**ANGLES AND TRIANGLES**

**Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Graph angles.

2. Measure angles.

3. Name angles.

4. Work with special angles (right, straight, acute, obtuse, complementary, supplementary).

6. Memorize the general characteristics of triangles.

7. Work with several types of triangles (right, equilateral, isosceles, right isosceles, scalene, acute, obtuse).

Definition of a RayA ray is a line that starts at one point and extends forever in one direction.

Definition of an AngleAn angle is determined by rotating a ray a specific distance about its starting point. The starting position of the ray is called the

initial sideof the angle. After the rotation we end up with another ray and it is called theterminal sideof the angle. The point where the initial and the terminal side meet is called thevertexof 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.

Angle MeasureMost commonly, angles are measured in degrees. We will know that angles are measured in degrees when there is little circle in the upper right-hand corner of a number. For example,

which is then pronounced 45 degrees.45^{o}Degrees can further be divided into minutes ( ' ) and seconds ( " ). That is,

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

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

Example 1:Change

(45 degrees and 14 minutes and 39 seconds) to decimal degree form. Round to two decimal places.45^{o}14' 39"Calculate . Type the entire calculation into your calculator. Do not round until you have the final answer:

45^{o}14' 39" 45.24^{o}

Example 2:Change

to degrees, minutes, and seconds rounded to whole numbers.57.68^{o}1. The

is a fraction of a degree. Take0.68away from0.68and convert it to minutes as follows:57.68

0.68(60') = 40.8'2. The

from Step 1 is a fraction of a minute. Take0.8away from0.8and convert it to seconds as follows:40.8

0.8(60") = 48"Thus,

57.68^{o}= 57^{o}40' 48"

Naming AnglesAngles 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

, and then name the angle1using the symbol for angle. We pronounce this "angle 1".1We can also use the letters of points on the rays together with the vertex point. That is,

orABC, either way, as long as the letter for the vertex point is in the middle.CBAFinally, 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.Sometimes Greek letters are used to represent angles. Most common are the following letters:

Theta: Alpha: Beta: Gamma:

Introducing Some Special Angles

Right AnglesAngles 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 AnglesAngles whose measure is exactly

.180^{o}

Acute AnglesAngles whose measure is greater than

but less than0^{o}.90^{o}

Obtuse AnglesAngles whose measure is greater than

but less than90^{o}.180^{o}

Complementary AnglesTwo angles are called

complementarywhen their sum is90.^{o}

Supplementary AnglesTwo angles are called

supplementarywhen their sum is180.^{o}

Drawing and Measuring AnglesWe use a protractor to draw and measure angles in degrees. Below is a picture of one type. Note specifically the

Base,theCenter, and theScale. When measuring angles, theCenteris placed at the vertex of the angle with theBasealong its initial side.Note there are two scales, one for measuring acute angles and one for measuring obtuse angles.

Example 3:Measure the angle

Oin the picture below.We place the

Baseof the protractor along the terminal side of the angle with itsCenterat the vertex.We read the measurement where the terminal side crosses the scale of the protractor.

The measure of angle

Ois40. NOTE: Since it is an acute angle, it is certainly not^{o}140!!!^{o}

General Characteristics of TrianglesA triangle is a closed 2-dimensional shape with 3 sides, 3 interior angles, and 3 vertices (singular: vertex).

Interior Angles of a TriangleThe

interior anglesof a triangle are the three angles inside the triangle. The sum of the threeinterior anglesin a triangle is always.180^{o}

Sides of a TriangleThe longest side of a triangle is opposite the largest angle and vice versa. The shortest side of a triangle is opposite the smallest angle and vice versa.

Vertices (singular: vertex)The point of intersection of any two sides of a triangle is known as a vertex. A triangle has three vertices. In the triangle above, the vertices are

,A, andB.C

Introducing Several Types of Triangles

The Right TriangleA

right trianglehas two sides that form aangle which is also called a90^{o}rightangle. These two sides are called thelegsof the triangle. The side opposite the right angle is called thehypotenuse.

Please note that the RIGHT ANGLE is usually indicated by a rectangle drawn between the legs.

The Equilateral TriangleAn

equilateral triangleis a triangle with three equal sides. The three angles of an equilateral triangle are also equal.In the triangle below,

equalsA.equalsBequalsC.60^{o}

The Isosceles TriangleAn

isosceles triangleis a triangle with exactly two equal sides. The angles opposite the two equal sides are also equal.In the triangle below,

equalsA.B

The Right Isosceles TriangleA

right isosceles triangleis a triangle with exactly two equal sides. However, one of the angles is. This means that the angles opposite the two equal sides each measure exactly90^{o}.45^{o}In the triangle below,

equalsAequalsB.45^{o}

The Scalene TriangleA triangle having sides of different lengths and angles of different measure.

Acute and Obtuse TrianglesIn an acute triangle all angles are greater than

but less than0^{o}. An obtuse triangle has one angle whose measure is greater than90\^{o}but less than90^{o}.180^{o}