Download Basic Geometry Terms

Basic Geometry Terms
Points
In geometry, we use points to specify exact locations. They
are generally denoted by a number or letter. Because points
specify a single, exact location, they are zero-dimensional. In
other words, points have no length, width, or height. It may
be helpful to think of a point as a miniscule “dot” on a piece
of paper.
Lines
Lines in geometry may be thought of as a “straight” line that
can be drawn on paper with a pencil and ruler. However,
instead of this line being bounded by the dimensions of the
paper, a line extends infinitely in both directions. A line is onedimensional, having length, but no width or height. Lines are
uniquely determined by two points.
Line Segments
Consider the task of drawing a “straight” line on a piece of paper (as
we’ve done when thinking about lines). What you’ve actually done is
create a line segment. Because our piece of paper has defined
dimensions and we cannot draw a line infinitely in any direction, we
have constructed a segment that begins somewhere and ends
somewhere. Note that the notation for lines and line segments differ
because a line segment has a defined length, whereas a line does not.
Rays
A ray is a “straight” line that begins at a certain point and extends
infinitely in one direction. A ray has one endpoint, which marks the
position from where it begins. This notation shows that the ray begins at
point A and extends infinitely in the direction of point B.
Endpoints
Endpoints mark the beginning or end of a line segment or ray. Line
segments have two endpoints, giving them defined lengths, whereas
rays only have one endpoint, so the length of a ray cannot be measured.
Midpoints
The midpoint of a line segment marks the point at which the segment is
divided into two equal segments. In other words, the lengths of the
segments from either endpoint to the midpoint are equal. Note that
neither lines nor rays can have midpoints because they extend infinitely
in at least one direction. It would be impossible to find the middle of a
line or ray that never ends!
Intersection
When we have lines, line segments, or rays that meet, or
cross at a certain point, we call it an intersection point. In
other words, those figures intersect somewhere.
Parallel
Two lines that will never intersect are called parallel lines. In the case of
line segments and rays, we must consider the lines that they lie in. In
other words, we must consider the case that the line segments or rays
were actually lines that extend infinitely in both directions. If the lines
they lie on never intersect, they are called parallel.
Transversal
A transversal is a type of line that intersects at least two other lines. The
lines that a transversal crosses may or may not be parallel.
In both figures, the red line is a transversal.
Planes
A plane can be thought of as a two-dimensional flat surface,
having length and width, but no height. A plane extends
indefinitely on all sides and is composed of an infinite number
of points and lines. One way to think about a plane is as a
sheet of paper with infinite length and width.
Space
Space is the set of all possible points on an infinite number of
planes. Thus, space covers all three dimensions – length,
width, and height.
Angles
Angle
An angle is formed when two rays meet at a common
endpoint, or vertex. The two sides of the angle are the rays,
and the point that unites them is called the vertex.
The vertices are shown in red in the diagram above
Naming Angles
Angles can be named in various ways. One way is to use the ∠
symbol accompanied by three letters. The first and third
letters indicate points on the two rays. The letter in the
middle is the vertex.
The angle above can be called ∠ABC, ∠CBA, ∠B, or ∠α
Classifications of Angles
Angles can be measured in degrees or radians. For the
time being, we will strictly talk about angles in terms
of their degree measure. The symbol for degrees is °.
Angles can measure from 0° up to 360°. Angles with
no measure are called zero angles, while angles of
360° are full rotations. For our study of geometry, we
will primarily focus on three important classifications
of angles:
• Acute
• Obtuse
• Right
Right Angle
A right angle is an angle whose measure is exactly 90°. An easy way to
determine whether an angle is a right angle is by considering whether a
small square could fit perfectly in the corner of the intersection of the
two lines that form the angle. While you would need a protractor to give
a more precise measurement, this can give you an approximation of
whether or not an angle is close to 90°.
Acute Angle
An acute angle is an angle whose measure is less than 90°. For these
kinds of angles, a square could not fit perfectly at the intersection of the
two lines that form them.
Obtuse Angle
Obtuse angles have measures greater than 90° but less than 180°.
Straight Angle
If an angle’s measure is 180°, it is called a straight angle. Straight angles
are just lines with three points on them.
Pairs of Angles
Complementary Angles
Two angles are complementary angles if their degree measurements
add up to 90°. That is, if we attach both angles and fit them side by side
(by putting the vertices and one side on top of each other), they will
form a right angle. We can also say that one of the angles is the
complement of the other.
Complementary angles are angles whose sum is 90°
Supplementary Angles
Another special pair of angles is called supplementary angles.
One angle is said to be the supplement of the other if the sum
of their degree measurements is 180°. In other words, if we
put the angles side by side, the result would be a straight line.
Supplementary angles are angles whose sum is 180°
Vertical Angles
Vertical angles are the angles opposite of each other at the intersection
of two lines. They are called vertical angles because they share a
common vertex. Vertical angles always have equal measures.
∠JKL and ∠MKN are vertical angles. Another pair of vertical angles in the picture is ∠JKM and
∠LKN.
Corresponding Angles
Corresponding angles are the pairs of angles on the same side of the
transversal and on corresponding sides of the two other lines. These
angles are equal in degree measure when the two lines intersected by
the transversal are parallel.
It may help to draw the letter "F" (forwards and backwards) in order to
help identify corresponding angles. This method is illustrated below.
Drawing the letter "F" backwards
helps us see that ∠ADH and ∠EHF are
corresponding angles. We have three
other pairs of corresponding angles in
this figure.
Triangles
Acute Triangle
A triangle whose three angles are acute is called an acute triangle. That
is, if all three angles of a triangle are less than 90°, then it is an acute
triangle.
Every angle in these triangles is acute.
Obtuse Triangle
An obtuse triangle is a triangle that has one
obtuse angle.
The obtuse angles in the triangles above are at vertex H and K, respectively.
Right Triangle
A triangle that has one angle that is a right angle is called a
right triangle. In other words, if one angle of a triangle is 90°,
then it is a right triangle.
Equiangular Triangle
If all three angles of a triangle are congruent, then the triangle is an
equiangular triangle. Later on, we will learn why the only angle measure
possible for equiangular triangles is 60°.
Classifying Triangles by Sides
The sum of the measures of the interior angles of a triangle is 180.
Equilateral Triangle
A triangle with three congruent sides is called an equilateral triangle.
The tick marks indicate congruence between all three sides.
Isosceles Triangle
If a triangle has at least two congruent sides, then the triangle is an
isosceles triangle. Note that, by definition, equilateral triangles can also
be classified as isosceles.
Scalene Triangle
A triangle that has no congruent sides is called a scalene triangle.
No two sides of the triangle above are congruent.