BASIC GEOMETRIC

ELEMENTS

POINTS AND LINES

POINTS We may think of a

point as a "dot" on a piece of paper.

We identify this point with a number or a CAPITAL letter.

A point has no length or width, it just specifies an exact location.

Intersection

The term intersect is used when lines, rays, line segments or figures meet, that is, they share a common point.

IN THIS CASE THE POINT OF INTERSECTION IS D

LINES STRAIGHT LINES don’t have a

beginning or an end. We usually name these lines with small

letters like r,s,t…

r

RAYS (Semirectas) We may think of a ray as a straight line

that begins at a certain point and extends forever in one direction.

B

LINE SEGMENT It has a beginning point and an

endpoint

A

B

LINES

CURVED LINES

POLYGONAL

PARALLEL LINES

rr1

SECANT STRAIGHT LINES

PERPENDICULAR LINES

ANGLES

What is an Angle?

Two rays that share the same endpoint form an angle.

The point where the rays intersect is called the vertex of the angle.

The two rays are called the sides of the angle.

We usually specify an angle using Greek letters like these a, b, g

We can also specify an angle with the letter of its vertex adding the symbol of angle like this A

AA

Measuring Angles We measure the size of an angle using

degrees. ACUTE < 90º RIGHT= 90º OBTUSE > 90º

FLAT = 180º FULL= 360º

CLASIFICATION BYMEASUREMENT

PAIRS OF ANGLES Complementary

Angles: Two angles are

called complementary angles if the sum of their degree measurements equals 90 degrees.

a

b

a+b = 90º

PAIRS OF ANGLES Supplementary Angles: Two angles

are called supplementary angles if the sum of their degree measurements equals 180 degrees.

ab

a+b= 180º

Angle Bisector An angle bisector is a ray

that divides an angle into two equal angles.

POLYGONS

A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.

The figure below is not a polygon, since it is not a closed figure:

The figure below is not a polygon, since it is not made of line segments:

The figure below is not a polygon, since its sides do not intersect in exactly two places each:

We’ve got two kinds of polygons:REGULAR AND IRREGULAR

examples of regular polygons examples of irregular

polygons

ANOTHER CLASIFICATION CONVEX POLYGONS: A figure is convex if

every line segment drawn between any two points inside the figure lies entirely inside the figure.

THESE FUGURES ARE CONVEX

The following figures are concave. Note the red line segment drawn between two points inside the figure that also passes outside of the figure.

Note the red line segment drawn between two points inside the figure that also passes outside the figure.

ELEMENTS OF A POLYGON

CLASIFICATION OF POLYGONS BY THE NUMBER OF SIDES

The sum of the angles of a triangle is 180 degrees.

3 SIDES (TRIANGLES)

Equilateral TriangleA triangle that has three sides of equal length. The angles of an equilateral triangle all measure 60 degrees.

Isosceles Triangle A triangle that has two sides of equal

length. Therefore, it has two equal angles.

Scalene Triangle A triangle that has three sides of

different lengths. So therefore, it has three different angles.

CLASIFICATION OF THE TRIANGLES BY THEIR ANGLES

Acute Triangle : A triangle that has three acute angles.

Obtuse Triangle A triangle that has an obtuse angle.

One of the angles of the triangle measures more than 90 degrees.

Right TriangleA triangle that has a right angle. One of

the angles of the triangle measures 90 degrees.

Quadrilateral A four-sided polygon. The sum of the

angles of a quadrilateral is 360 degrees.

CLASIFICATION

OTHER POLYGONS