1.4 Angles 1.4 Angles and Their and Their MeasuresMeasures

Goal 1: Using Angle PostulatesGoal 1: 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

Example 1: Naming AnglesExample 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.

NoteNote

• 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 . . . 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°

Reminder: Geometry doesn’t use Reminder: Geometry doesn’t use equal signs like Algebraequal 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 PostulatePostulate 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

A

D

E

Interior/ExteriorInterior/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.

Postulate 4: Angle Addition Postulate 4: Angle Addition PostulatePostulate

• If P is in the interior of RST, then

mRSP + mPST = mRST

R

S

T

P

Example 2: Calculating Angle Example 2: Calculating Angle MeasuresMeasures

• 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.

SolutionSolution

You can use the Angle Addition Postulate.

Goal 2: Classifying AnglesGoal 2: 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°.

Example 3: Classifying Angles in Example 3: Classifying Angles in a Coordinate Planea 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

SolutionSolution

• Begin by plotting the points. Then use a protractor to measure each angle.

SolutionSolution

• 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.

Example 4: Drawing Adjacent Example 4: Drawing Adjacent AnglesAngles

• Use a protractor to draw two adjacent acute angles RSP and PST so that RST is (a) acute and (b) obtuse.

Example 4: Drawing Adjacent Example 4: Drawing Adjacent AnglesAngles

• Use a protractor to draw two adjacent acute angles RSP and PST so that RST is (a) acute and (b) obtuse.

Example 4: Drawing Adjacent Example 4: Drawing Adjacent AnglesAngles

• Use a protractor to draw two adjacent acute angles RSP and PST so that RST is (a) acute and (b) obtuse.

Solution:

Closure QuestionClosure 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.