Download PH_Geo_1-6_Measuring_Angles[1].

Sec. 1 – 6
Measuring Angles
Objectives:
1) Find the measures of
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 number 1
Or by 3 Points BAC
Vertex has to be in the middle
1
A
C
Ex.5: 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°
Protractor Postulate
P (1 – 7) 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
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
Ex.6: Finding  measures
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
Name the angle below in four ways.
The name can be the number between the sides of the angle:
The name can be the vertex of the angle:
3.
G.
Finally, the name can be a point on one side, the vertex, and a point
on the other side of the angle: AGC, CGA.
1-4
Suppose that m
1 = 42 and m
ABC = 88. Find m
2.
Use the Angle Addition Postulate to solve.
m
ABC Angle Addition Postulate.
1+m
2=m
42 + m
2 = 88
Substitute 42 for m
m
2 = 46
Subtract 42 from each side.
1-4
1 and 88 for m
ABC.
Name all pairs of angles in the diagram that are:
a. vertical
Vertical angles are two angles whose sides are opposite
rays. Because all the angles shown are formed by two
intersecting lines, 1 and 3 are vertical angles, and 2
and 4 are vertical angles.
b. supplementary
Two angles are supplementary if the sum of their measures is 180.
A straight angle has measure 180, and each pair of adjacent
angles in the diagram forms a straight angle. So these pairs of
angles are supplementary: 1 and 2, 2 and 3, 3 and 4,
and 4 and 1.
(continued)
c. complementary
Two angles are complementary if the sum of their
measures is 90. No pair of angles is
complementary.
1-6
Use the diagram below. Which of the
following can you conclude: 3 is a right angle, 1 and 5
are adjacent,
3 5?
You can conclude that 1 and 5 are adjacent
because they share a common side, a common
vertex, and no common interior points.
Although 3 appears to be a right angle, it is not marked
with a right angle symbol, so you cannot conclude that 3
is a right angle.
3 and 5 are not marked as congruent on the diagram. Although
they are opposite each other, they are not vertical angles. So you
cannot conclude that 3 5.