1.4 Angles and Their Measures

GeometryMrs. SpitzFall 2005

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

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.

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.