introduction to angles and triangles
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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
Parallel Lines and Transversals
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.
Parallel Lines and Transversals
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
Parallel Lines and Transversals
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
Parallel Lines and Transversals
Alternate interior angles are _________.
1 234
57
68
64
53
congruent
Parallel Lines and Transversals
1 2
34
576
8
Alternate exterior angles is _________.congruent
71
82
Parallel Lines and Transversals
1 2
34
576
8
consecutive interior angles is _____________.supplementary
18054
18063
Transversals and Corresponding Angles
corresponding angles is _________.congruent
Transversals and Corresponding Angles
ConceptSummary
Congruent Supplementary
alternate interior
alternate exterior
corresponding
consecutive interior
Types of angle pairs formed when a transversal cuts two parallel lines.
Transversals and Corresponding Angles
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
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