Regents Examination in Geometry: The Basics of Geometry

Key Ideas
The building blocks of geometry are the point, line, and plane. The definitions of the other geometric figures can all be traced back to these three.

We can think of the point, line, and plane as analogous to the elements in chemistry. All compounds are built up from the elements in the same way that the geometric figures are built up of points, lines, and planes. Along with definitions, we also look at the notation used for each. The ability to interpret vocabulary and notation is important for success in geometry.


Point, Lines, and Planes
A point is location in space. It is zero dimensional, having no length, width, or thickness. Points are represented by a dot and named with a capital letter, as shown by Point A in the figure below. Don’t let the dot confuse you—points are infinitely small. Even the smallest dot you can draw is two-dimensional.

Points, lines, and planes
A line is a set of points extending without end in opposite directions. Lines can be curved or straight. In this book, we will use the term line to refer to straight lines. Lines are one dimensional. They have an infinite length but have no height or thickness. They are represented by a double arrow to indicate the infinite length.

They are named with any two points on the line as shown in above or with a lowercase letter as shown. Three or more points may also be used if we want to indicate the line continues straight through multiple points as above.

A plane is a set of points that forms a flat surface. Planes are two-dimensional. They have infinite length and width but no height. A tabletop or wall can represent a portion of a plane. Remember, though, that the plane continues infinitely beyond the boundaries of the tabletop or wall in each direction. Planes are named with any three points that do not lie on the same line, as shown above or with a capital letter, as shown.

Example 1
Name the following line in 7 different ways.

Solution:


Example 2
Name the plane in two different ways.

Solution: Plane QRS, plane Z

Example 3
How many points lie on

Solution: An infinite number. Every line contains an infinite number of points. We just show a few of them when representing and naming a line.

Rays and Segments
A ray is a portion of a line that has one endpoint and continues infinitely in one direction. A ray is named by the endpoint followed by any other point on the ray. When naming a ray, an arrow is used. The endpoint of the arrow is over the endpoint of the ray. Figure 1.2 illustrates ray

with endpoint A and ray

with endpoint B.

Rays

and

When two rays share an endpoint and form a straight line, the rays are called opposite rays. We say the union of the two rays forms a straight line.
A line segment is a portion of a line with two endpoints. It is named using the two endpoints in either order with an overbar. Figure 1.3 illustrates segment

or
.

The length of a segment is the distance between the two endpoints. The length of

can be referred to as FG or |FG|. In some situations, we may wish to specify a particular starting point and ending point for the segment by using a directed segment. For example, a person walking along directed segment FG would begin at point F and walk directly to point G.

Segment

or

Remember that an infinite number of points are on any line, ray, or segment even though they are not explicitly shown in a figure. Also remember that lines, rays, and segments can be considered to exist even though they are not explicitly shown in a figure.

Example 1
Name each segment and ray in the figure.

Solution: Segments

and
, rays

and


Example 2
If

has a length of 5, what is the length of
?
Solution:

also has a length of 5 because

and

are the same segment.

Angles
An angle is the union of two rays with a common endpoint. The common endpoint is called the vertex. Angles can be named using three points—a point on the first ray, the vertex, and a point on the second ray. The vertex is always listed in the middle.

Alternatively, one can use only the vertex point or a reference number. Figure 1.4 shows the different ways to name an angle.


Naming angles
Angles are measured in degrees. One degree is defined as

of the way around a circle. Halfway around the circle is 180°, and one-quarter around is 90°.

The measure of an angle can be specified using the letter m. For example, m∠RST = 30°.

Angles can be classified by their degree measure.
 
Acute angle—an angle whose measure is less than 90°.
Right angle—an angle whose measure is exactly 90°.
Obtuse angle—an angle whose measure is more than 90° and less than 180°.
Straight angle—an angle whose measure is exactly 180°.

This figure shows examples of each type of angle.

The square positioned at the vertex of the right angle is often used to specify a right angle.

Classification of angles


Math Fact
Our definition of the degree as

of a rotation around a center point has been used since ancient times. No one knows for sure why

was chosen. One theory is that it originated with ancient Babylonian mathematicians, who used a base-60 number system instead of the base-10 system we use today. They divided a circle into 6 congruent equilateral triangles with 60° central angles. Then the ancient Babylonians subdivided each central angle into 60 parts. Another theory is that the circle was divided into 360 parts because one year is approximately 360 days.

Either way, 360 is a convenient number to partition the circle with because 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, and 10.




Example 1
Name one angle and two rays.
Solution:
,
,


Example 2
Name each angle in 3 ways, and classify each angle.
Solution:
∠R, ∠SRT, ∠TRS; acute angle
∠E, ∠DEF, ∠FED; obtuse angle
∠I, ∠HIJ, ∠JIH; right angle

Adjacent Angles
Angles that share a common ray and vertex but no interior points are adjacent angles. In Figure 1.6, ∠ABC and ∠CBD are adjacent angles. ∠ABC and ∠ABD are not to be considered adjacent because they share interior points in the region of ∠CBD.

To avoid confusion, always use three vertices or a reference number when naming adjacent angles. Using the vertex alone would be ambiguous.

Adjacent angles ∠ABC and ∠CBD

Example 1
Name 3 pairs of adjacent angles.

 
Solution: ∠AOB and ∠BOC, ∠BOC and ∠COA, ∠COA and ∠AOB

Polygons
A polygon is a closed figure with straight sides. They are named for the number of sides.


Be familiar with these common polygons.
Triangle—3 sides
Quadrilateral—4 sides
Pentagon—5 sides
Hexagon—6 sides
Octagon—8 sides
Decagon—10 sides

The intersection of two sides in a polygon is called a vertex (plural is vertices). The vertices are used to name specific polygons by listing the vertices in order around the polygon. They can be called out either clockwise or counterclockwise but must always be stated in continuous order—no skipping allowed.

This figure shows some polygons with their names. We can list the vertices of the triangle in any order since it would be impossible to skip a vertex. For the quadrilateral, the name EFGD is valid but EFDG is not. For triangles, we often precede the vertices with the triangle symbol, ∆, so triangle ABC would be referred to as ∆ABC.


Triangle ABC (or ΔABC), quadrilateral DEFG, pentagon HIJKL

When all the sides of a polygon are congruent to one another (equilateral) and all the angles of the polygon are congruent to one another (equiangular), we refer to that polygon as regular. So a square is an example of a regular quadrilateral, while a rectangle may have two sides with lengths different from the other two.

Example
Sketch hexagon RSTUVW. Name each side and each angle.

Solution:

Sides

Angles ∠R, ∠S, ∠T, ∠U, ∠V, ∠W

Classifying Triangles
Triangles can be classified by their angle lengths and measures as shown in the Figure below.

Triangle classifications


Classifying triangles by sides:
Scalene—no congruent sides, no congruent angles
Isosceles—at least two congruent sides, two congruent angles
Equilateral—three congruent sides, three congruent angles

Classifying triangles by angles:
Acute—all angles are acute
Right—one right angle
Obtuse—one obtuse angle

Example 1
∆ABC has side lengths AB = 2, BC = 1, and AC = 1.7. ∠A measures 30°, ∠B measures 60°, and ∠C measures 90°. Classify the triangle.
Solution: ∆ABC is a right acute triangle.

Four special segments can be drawn in a triangle, and every triangle has three of each. These special segments are the altitude, median, angle bisector, and perpendicular bisector, shown in this figure.

Special segments in triangles
 

Altitude—a segment from a vertex perpendicular to the opposite side
Median—a segment from a vertex to the midpoint of the opposite side
Angle bisector—a line, segment, or ray passing through the vertex of a triangle and bisecting that angle
Perpendicular bisector—a segment, line, or ray that is perpendicular to and passes through the midpoint of a side

Example 2
In ∆ABC,

is drawn such that D lies on

and

is perpendicular to
.

What special segment is
?
Solution:

is an altitude. It has an endpoint at a vertex and is perpendicular to the opposite side of the triangle.

Example 3
In triangle ∆ABC,

is drawn such that ∠BAD and ∠CAD have the same measure. What special segment is
?
Solution:

is an angle bisector of ∆ABC.