Download 3.6-3.7-congruent-angles

Warm Up
Find the
area of the
green
region.
Assume all
angles are
right
angles.
2 ft.
2 ft.
2 ft.
1 ft.
4 ft.
1 ft.
1 ft.
1 ft.1 ft.
2 ft.
1 ft.
3 ft.
Sections 3.6 and 3.7
Congruent Angles
• Definition: they have the same degree
measure.
• Symbols: angle A congruent angle B if and
only if m a and m  b
• Picture:
30°
•
B
30°
A
The first relationship we are going to talk about
Definition: Two angles are vertical angles if
their sides form two pairs of opposite rays
Vertical angles are always congruent.
Angles 1 and 2 are vertical angles
Angles 3 and 4 are also vertical angles
1
3
4
2
Theorem 3.1-Vertical Angle Thm
• Definition: vertical angles are congruent
• Picture:
1
2
3
4
• Symbols: Angles 1 and 3 are congruent
• Angles 2 and 4 are congruent
What is the measure of the angle?
5y – 50
What type
of angles
are these?
4y – 10
5y – 50 = 4y – 10
y = 40
Plug y back into our angle equations and we get
150
Find the value of x in each figure
• 1.
2.
130°
5x°
25°
x°
• 3.
4.
125 °
x°
40°
(x – 10)°
Theorem 3-2:
• If two angles are congruent, then their
complements (90 °) are congruent.
• Picture:
•
60°
60°
• The measure of angles complementary to A and
B is 30.
Identify each pair of angles as adjacent, vertical,
complementary, supplementary, and/or as a linear pair.
Example 1:
1 and 2
ADJACENT
2
1
3
4
5
Identify each pair of angles as adjacent, vertical,
complementary, supplementary, and/or as a linear pair.
Example 2:
1 and 4
VERTICAL
2
1
3
4
5
Identify each pair of angles as adjacent, vertical,
complementary, supplementary, and/or as a linear pair.
Example 3:
3 and 4
ADJACENT,
COMPLEMENTARY
2
1
3
4
5
Theorem 3-3:
• If two angles are congruent, then their
supplements (180 °) are congruent.
• 70° 110°
110° 70°
• Angle 1 is congruent to angle 4 (measure of
70°)
Theorem 3-4:
• If two angles are complementary to the
same angle, then they are congruent.
3
4
5
• Angle 3 is complementary to angle 4, angle
5 is complementary to angle 4. Angle 3 is
congruent to angle 5.
Theorem 3.5:
• If two angles are supplementary to the same angle,
then they are congruent.
3
1
2
• Angle 1 is supplementary to angle 2. Angle 3 is
supplementary to angle 2. Angle 1 congruent to
angle 3.
Identify each pair of angles as adjacent, vertical,
complementary, supplementary, and/or as a linear pair.
Example 4:
1 and 5
ADJACENT,
SUPPLEMENTARY,
LINEAR PAIR
2
1
3
4
5
Find x, y, and z.
Example 5:
x = 129,
y = 51,
z = 129
51
x
z
y
L
Example 6:
Find x.
(5x - 15)
P
(5x - 15) = (3x + 1)
5x  15  3x  1
2 x  15  1
2x  16
X=8
(20x - 5)
A
T
(3x + 1)
O
L
Example 7:
(5x - 15)
Find
mLAT
P
Since we have already found the
value of x, all we need to do now is to
plug it in for LAT.
20x  5  20(8)  5
160  5 
155
(20x - 5)
A
T
(3x + 1)
O
Theorem 3-6:
• If two angles are congruent and
supplementary, then each is a right angle.
•
1
2
Theorem 3-7:
• All right angles are congruent.
• These angles are congruent
Moving right along…
• Section 3-7- Perpendicular Lines
• This is the last section of the chapter!
YEAH.
Lines that intersect
to form four right
angles are
perpendicular lines.
Symbol:
m
l is read as m
perpendicular to l
m
1
4
2
3
l
Clipper- Flying Cloud Ship
• The main mast and the
frame for the sails are
examples of perpendicular
line segments. The main
mast is perpendicular to
the sail frame, the
likewise, the frame for the
sail is perpendicular to the
main mast.
Flying Cloud
• One common
nineteenth century
ship was the clipper.
This ship, which had
many sails, was
designed for speed. In
fact, it was named a
clipper because of the
way it “clipped off”
the miles.
Theorem 3-8:
• If two angles are perpendicular, then they
form four right angles.
Each angle is 90 degrees.
Each angle is congruent.
GV  AE
F
A
5
6
G
B
V
1
2
8 3
C 4
E
AE  FV
F
A
5
6
G
B
V
1
2
8 3
C 4
E
4 1
F
A
5
6
G
B
V
1
2
8 3
C 4
E
3
4
F
A
5
6
G
B
V
1
2
8 3
C 4
E
1  2  90
F
A
5
6
G
B
V
1
2
8 3
C 4
E
GVA is a
right 
F
A
5
6
G
B
V
1
2
8 3
C 4
E
6 & 3 are
supplementary
F
A
5
6
G
B
V
1
2
8 3
C 4
E
6 & 2 are
complementary
F
A
5
6
G
B
V
1
2
8 3
C 4
E
Theorem 3.9:
• If a line m is in a plane and point T is a
point on m, then there exists exactly one
line in that plane that° is perpendicular to m
at T.
m
°
T
Homework: workbook page 17 and 18 ALL