Transcript Angles-and-Their-Measures

Angles and
Their
Measures
Standard/Objectives:
Standard: Students will understand
geometric concepts and applications
Benchmark: Use visualization, spatial
reasoning, and geometric modeling to
solve problems.
Standard/Objectives:
Performance Standard: Solve problems
involving complementary, supplementary
and congruent angles.
Objectives:
• Use angle postulates
• Classify angles as acute, right, obtuse, or
straight.
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. vertex
The point A is the vertex
of the angle.
C
sides
B
A
Ex.1: Naming Angles
• Name the angles in
the figure:
S
SOLUTION:
There are three
Q
different angles.
R
• PQS or SQP
You should not name any of
• SQR or RQS
these angles as Q because
• PQR or RQP
all three angles have Q as their
P
vertex. The name Q would
not distinguish one angle from
the others.
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
B
mBAC = 50°.
A
C
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
50°
congruent.
E
D
F
Note – Geometry doesn’t use
equal signs like Algebra
MEASURES ARE EQUAL
ANGLES ARE CONGRUENT
mBAC = mDEF
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).
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 1180.
• The measure of AOB is
equal to the absolute
value of the difference
between the real
numbers for OA and OB.
A
O
B
Postulate 4: Angle Addition
Postulate
• If P is in the interior
of RST, then
mRSP + mPST =
mRST
R
P
S
T
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.
Solution:
You can use the Angle Addition Postulate.
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°.
Ex. 3: Classifying Angles in a
Coordinate Plane
•
a.
b.
c.
d.
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.
LMN
LMP
NMQ
LMQ
Solution:
• Begin by plotting the points. Then use a
protractor to measure each angle.
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.
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.
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.
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:
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.