Ch. 1-3: Measuring Angles

SOL: G4Objectives:• Measure and classify angles.• Identify special angle pairs.• Use the special angle pairs to find angle measures.

Angle formed by two noncollinear rays that

have a common endpoint.

A

BNoncollinear

rays

Common

EndpointC

A

Sides of an Angle The rays that make the angle

Side AB

Side AC

A

B

C

A

Vertex of an Angle The common endpoint

Vertex A

Symbols When we name an angle, the vertex

point is always in the middle

Note the vertex is the middle point listed

The following are the different ways we can name this angle:

A, BAC, CAB, 4

A

B

C

4

Interior of an Angle Any point inside the angle

Interior

Exterior of an Angle Any point outside the angle

Exterior

Exterior

Equal vs. Congruent

Just like with segments, there is only one way to measure an angle.

To measure an angle, we find how many degrees there are from one side to the other. (360° in a circle)

∠A ≅ ∠C m∠A = m∠C = 70°

A

B

C70° 70°

40°

Example 1:Use the diagram to answer a, b, and c.

a.) Name all angles that have B as a vertex.

b.) Name the sides of 5.

c.) Write another name for 6.

∡ABG, ∡ABD, ∡DBE or ∡DBF,

∡EBG or ∡FBG, ∡5, ∡6, ∡7

BG and BE or BF

∡DBE or ∡DBF

Measuring AnglesTo measure an angle, you use a protractor.

The protractor has two scalesrunning from 0 to 180 degrees

in opposite directions. These are the scales we use to determine the

measure of the angle

Place the center point of theprotractor on the vertex

Align the 0 on either sideof the scale with one side

of the angle. (Paying attention to which

direction the angle is opening

Example 2:Find the measure of PQR.

P Q

R

Since QP is aligned with the 0on the outer scale, use the outer scale to find that QR intersects

the scale at 65 degrees.

65°

Classify Angles by Angle Measure

Measures 90 Written as mA = 90

Right Angle

Acute Angle

This symbol means, right

angle, perpendicular

A B

Measures less than 90

Written as mB < 90

Obtuse Angle Straight Angle (Line)

Measures greater than 90

Written as mC > 90

C

A B C

Classify Angles by Angle Measure

Measures 180

Example 3:Measure the angle and classify it.

12°, Acute

Example 4:Measure the angle and classify it.

99°, Obtuse

Example 5:Measure the angle and classify it.

70°, Acute

Congruent Angles Angles that have the same measure

Symbols: NMP QMR

Angle Bisector

An angle bisector is a segment, line, or ray that splits an angle into two congruent angles.

In the picture, rayBD bisects ∠ABC. Therefore, we know∠ABD ≅ ∠DBC.

Postulate 1.8: Angle Addition Postulate If point B is in the interior of ∠AOC , then

m∠AOB + m∠BOC = m∠AOC

Example 6:Apply the angle addition postulate.

A

D

C

B

23°

41°

What is the m∡ABC?

E

H

G

F

If m∠EFG = 23°, what is the m∠EFH?

11°If m∠KJL = 117°,

what is the m∠KJM?

K

M

L

J

68°

m∡ABD + m∡CBD = m∡ABC23° + 41° =

64°

64°

12°m∡EFG - m∡GFH =

m∡EFH23° - 11° = 12°

49°m∡KJL - m∡LJM = m∡KJM

117° - 68° = 49°

Example 7: If m∠RQT = 155°, what are the m∠RQS and m∠SQT?

m∡RQS + m∡TQS = m∡RQT

(4x – 20) + (3x + 14) = 155°

7x – 6= 155°+ 6 + 6

7x = 161°

7x = 161°7 7x = 23

m∡RQS = 4x – 20 = 4(23) – 20 = 72°m∡TQS = 3x + 14 = 3(23) + 14 = 83°

Check:72° + 83° =

155°

Example 8: ∠DEF is a straight angle. What are the m∠DEC and m∠CEF?

m∡DEC + m∡FEC = m∡DEF

(11x – 12) + (2x + 10) = 180° 13x – 2 =

180°+ 2 + 2

13x = 182°

13x = 182°13 13x = 14°

m∡DEC = 11x – 12 = 11(14) – 12 = 142°m∡FEC = 2x + 10 = 2(14) + 10 = 38°

Check:142° + 38° =

180°