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Lines, Rays, and Angles

This fourth grade geometry lesson teaches the definitions for a line, ray, angle, acute angle, right angle, and obtuse angle. We also study how the size of the angle is ONLY determined by how much it has "opened" as compared to the whole circle. The lesson contains many varied exercises for students.



This is point A.
Points are named
with capital letters.
When two points are connected with a straight
line, we get a line segment. We call this line
segment AB or line segment AB (note the bar on top).
The sides of a triangle
are line segments.

A line has no beginning point or end point. Imagine it continuing indefinitely in both directions.
We can illustrate that by  little arrows on both ends.


We can name a line using two points on it. This is line EF or line (note the arrowheads).
Or, we can name a line using a lowercase letter: this is line s.

A ray starts out at a point and continues off to infinity. We can show
that by drawing an arrow at one end of the ray. Think of the sun's rays:
they start at the sun and go on indefinitely.

We can name a ray using its starting point and one other point that is
on the ray: this is ray QP or ray (note the one arrowhead). Or, we can
name a ray using a lowercase letter: this is ray r.


What is an angle? Many people think that an angle is some kind of
slanted line. But in geometry an angle is made up of two rays that
have the same beginning point

That point is called the vertex and the two rays are called the sides
of the angle.

To name an angle, we use three points, listing the vertex in the middle.
This is angle DEF or ∠DEF. We can use the symbol ∠ for angle.


1. Write if each figure is a line, ray, line segment, or an angle, and name it.


a.  _______________________

b.  _______________________


c.  _______________________

d.  _______________________

e.  _______________________

f.  _______________________

2. a. Find the angle formed by the rays DE and DF.
        How do we name it?

    b. Find the angle formed by the rays CA and CE.
        How do we name it?

    c. What is BD? (a line, a line segment, or a ray)?


3. a. Draw two points, D and E. Then draw line DE.

    b. Draw point Q not on the line.

    c. Draw rays DQ and EQ.

    d. Find angles EDQ and DEQ in your drawing.

     Imagine that the two sides of the angle start side by side, and then
open up to a certain point. When the two sides “open up”, they draw
an imaginary arc of a circle. (You can illustrate this with two pencils as the
two sides of an angle. Keep one pencil stationary while you rotate the other.)

If the angle opens up to a full
, we say the angle is
360 degrees


This angle is half of the full circle,
so it measures 180°. It is called
the straight angle.

Your two pencils (rays) are lying
down flat or straight on the floor.


This is one-fourth of the
full circle, so it is 90°.

This is called the right
Table and book
corners are right angles.


In each of these pictures the angle is opened more and more and keeps getting bigger. The arc of the circle is larger.

These angles are acute angles, which means they are less than a right angle (less than 90°). Think of acute angles as sharp angles. If someone stabbed you with the vertex of an acute angle, it would feel sharp.


The angle is opened even
more now. It is an obtuse
: an angle that is
more than a right angle,
yet less than a straight

Think of obtuse angles as
dull angles.


Here's another way of thinking about angles. Think of a sun rising in the morning in the horizon, gradually getting higher, and traveling through the sky along an arc of a circle.

How big is the angle?

It does not matter how long the sides of the angle are. Remember, they are rays, and rays go on indefinitely. But when we draw them on paper, we have to draw them as ending somewhere. 

The sides of the angle might even seem to have different lengths. That doesn't matter either. The size of the angle is ONLY determined by how much it has “opened” as compared to the whole circle. Think how big an arc of a circle the sides have drawn, as compared to a whole circle. 

Which of these two angles is bigger?
Look at how much the angle has opened?
How big a part of a circle have the sides drawn?
The second angle (on the right) is bigger.
Many times the arrows are omitted from the rays, and the
arc of the circle is drawn as a tiny arc near the vertex.
Even that is not necessary. Which of these is a bigger angle?
Again, the second one.

4. Which angle is bigger?

a.    OR  
b.  OR   
c. OR   
d. OR  
e.   OR  
f.    OR

5. a. Sketch three different
        acute angles.

    b. Sketch three different
        obtuse angles.

    c. Sketch a right angle
        and a straight angle.

6. Label the angles as acute, right, obtuse, or straight. To help, make these angles with two pencils,
    checking how much you need to open up the angle.



7. A triangle has three angles. In fact, the word tri-angle means a three-angled shape.

    Which of the triangles 
    a, b, or c has one
    obtuse angle?

    Which has one right angle?

a.  b.  c. 


8. (Optional) Make a geometry notebook where you write down each new term and draw a picture or
    pictures that illustrate the term. Use colors and tidy writing. It is like your personal geometry
    dictionary. You can also do any drawing problems from the lessons in it. Drawing and writing
    yourself, instead of just reading, can help you remember the terms better!

New Terms
  • a line
  • a line segment
  • a ray
  • an angle
  • an acute angle
  • a right angle
  • an obtuse angle
  • a straight angle

This lesson is taken from Maria Miller's book Math Mammoth Geometry 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

Math Mammoth Geometry 1

A self-teaching worktext for 4th-5th grade that covers angles, triangles, quadrilaterals, cirlce, symmetry, perimeter, area, and volume. Lots of drawing exercises!

Download ($6.90). Also available as a printed copy.

=> Learn more and see the free samples!

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