mathematics

1. Angles
and Their
Measures

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 ∠BAC, ∠CAB, ∠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.
∀ ∠PQS or ∠SQP
∀ ∠SQR or ∠RQS
∀ ∠PQR or ∠RQP
Q
P
S
R
You should not name any of
these angles as ∠Q because
all three angles have Q as their
vertex. The name ∠Q would
not distinguish one angle from
the others.

4. Note:
• The measure of ∠A is denoted by m∠A.
The measure of an angle can be
approximated using a protractor, using
units called degrees(°). For instance,
∠BAC has a measure of 50°, which can
be written as
m∠BAC = 50°.
B
A
C

5. more . . .
• Angles that have the
same measure are
called congruent
angles. For instance,
∠BAC and ∠DEF
each have a measure
of 50°, so they are
congruent.
D
E
F
50°

6. Note – Geometry doesn’t use
equal signs like Algebra
MEASURES ARE EQUAL
m∠BAC = m∠DEF
ANGLES ARE CONGRUENT
∠BAC ≅ ∠DEF
“is equal to” “is congruent to”
Note that there is an m in front when you say
equal to; whereas the congruency symbol ≅ ;
you would say congruent to. (no m’s 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 ∠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 ∠RST, then
m∠RSP + m∠PST =
m∠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.

11. Solution:
You can use the Angle Addition Postulate.

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.
α. ∠LMN
β. ∠LMP
χ. ∠NMQ
δ. ∠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 ∠RSP and ∠PST so that
∠RST is (a) acute and (b) obtuse.

17. Ex. 4: Drawing Adjacent
Angles
• Use a protractor to draw two adjacent
acute angles ∠RSP and ∠PST so that
∠RST is (a) acute and (b) obtuse.

18. Ex. 4: Drawing Adjacent
Angles
• Use a protractor to draw two adjacent acute
angles ∠RSP and ∠PST so that ∠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.