What is discussed in this lecture? 

1.   Introduction to Geometry
2.   Some Definitions
3.   Introduction to Angles
4.   Angle Measure
5.   Naming of Angles
6.   Special Angles
7.   Measuring Angles
8.   Relationships Among Angles Formed by Intersecting Lines

1.   Introduction to Geometry

Geometry, from the Ancient Greek "geo" (earth) and "metron" (measurement), is a branch of mathematics concerned with questions of shape and space. The fundamental objects of geometry are points, lines, and planes.  Angles and curves are also critical to the study of geometry because they play an important role in creating three-dimensional shapes.  Studying the shape of our world will provide us with an abundance of practical applications.  You can tile your house, buy carpet, build a brick wall, or determine how much money to save before purchasing that granite countertop for your new kitchen.  Geometry is found everywhere: in construction, landscaping, art, architecture, astronomy, engineering, surveying, machines, tool making, cars and much more.

2.   Some Definitions

Point

A point is a small dot.  It has no length, width, or thickness.

Plane

A plane is a flat surface with no boundaries and no thickness.

Line

A line extends indefinitely in both directions indicated by arrows.  A line can be identified by naming any two points on the line, for example, A and B.  The notation for a line that extends through points A and B is (in alphabetic order).

Parallel Lines

Lines that never intersect.

Intersecting Lines

Lines that have a point in common.

Perpendicular Lines

Lines that intersect at a 90angle.

Line Segment

A line segments starts and stops at distinct points called endpoints.  A line segment can be identified by its endpoints, for example, A and B.  The notation for a line segment that includes the points A and B is (in alphabetic order).

 

Ray

A ray consists of a point on a line and all subsequent points on one side of the point.  The point from which the ray originates is called the endpoint.  A ray is named by its endpoint and any other point on the ray, for example, A and B.  The notation for a ray that includes the points A and B is (in alphabetic order)

3.  Introduction to Angles

An angle is determined by rotating a ray about its endpoint.  The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side of the angle.  The terminal and initial side are also called arms of the angle.  The point where the initial and the terminal side meet is called the vertex of the angle.

 

NOTE:  When drawing an angle, the rotation of the ray about its endpoint is usually indicated with an arc in between the initial and terminal sides.

4.   Angle Measure

Most commonly, angles are measured in degrees.  This is indicated by the degree sign ^o next to a number.  For example, 45^o. 

Degrees can further be divided into minutes ( ' ) and seconds ( " ).  That is,

1^o = 60' (minutes) using the apostrophe on the computer keyboard

1' = 60" (seconds) using the quotation mark on the keyboard

Angles can be named in several ways.  In this course we are going to use three different ways.  We'll show them by using the following picture:

 

• We can place a number or letter in between the two rays, say 1, and then name the angle 1 using the symbol for angle.  We pronounce this "angle 1".
• We can also use the letters of points on the rays together with the vertex point.  That is, ABC or CBA, either way, as long as the letter for the vertex point is in the middle.
• Finally, we can use the letter for the vertex point alone as long as it is perfectly clear which angle is designated by this letter.  In the picture above, it is quite clear which angle we mean when we say B

Example 1:

 

a.  Express the angle indicated above in two different ways using three capital letters.

 DEF or FED  The vertex point is in the middle!

b.  Express the angle indicated above using one capital letter.

E  using the letter of the vertex point!

c.  Express the angle indicated above using the Greek letter θ (theta).

  θ

6.   Special Angles

0^o (Zero)-Degree Angle

One ray indicates the initial and the terminal side.

 

360-Degree Angle

 

NOTE:  Unlike with the 0^o angle, there is a curved angle indicator showing that the terminal side moved through a rotation of 360^o.

Right Angles

Angles whose measure is exactly 90^o.

 

Please note that the RIGHT ANGLE is usually indicated by a rectangle drawn between the terminal and initial side.

Straight Angles

Angles whose measure is exactly 180^o.

Acute Angles

Angles whose measure is greater than 0^obut less than 90^o.

Obtuse Angles

Angles whose measure is greater than 90^o but less than 180^o.

Reflex Angles

Angles whose measure is greater than 180^o

Example 2:

Classify the given angles as right, straight, acute, obtuse, reflex:

a.  38^o - acute angle because the measure is greater than 0^o but less than 90^o

b.  95^o - obtuse angle because the measure is greater than 90^o but less than 180^o

c.  90^o - right angle

d.  153^o - obtuse angle because the measure is greater than 90^o but less than 180^o

e.  10^o - acute angle because the measure is greater than 0^o but less than 90^o

f.  180^o - straight angle

g. 270^o - reflex angle

Complementary Angles

Two angles are called complementary when their sum is 90^o. 

Supplementary Angles

Two angles are called supplementary when their sum is 180^o. 

Example 3:

Tell whether the angle pairs are complementary, supplementary, or neither.

a.  42^o, 80^o - sum equals 122^o, which is neither complementary (90^o) nor supplementary (180^o)

b.  17^o, 73^o - sum equals 90^o, therefore, angles are complementary

c.  38^o, 142^o - sum equals 180^o, therefore, angles are supplementary

d.  52^o, 48^o - sum equals 100^o, which is neither complementary (90^o) nor supplementary (180^o)

e.  60^o, 30^o - sum equals 90^o, therefore, angles are complementary

f.  110^o, 70^o - sum equals 180^o, therefore, angles are supplementary

7.   Measuring Angles

We use a semicircular protractor to measure angles.  Below is a picture of one type. 

Measuring Procedure:

• Place the base of the protractor along one side of the angle with the point on your protractor that indicates its "center" on the vertex of the angle.  Please note that the "center" of the protractor is somewhere along or close to the base of the protractor.  Be very careful! Not all protractors have user-friendly centers.
• Depending on the location of the other side of the angle, either choose the scale that has the zero-degree reading on the right side of the protractor or on the left side.  Read the measurement where this side crosses the scale of the protractor.

  

The measure of angle O is 40^o. 

8.   Relationships Among Angles Formed by Intersecting Lines

Below is the picture of two intersecting lines. 

 

In the picture above, 1 and 3 are called opposite (vertical) angles.  Likewise, 2 and 4 are called opposite (vertical) angles.

The measures of vertical angles are equal. 

When we discuss the measure of an angle, traditionally the letter m is placed before the angle symbol.

Therefore,

m1 = m3

m2 = m4

The sum of the measures of adjacent angles equal 180^o. 

Therefore,

m1 + m2 = 180^o

m3 + m4 = 180^o

m1 + m4 = 180^o

m2 + m3 = 180^o

Example 4:

In the figure shown below, you are given the measure of two angles.  Find the measure of the remaining angles.

 

Angles a and d are opposite angles.  Therefore, ma equals md .  The measure of d must be 90^o.  This indicates that the angle b + c must have a measure of 90^o because the angles a and b + c are adjacent angles who are supplementary!

Angles b and c are complementary angles.  Therefore, mb + mc = 90^o .  The measure of c must be 90^o 55^o = 35^o.

Angles b and e are opposite angles.  Therefore, mb equals me .  The measure of e must be 55^o.

Angles c and f are opposite angles.  Therefore, mc equals mf .  The measure of f must be 35^o.