Building Blocks of Geometry
Point: Undefined term. It is analogous to dot on paper, a speck of dust, or a grain of
sand. A point has no size only location. A point is labeled with a capital letter.
Line: A line is a straight continuous arrangement of infinitely many points. It has infinite
length but no thickness. A line is labeled by naming two points on the line. So the line
containing points A and B could be labeled either AB or BA .
Line segment: The line segment AB is the subset of the line AB consisting of the points
A and B, called endpoints, and all points in between.
Ray: Is a subset of a line consisting of a point A called the endpoint and all points on
one side of point A.
Plane: A flat surface that has infinite length and width but no thickness.
Angle: An angle is formed when two rays intersect at a common point called the vertex.
When you name an angle you use three letters, one on each ray and the vertex angle.
Note that the vertex angle must be in the middle. This angle can be referred to as either
TAP or PAT .
The measure of an angle is the smallest amount of rotation about the vertex from one ray
to the other. The most common way of measuring angles is in degrees.
A protractor is a device used to measure angles in degrees.
Angles can be classified according to their measure as




Acute - measures between 00 and 900
Right - measures exactly 900
Obtuse - measures between 900 and 1800
Straight - measures exactly 1800
Angles formed by two intersecting lines are called vertical angles.
Vertical Angle Theorem: Vertical Angles have the same measure.
Consider the situation when two lines are both intersected by a third line called a
transversal. This creates 4 pairs of corresponding angles.
If two lines in a plane have no points in common they are said to be parallel.
Of special importance is the situation when the two lines are parallel. So suppose lines l
and m in the figure are parallel ( written l m ).
Corresponding Angle Property:
 If two parallel lines are cut by a transversal, then corresponding angles are equal.
 If two lines in the same plane are cut by a transversal and some pair of corresponding
angles have the same measure then the lines are parallel.
The following chart illustrates the notions of alternate interior and alternate exterior
angles. Alternate angles are on opposite sides of the transversal.
Alternate Interior Angle Theorem: Two lines cut by a transversal are parallel if, and
only if, a pair of alternate interior angles are equal.
Curve: Informally a curve is a continuous path connecting two points.
A simple curve does not intersect itself except possibly at the endpoints
A closed curve has the starting and ending point the same.
Polygons are simple closed curves formed by line segments.
Convex curves are simple, closed curves with no "indentations".
Concave curves are simple and closed but not convex.
The next chart illustrates a variety of curves and polygons.
Polygons are classified according to the number of sides.
When two line segments or angles have the same measure we say they are congruent.
We write
AB  CD to mean that the segments have the same length and
A  B to mean that both angles have the same measure.
Triangles can be classified according to either the length of the sides or the measure of
the angles.
Quadrilaterals are classified in a variety of ways including whether they have parallel
sides, congruent sides, or right angles.
The next table defines and illustrates how both triangles and quadrilaterals are classified.
The following flow chart can be used to summarize how triangles and quadrilaterals are
classified.
The following page is taken from a 6th grade textbook.
Sum of the Measures of the Angles in a Triangle
Perhaps the most simplistic way of illustrating that the sum of the measures of the interior
angles of a triangle is 1800 is to tear the corners of a triangle and rearrange them to form
a straight angle.
Deductive reasoning can also be used to prove the sum of the angles of a triangle is 1800 .
Draw line l parallel to AC through vertex B of triangle ABC.
Since alternate interior angles are congruent it follows that
Thus:
1 2  3 
1  4 and
4  2  5  1800
Theorem: The sum of the interior angles of a triangle is 1800 .
3 5.
An alternative approach to show that the sum of the interior angles of a triangle is 1800 is
to pretend a turtle is going to walk the boundary of triangle ABC, starting at point A
facing point B. He finishes his walk by returning to point A and facing in the same
direction that he started.
Now the sum
4  5  6  3600
since this represents the turtle making one full rotation.
Also:
1 4  1800
2  5  1800
3  6  1800
so
1  2  3  4  5  6  3(1800 )
so
1 2  3  3(1800 )  ( 4  5  6)  3(1800 )  3600  1800