introduction to angles and triangles

Introduction to Angles and Triangles

Introduction to angles

Math is a languageLine – extends indefinitely, no thickness or widthRay – part of a line, starts at a point, goes indefinitelyLine segment – part of a line, begin and end point Angle - two lines, segments or rays from a common point Vertex - common point at which two lines or rays are joined

Degrees: Measuring AnglesWe measure the size of an angle using degrees. Example: Here are some examples of angles and their degree measurements.

Acute Angles

An acute angle is an angle measuring between 0 and 90 degrees.

Example:

Right Angles

A right angle is an angle measuring 90 degrees.

Example:

90°

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

Example: These two angles are complementary.

58°

32°

Together they create a 90° angle

Obtuse Angles

An obtuse angle is an angle measuring between 90 and 180 degrees.

Example:

Straight AngleA right angle is an angle measuring 180 degrees.

Examples:

Supplementary AnglesTwo angles are called supplementary angles if the sumof their degree measurements equals 180 degrees.

Example: These two angles are supplementary.

139°

41°

These two angles sum is 180° and together

the form a straight line

ReviewState whether the following are acute, right, or obtuse.

1.

2.

3.

4.

5.

?

?

acuteobtuse

right

obtuseacute

Complementary and Supplementary

1. Two angles are complementary. One measures 65 degrees.

2. Two angles are supplementary. One measures 140 degrees.

Find the missing angle.

Answer : 25

Answer : 40

Complementary and SupplementaryFind the missing angle. You do not have a protractor.Use the clues in the pictures.

1. 2.

x55 165

x

X=35X=15

1.

90xy z

x =

y =

z =

2.

110xy z

x =

y =

z =

9090

90

11070

70

70110

9090

Vertical Angles are angles on opposite sides of intersecting lines

90xy z

1. 90 and y are vertical angles

x and z are vertical angles

110xy z

2.

909090 90

11070110 70

The vertical angles in this case are equal, will this always be true?

110 and y are vertical angles

x and z are vertical angles

Vertical angles are always equal

Vertical Angles

Find the missing angle.

Use the clues in the pictures.

58 x X=58

Can you find the missing angles?

20

C

J

D

EF

G

H

70

907020

90

Can you find these missing angles

52

B

A

F E

D

C

60

G

68

686052

18

Parallel lines transversalsand their angles

Parallel Lines

In geometry, two lines in a plane that never intersect ,have the same slope, are called ____________.parallel lines

parallel lines are always the same distance apart

You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel linesand a transversal.

Parallel Lines and Transversals

The lines cut by a transversal may or may not be parallel.

l

m

1 2

34

576

8

ml

Parallel Lines

t is a transversal for l and m.

t

1 234

5

7

6

8

b

c

cb ||

Nonparallel Lines

r is a transversal for b and c.

r

In geometry, a line, line segment, or ray that intersects two or more lines atdifferent points is called a __________transversal


l

m

B

A

AB is an example of a transversal. It intercepts lines l and m.

1 2

34

5

76

8

• We will be most concerned with transversals that cut parallel lines.

• When a transversal cuts parallel lines, special pairs of

angles are formed that are sometimes congruent and sometimes supplementary.


Two lines divide the plane into three regions.

The region between the lines is referred to as the interior.

The two regions not between the lines is referred to as the exterior.

Exterior

Exterior

Interior

l

m

1 2

34

576

8


When a transversal intersects two lines, _____ angles are formed.eight

These angles are given special names.

t

1. Interior angles , 3,4,5,6lie between the two parallel lines.

2. Exterior angles 1,2,7,8lie outside the two lines.

3. Alternate Interior angles 4&6, 5&3 opposite sides of the transversal andlie between the parallel lines 5. Consecutive Interior angles 4&5,

3&6on the same side of the transversaland are between the parallel lines

4. Alternate Exterior angles 1&7, 2&8Are on the opposite sides of the transversal and lie outside thetwo lines

6. Corresponding angles 1&5, 4&8, 2&6, 3&7on the same side of the transversalone is exterior and the other is interior

Name the pairs of the following angles formed by a transversal.

Line MBA

Line ND E

P

Q

G

F

Line L

Line MBA

Line ND E

P

Q

G

F

Line L

Line MBA

Line ND E

P

Q

G

F

Line L

500

1300

Congruent: Same shape and size

The symbol means that the shapes, lines or angles are congruent

two shapes both have an area of 36 in2 , are they congruent?

6 in

6 in

Area is 36 in2

9 in

4 in

Area is 36 in2

Numbers, or expressions can have equal value…..In Geometry, we use “congruent” to describe two or more objects, lines or angles as being the same


Alternate interior angles are _________.

1 234

57

68

64

53

congruent


1 2

34

576

8

Alternate exterior angles is _________.congruent

71

82


1 2

34

576

8

consecutive interior angles is _____________.supplementary

18054

18063

Transversals and Corresponding Angles

corresponding angles is _________.congruent


ConceptSummary

Congruent Supplementary

alternate interior

alternate exterior

corresponding

consecutive interior

Types of angle pairs formed when a transversal cuts two parallel lines.


s t

c

d

1 2 3 45 6 7 8

9 10 11 12

13 14 15 16

s || t and c || d.

Name all the angles that arecongruent to 1.Give a reason for each answer.

3 1 corresponding angles

6 1 vertical angles

8 1 alternate exterior angles

9 1 corresponding angles

11 9 1 corresponding angles

14 1 alternate exterior angles

16 14 1 corresponding angles

Let’s Practicem<1=120°Find all the remaining angle

measures.1

4

2

65

7 8

3

60°

60°

60°

60°

120°

120°

120°

120°

Another practice problem

Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements.

40°

120°

120°60°

60°

40°60°

60°180-(40+60)= 80°

80°

80°

80°

100°

100°

123

45

6

9

7

8

10

11 1

2

introduction to angles and triangles

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