Transcript angle

1.6 Angle Relationships 9/10/12
• Pairs of Angles
– Adjacent Angles
– Vertical Angles
– Linear Pair
– Complementary
– Supplementary
CCSS: G-CO1
Know precise definitions of
angle, circle, perpendicular line,
parallel line, and line segment,
based on the undefined notions
of point, line, distance along a
line, and distance around a
circular arc.
Mathematical Practice
1. Make sense of problems, and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments, and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for, and make use of, structure.
8. Look for, and express regularity in, repeated reasoning.
E.Q:
• 1. What are characteristics of
complementary, supplementary, adjacent,
linear and vertical angles?
• 2. How do we use the formulas for area
and perimeter of 2-D shapes to solve real
life situations?
Adjacent Angles
• Adjacent angles – two angles that lie in the same
plane, have a common vertex, and share a common
side, but NO common interior points.
C
A
C
B
A
B
D
ABC and CBD
D
Vertical Angles
• Vertical angles – two nonadjacent angles
formed by two intersecting lines.
B
A
E
C
D
AEBand CED
AEDand BEC
Linear Pair
• A linear pair – a pair of adjacent angles whose
non-common sides are opposite rays.
B
D
E
C
BEDand BEC
Identify Angle Pairs
• Name an angle pair that satisfies each condition.
a. Two obtuse vertical angles. VZXand YZW
b. Two acute adjacent angles. VZYand YZT
YZTand TZW
TZWand WZX
X
V
115°
Z
65°
65°
50° 65°
Y
T
W
Angle Relationships
• Complementary Angles
– Two angles whose measures have a sum of 90°.
A
B
E
50°
40°
1
2
D
C
1and 2
ABCand DEF
F
Angle Relationships
• Supplementary Angles
– Two angles whose measures have a sum of 180°.
A
C
D
80°
M
B
N
100°
M and N
ABDand CBD
Angle Measures
• Find the measure of two complementary angles if the
difference of the measures is 12.
x°
A
B
Angle Measures
• Find the measure of two complementary angles if the
difference of the measures is 12.
mB  mA  12
(90  x)  x  12
90  2x  12
2x  78
x  39
mA  39
mB  90  39
 51
Perpendicular Lines
• Perpendicular lines – lines that form right angles
– Intersect to form four right angles.
– Intersect to form congruent adjacent angles.
– Segments and rays can be perpendicular to lines or to the
other line segments and rays.
– The right angle symbol in the figure indicates that the lines
are perpendicular.
Y
•
• XZ
is read “is perpendicular to”.
WY
X
Z
W
Perpendicular Lines
• Find x and y so that BE and AD are perpendicular.
mBFD  mBFC  CFD
90  6 x  3x
B
90  9x
10  x
6x°
3x°
A
mAFE  12 y  10
90  12 y  10
100  12 y
25
 y
3
C
(12y – 10)°
F
E
D
1.7 Introduction to Perimeter,
Circumference, & Area
9/10/12
Rectangle
W
(width)
• Perimeter
P=2l +2w
l
(length)
• Area
A=lw
Square
• Perimeter
P=4s
S
(side)
• Area
A=s2
Example: Find the perim. & area of the figure.
• P=2l +2w
5 in
P=2(5in)+2(3in)
P=10in+6in
P=16in
3 in
• A=lw
A=(5in)(3in)
A=15in2
Ex: Find the perim. & area of the figure.
• P=4s
P=4(20m)
P=80m
20 m
• A=s2
A=(20m)2
A=400m2
Triangle
• Perimeter
P=a+b+c
a
c
h
(height)
b
• Area
A= ½ bh
(base)
Ex: Find the perim. & area of the figure.
• P=a+b+c
P=5ft+7ft+6ft
P=18 ft
5 ft
4 ft
7 ft
6 ft
• A= ½ bh
A= ½ (7ft)(4ft)
A= ½ (28ft2)
A= 14ft2
Circle
• Diameter
d=2r
• Circumference
C=2πr
• Area
A=πr2
diameter
** Always use the π button on the calculator;
even when the directions say to use 3.14.
Ex: Find the circumference & area of the
circle.
• C=2πr
C=2π(5in)
C=10π in
C≈31.4 in
• A=πr2
A=π(5in)2
A=25π in2
A≈78.5 in2
GROUP WORK
CHAMPs #2