+

Introduction to Angles

+ Introduction to LessonThe purpose of this tutorial is to introduce angles and the various relationships they have. Upon completion of this tutorial, you will be able to:

to recognize basic types of angle vocabulary

to recognize and apply characteristics of basic angle relationships (supplementary, linear pair, complementary, adjacent, vertical) to solve angle problems.

to recognize and apply characteristics of angle relationships on parallel lines to solve angle problems.

A BRAY: A ray, AB, starts at point A and

goes on forever in one direction.

ANGLE: An angle is formed by two rays with a common endpoint. B

A

C

Angles can be named several ways . . .

1 means 1 A means ABC means

A

BC

This 3rd method is used when more than two rays form a vertex.

VERTEX: A vertex is the point where two or more rays (or segments in a polygon) intersect.

B

A

C

A

B C

D This would be too confusing!Therefore we use:

ABC, ABD or CBD

A

B

C

Which is B?

When we want to indicate that two or more segments or angles are equal in a figure, we use an equal number of slash marks like the figures below.

We can classify the different types of angles.

Acute Angle: Any angle with a measure between (but not including) 0º and 90º.

Right Angle: Any angle which measures 90º.

Obtuse Angle: Any angle with a measure between (but not including) 90º and 180º.

Straight Angle: Straight Angles (or lines) have a measure of 180º.

We will classify some of the different RELATIONSHIPS of angles.

Any pair of angles whose sum is 180º.

30º 150ºAB

Any pair of angles whose sum is 90º.

50º40º 60º

30º

Supplementary Angles:

Complementary Angles:

LINEAR PAIR: A supplementary pair of angles that form a straight angle.

A O B20º160º

D

Transversal: a line that intersects two or more lines in the same plane.

Parallel lines: lines that are in the same plane and do not intersect (cross).

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Exterior means the angles are outside the parallel lines.

Interior means the angles are between the parallel lines.

When we classify angle relationships, we need to look at the POSITION of the angles in relation to the parallel

lines AND the transversal.

Two angles can be on the same side of the transversal or they can be on alternate sides of the transversal.

Same side

Alternate sides

+Now, we put these two ideas together to classify angle relationships.

21

34

56

78

Angle 4 & 5 are on alternating sides of the transversal, and are interior of the parallel lines.

So, we say that angle 4 & angle 5 are Alternate Interior

Angles.4

5

**Note when we are classifying angles on parallel lines, we are dealing with PAIRS of angles classified by their positions in relation to two things: the parallel lines AND the transversal. We do NOT say that 8 is an exterior angle (this will be defined later as a single classification).

Alternate Interior Angles

alternate sides of the transversal and interior of the parallels

Same Side Interior Angles

Same side of the transversal and interior of the parallels

Alternate Exterior Angles

alternate sides of the transversal and exterior of the parallels

Same Side Exterior Angles

Same side of the transversal and exterior of the parallels

The angle relationships when parallel lines are cut by a transversal are…

There is one other important type of angle relationship on parallel lines cut by a transversal but it doesn’t quite follow the same ideas as the first four…

Corresponding Angles

Angles on the same side of the transversal and in the same position on the parallel lines (either the angles are

both above or both below the parallel lines.)

Since the five types of angles describe angle relationships, we can only classify PAIRS of angles as Alternate Interior Angles,

Same Side Interior Angles, Alternate Exterior Angles, Same Side Exterior Angles, or Corresponding Angles.

Pairs of angles can be classified as one of these relationships even if the lines being cut by the transversal are NOT parallel.

Find pairs of angles that are:

Alternate exterior angles

Same side interior angles

Same side exterior angles

12

34

56

78

(1 & 8) or (2 & 7)

(3 & 5) or (4 & 6)

(2 & 8) or (1 & 7)

1

8

2

7

35

46

2

8

1

7Alternate interior angles

(3 & 6) or (4 & 5)

35

46

Corresponding angles:

(1 & 5) or (2 & 6)

or (3 & 7) or (4 & 8)

ab

cd

ef

gh

Some pairs are equal and some are supplementary.

If (and only if) the lines cut by the transversal are PARALLEL, then we can make some conclusions based on the angle relationships.

Alternate Interior Angles

Same Side Interior Angles

Alternate Exterior Angles

Same Side Exterior Angles

Corresponding Angles

So, if the lines are parallel, what can you conclude about:

Equal

Supplementary

Equal

Supplementary

Equal

Vertical Angles: The opposite angles formed by two intersecting lines. This is a relationship between PAIRS of angles…one angle cannot be vertical.

When two lines intersect, the resulting pairs of vertical angles are ________.

Conjecture:equal

Adjacent Angles: Angles that are “next to” each other. To be adjacent angles, they MUST satisfy 3 conditions: share a common side, share a common vertex, No overlap … the common side is

IN BETWEEN the two angles.