints, so either order of letters names it.
$\text{line } CD \text{ or line } DC$
The stroke through $F$ starts at the dot $E$ , passes $F$ and ends in one arrowhead. One endpoint and one arrowhead make a ray. Write the endpoint first. The name ray $FE$ would describe a ray that starts at $F$ .
$\text{ray } EF$
The stroke through $G$ also starts at the dot $E$ and has one arrowhead, past $G$ . So it is a second ray with endpoint $E$ , and again the endpoint comes first.
$\text{ray } EG$

Answer

The figure shows segment $AB$ , line $CD$ , ray $EF$ and ray $EG$ , and the two rays share the endpoint $E$ .

Two rays that share an endpoint form an angle. The shared endpoint is the angle's vertex, and the two rays are its arms. An arc drawn between the arms shows which opening is meant. In the last figure, rays $EF$ and $EG$ share the endpoint $E$ , so they form an angle with vertex $E$ .

Name an angle with three letters: a point on one arm, then the vertex, then a point on the other arm. The vertex always goes in the middle, so that angle is angle $FEG$ or angle $GEF$ . You may use the vertex letter alone, angle $E$ , only when $E$ is the vertex of just one angle in the figure. Otherwise, the single letter could mean more than one angle.

The size of an angle is how far one arm turns to reach the other. It is given in degrees, written °. A square corner measures $90^\circ$ . Two arms that point in opposite directions along one line make $180^\circ$ . In this lesson, every measure is stated in the text, and nothing is measured from a drawing.

The four classes of angles and the givens that show each one

Class	Measure	A given that shows it
Acute	more than $0^\circ$ and less than $90^\circ$	a ray inside a marked right angle cuts off acute angles, or a stated measure below $90^\circ$
Right	exactly $90^\circ$	a square right-angle mark at the vertex, or a stated measure of $90^\circ$
Obtuse	more than $90^\circ$ and less than $180^\circ$	a right angle plus more, with arms not on one line, or a stated measure between $90^\circ$ and $180^\circ$
Straight	exactly $180^\circ$	two opposite rays on one line, or a stated measure of $180^\circ$

Key concept

Classify an angle from its givens, not its looks

A figure is not drawn to scale, so an angle's class comes only from what is given. A given can be a right-angle mark, opposite rays on one line, a ray inside a marked right angle, a right angle plus more that stops short of a line, or a measure stated in the text.

For a stated measure, compare it with $90^\circ$ and $180^\circ$ . Below $90^\circ$ is acute, exactly $90^\circ$ is right, between the two is obtuse, and exactly $180^\circ$ is straight.

Rays

BA

and

BC

share the vertex

B

, and the text states that angle

ABC

measures

128^\circ

.

Worked example

Classify from a stated measure

Angle $ABC$ in the figure above measures $128^\circ$ . Give every correct name for the angle, then classify it.

Find the vertex. Rays $BA$ and $BC$ share the endpoint $B$ , so $B$ is the vertex and goes in the middle of a three-letter name.
$\text{angle } ABC \text{ or angle } CBA$
$B$ is the vertex of only one angle in this figure, so the single letter cannot mean any other angle. That means angle $B$ also names it.
$\text{angle } B$
Compare the stated measure with a right angle. Subtracting shows that it is $38^\circ$ more than $90^\circ$ .
$128^\circ - 90^\circ = 38^\circ$
Compare the stated measure with a straight angle. Subtracting shows that it is $52^\circ$ short of $180^\circ$ .
$180^\circ - 128^\circ = 52^\circ$
The measure is more than $90^\circ$ and less than $180^\circ$ , which is the obtuse range.
$90^\circ < 128^\circ < 180^\circ$

Answer

Angle $ABC$ , also called angle $CBA$ or angle $B$ , measures $128^\circ$ , so it is an obtuse angle.

Square marks show that angles

ABD

and

DBC

are right angles, and arcs mark angles

EBC

and

ABE

.

Worked example

Classify from marks and rays

In the figure above, points $A$ , $B$ and $C$ lie on one line, with $B$ between $A$ and $C$ . Square marks show that angles $ABD$ and $DBC$ are right angles, and ray $BE$ lies inside angle $DBC$ . Name and classify angle $EBC$ and angle $ABE$ .

Four rays start at $B$ , so $B$ is the vertex of several angles. The name angle $B$ would not say which one you mean. Use three letters with $B$ in the middle.
Ray $BE$ is inside the right angle $DBC$ , so it splits that $90^\circ$ into angle $DBE$ and angle $EBC$ . Each part has some size, so each part is less than the whole $90^\circ$ . That makes angle $EBC$ acute.
$\text{angle } EBC < 90^\circ$
The arms of angle $ABE$ are rays $BA$ and $BE$ . Turning from $BA$ to $BE$ passes through all of the right angle $ABD$ and then the extra angle $DBE$ . So angle $ABE$ is more than $90^\circ$ .
$\text{angle } ABE = 90^\circ + \text{angle } DBE > 90^\circ$
Rays $BA$ and $BC$ are opposite rays on one line, so angle $ABC$ is straight. Ray $BE$ lies inside angle $DBC$ , not along ray $BC$ . So arms $BA$ and $BE$ do not form one line, and angle $ABE$ stops short of straight.
$\text{angle } ABE < 180^\circ$
Angle $ABE$ is more than $90^\circ$ and less than $180^\circ$ , so it is obtuse.
$90^\circ < \text{angle } ABE < 180^\circ$

Answer

Angle $EBC$ (or $CBE$ ) is acute and angle $ABE$ (or $EBA$ ) is obtuse. Neither can be called angle $B$ , because $B$ is the vertex of several angles in this figure.

Takeaway: read a figure by its ends and its givens. A segment has two endpoints. A ray has one endpoint, which you write first. A line has an arrowhead at each end.

An angle is two rays from one vertex, and you name it with the vertex letter in the middle. Its class (acute, right, obtuse or straight) comes from a mark, opposite rays, or a stated measure compared with $90^\circ$ and $180^\circ$ . It never comes from how the angle looks.

Next, you will use these ideas to identify parallel, perpendicular and intersecting lines.