1. Introduction to Angles and Triangles

2. Math is a language
Line – extends indefinitely, no thickness or width Ray – part of a line, starts at a point, goes indefinitely Line 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

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

4. Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
                                                               

5. Right Angles A right angle is an angle measuring 90 degrees. Example:
                                                               
90°

6. Two angles are called complementary angles if the
sum of their degree measurements equals 90 degrees.
Example:
These two angles are complementary.
                                                      
                        
32°
58°
Together they create a 90° angle

7. Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
                                                                 

8. Straight Angle A right angle is an angle measuring 180 degrees.
Examples:
                                      

9. Two angles are called supplementary angles if the sum
of 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
                                                

10. Review State whether the following are acute, right, or obtuse. acute3. 5. 1.
right
obtuse ? 4. 2.
acute ?
obtuse

11. Complementary and Supplementary
Find the missing angle.
1. Two angles are complementary. One measures 65 degrees.
2. Two angles are supplementary. One measures 140 degrees.
Answer : 25
Answer : 40

12. Complementary and Supplementary
Find the missing angle. You do not have a protractor.
Use the clues in the pictures. 2. 1. x x 55
165
X=35
X=15

13. 1. x
x = 90 y z
y =
z =
x = 2. x
110 y z
y =
z =

14. Vertical Angles are angles on opposite sides of intersecting lines
                    1. 90 x y z 90
90 and y are vertical angles
x and z are vertical angles
The vertical angles in this case are
equal, will this always be true?
110 70
110 x y z
110 and y are vertical angles 2.
x and z are vertical angles
Vertical angles are always equal

15. Vertical Angles Find the missing angle. Use the clues in the pictures.x
X=58 58

16. Can you find the missing angles?20 90 D 70 G C 70 90 20 J H

17. Can you find these missing anglesB C 68 A 52 G 60 D 60 52 68 F E

18. Parallel lines transversals
and their angles

19. Parallel Lines
What You'll Learn
You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal.
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

20. Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
different points is called a __________
transversal
The lines cut by a transversal may or may not be parallel. l m 1 2 3 4 5 7 6 8
Parallel Lines
t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c
Nonparallel Lines
r is a transversal for b and c. r

21. Parallel Lines and Transversals
is an example of a transversal. It intercepts lines l and m.
We will be most concerned with transversals that cut parallel lines. B A l m 1 2 4 3
When a transversal cuts
parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary. 5 6 8 7

22. 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
Interior

23. Parallel Lines and Transversals
When a transversal intersects two lines, _____ angles are formed.
eight
These angles are given special names.
1. Interior angles , 3,4,5,6
lie between the two parallel lines. l m 1 2 3 4 5 7 6 8
2. Exterior angles 1,2,7,8
lie outside the two lines.
3. Alternate Interior angles 4&6, 5&3
opposite sides of the transversal and
lie between the parallel lines t
5. Consecutive Interior angles 4&5, 3&6
on the same side of the transversal
and are between the parallel lines
4. Alternate Exterior angles 1&7, 2&8
Are on the opposite sides of the transversal and lie outside the
two lines
6. Corresponding angles
1&5, 4&8, 2&6, 3&7
on the same side of the transversal
one is exterior and the other is
interior

24. Name the pairs of the following angles formed by a transversal.
Line M B A
Line N D E P Q G F
Line L
Line M B A
Line N D E P Q G F
Line L
500
1300
Line M B A
Line N D E P Q G F
Line L
Alternate Interior angles
Corresponding angles
Consecutive Interior angles

25. 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
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

26. Parallel Lines and Transversals
Alternate interior angles are _________.
congruent 1 2 4 3 5 6 8 7

27. Parallel Lines and Transversals
Alternate exterior angles is _________.
congruent 1 2 3 4 5 7 6 8 ?

28. Parallel Lines and Transversals
consecutive interior angles is _____________.
supplementary 1 2 3 4 5 7 6 8

29. Transversals and Corresponding Angles
corresponding angles is _________.
congruent

30. Transversals and Corresponding Angles
Concept
Summary
Congruent
Supplementary
Types of angle pairs formed when
a transversal cuts two parallel lines.
alternate interior
consecutive interior
alternate exterior
corresponding

31. Transversals and Corresponding Angles1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
s || t and c || d.
Name all the angles that are congruent 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
14  1
alternate exterior angles
11  9  1
corresponding angles
16  14  1
corresponding angles

32. Let’s Practice m<1=120° Find all the remaining angle measures. 1 46 5 7 8 3
m<1=120°
Find all the remaining angle measures.
120°
60°
120°
60°
120°
60°
120°
60°

33. Another practice problem
40°
180-(40+60)= 80°
Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements.
60° 6 7 8 4
80° 5
60°
40°
80°
120°
100°
60° 11 9 2 1 10 12 3
80°
120°
60°
100°