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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. The point A is the vertex of the angle.
sides
vertex
C
AB
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.
Note:
• The measure of A is denoted by mA. 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 asmBAC = 50°.
B
AC
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
EF
50°
Note – Geometry doesn’t use equal signs like Algebra
MEASURES ARE EQUAL
mBAC = mDEF
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).
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
Postulate 4: Angle Addition Postulate
• If P is in the interior of RST, then
mRSP + mPST = mRST
R
S
T
P
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
• 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.
a. LMN
b. LMP
c. NMQ
d. 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.