Transcript Sec. 1 – 4 Measuring Segments & Angles

Measuring Angles
Geometry vs Algebra
Segments are Congruent
– Symbol [  ]
– AB  CD
– 1  2
Lengths of segments are equal.
– Symbol [ = ]
– AB = CD
– m1 = m2
Angles
Formed by 2 rays with the same endpoint
– Vertex of the Angle
B
Symbol: [  ]
Name it by:
– Its Vertex A
A
– A number 1
– Or by 3 Points BAC
- Vertex has to be in the middle
1
C
How many s can you find? Name them.
B
A
1 2
3 s
D
– ADB or BDA
– BDC or CDB
– ADC or CDA
Notice D (the vertex) is always in the middle.
Can’t use D
But 1 or 2 could be added.
C
Classifying Angles by their Measures
Acute 
x°
x < 90°
Straight 
Right 
Obtuse 
x°
x°
x = 90°
x > 90°
x°
x = 180°
Postulate 1-7
Protractor Postulate
Let OA & OB be opposite rays in a plane, &
all the rays with endpoint O that can be
drawn on one side of AB can be paired with
the real number from 0 to 180.
C
A
D
O
B
Postulate 1-8
Angle Addition Postulate
If point B is in the interior of MAD, then
mMAB + mBAD = mMAD
B
M
D
A
If MAD is a straight , then
mMAB + mBAD = mMAD = 180°
B
D
M
A
Finding  measures (m )
Find mTSW if
– mRSW = 130°
– mRST = 100°
W
T
R
mRST + mTSW = mRSW
100 + mTSW = 130
mTSW = 30°
S
 Addition
mXYZ = 150
x
1 = 3x - 15
2 = 2x - 10
m1 + m2 = mXYZ
(3x - 15) + (2x – 10) = 150
5x – 25 = 150
5x = 175
x = 35
Y
Z
Adjacent Angles
Adjacent angles – two coplanar angles
with a common side, a common vertex,
and no common interior points.
1and 2
3and 4
Vertical Angles
Vertical angles – two angles whose sides
are opposite rays.
1and 2
3and 4
Complementary Angles
Complementary angles – two angles
whose measures have a sum of 90°.
– Each angle is called the complement of the
other.
1and 2
Aand B
Supplementary Angles
Supplementary angles – two angles
whose measures have a sum of 180°.
– Each angle is called the supplement of the
other.
3and 4
Band C
Identifying Angle Pairs
Is the statement true or false?
a. BFDand CFD are adjacent angles.
b. AFBand EFD are vertical angles.
c. AFEand BFC are complementary.
F
F
T
Perpendicular Lines
Perpendicular lines – intersecting lines
that form right angles
Linear Pairs
A linear pair is a pair of adjacent angles whose
noncommon sides are opposite rays.
– The angles of a linear pair form a straight angle.
Finding Missing Angle Measures
KPLand JPL are a linear pair.
mKPL  2 x  24, andmJPL  4 x  36.
What are the measures of
KPLand JPL ?
Finding Missing Angle Measures
mKPL  mJPL  180
(2 x  24)  (4 x  36)  180
6 x  60  180
6 x  120
x  20
mKPL  2x  24  2(20)  24  40  24  64
mJPL  4 x  36  4(20)  36  80  36  116
Angle Bisector
An angle bisector is a ray that divides an angle
into two congruent angles.
– Its endpoint is at the angle vertex.
– Within the ray, a segment with the same endpoint is
also an angle bisector.
The ray or segment bisects the angle.
Using an Angle Bisector to Find Angle
Measures
AC bisectsDAB . If mDAB  58 ,
what is
mDAC ?
mCAB  mDAC
 58
mDAB  mCAB  mDAC
 58  58
 116