2. Using Angle Postulates
⢠An angle consists of two
different rays that have
the same initial point.
The rays are the sides of
the angle. The initial
point is the vertex of the
angle.
⢠The angle that has sides
AB and AC is denoted
by â BAC, â CAB, â A.
The point A is the vertex
of the angle.
sides
vertex
C
A
B
3. Ex.1: Naming Angles
⢠Name the angles in
the figure:
SOLUTION:
There are three
different angles.
â â PQS or â SQP
â â SQR or â RQS
â â PQR or â RQP
Q
P
S
R
You should not name any of
these angles as â Q because
all three angles have Q as their
vertex. The name â Q would
not distinguish one angle from
the others.
4. Note:
⢠The measure of â A is denoted by mâ A.
The measure of an angle can be
approximated using a protractor, using
units called degrees(°). For instance,
â BAC has a measure of 50°, which can
be written as
mâ BAC = 50°.
B
A
C
5. more . . .
⢠Angles that have the
same measure are
called congruent
angles. For instance,
â BAC and â DEF
each have a measure
of 50°, so they are
congruent.
D
E
F
50°
6. Note â Geometry doesnât use
equal signs like Algebra
MEASURES ARE EQUAL
mâ BAC = mâ DEF
ANGLES ARE CONGRUENT
â BAC â â DEF
âis equal toâ âis congruent toâ
Note that there is an m in front when you say
equal to; whereas the congruency symbol â ;
you would say congruent to. (no mâs in front of
the angle symbols).
7. Postulate 3: Protractor
Postulate
⢠Consider a point A on
one side of OB. The rays
of the form OA can be
matched one to one with
the real numbers from 1-
180.
⢠The measure of â AOB is
equal to the absolute
value of the difference
between the real
numbers for OA and OB.
A
O B
8. A
D
E
Interior/Exterior
⢠A point is in the
interior of an angle if
it is between points
that lie on each side
of the angle.
⢠A point is in the
exterior of an angle if
it is not on the angle
or in its interior.
9. Postulate 4: Angle Addition
Postulate
⢠If P is in the interior
of â RST, then
mâ RSP + mâ PST =
mâ RST
R
S
T
P
10. Ex. 2: Calculating Angle
Measures
⢠VISION. Each eye of
a horse wearing
blinkers has an angle
of vision that
measures 100°. The
angle of vision that is
seen by both eyes
measures 60°.
⢠Find the angle of
vision seen by the
left eye alone.
12. Classifying Angles
⢠Angles are classified as acute, right, obtuse,
and straight, according to their measures.
Angles have measures greater than 0° and less
than or equal to 180°.
13. Ex. 3: Classifying Angles in a
Coordinate Plane
⢠Plot the points L(-4,2), M(-1,-1), N(2,2),
Q(4,-1), and P(2,-4). Then measure and
classify the following angles as acute,
right, obtuse, or straight.
Îą. â LMN
β. â LMP
Ď. â NMQ
δ. â LMQ
14. Solution:
⢠Begin by plotting the points. Then use a
protractor to measure each angle.
15. Solution:
⢠Begin by plotting the points. Then use a
protractor to measure each angle.
Two angles are adjacent angles if they share a common vertex
and side, but have no common interior points.
16. Ex. 4: Drawing Adjacent
Angles
⢠Use a protractor to draw two adjacent
acute angles â RSP and â PST so that
â RST is (a) acute and (b) obtuse.
17. Ex. 4: Drawing Adjacent
Angles
⢠Use a protractor to draw two adjacent
acute angles â RSP and â PST so that
â RST is (a) acute and (b) obtuse.
18. Ex. 4: Drawing Adjacent
Angles
⢠Use a protractor to draw two adjacent acute
angles â RSP and â PST so that â RST is (a)
acute and (b) obtuse.
Solution:
19. Closure Question:
⢠Describe how angles are classified.
Angles are classified according to their
measure. Those measuring less than
90° are acute. Those measuring 90° are
right. Those measuring between 90°
and 180° are obtuse, and those
measuring exactly 180° are straight
angles.