1-4: Measure and Classify Angles

Solve each equation

1.) 5𝑥 + 6 + 2𝑥 − 14 = 90 3.) 𝑥 3+ 2 = 𝑥 − 5

2.) 4𝑥 − 30 + 𝑥 + 25 = 180

7𝑥 − 8 = 90

7𝑥 = 98

𝒙 = 𝟏𝟒

5𝑥 − 5 = 180

5𝑥 = 185

𝒙 = 𝟑𝟕

𝑥 3 = 𝑥 − 7

𝑥 = 3𝑥 − 21

−2𝑥 = −21

𝒙 = 𝟏𝟎. 𝟓

Objective: Students will be able to name, classify, and measure angles using postulates and a protractor.

Agenda

Vocabulary

Protractor Postulate

Measuring and Classifying Angles

Angle Addition Postulate

Finding Angle Measures

An Angle consists of two different rays with the same endpoint.

The rays that form the angle are known as the sides.

The endpoint that connects the rays is known as the vertex.

Sides

Vertex

AB

C The angle with sides 𝑨𝑩 and 𝑨𝑪 can

be named < 𝑩𝑨𝑪, < 𝑪𝑨𝑩, or < 𝑨

Name the three angles in the diagram.

XZ

W

Y

Solution:

< 𝑾𝑿𝒀

< 𝒀𝑿𝒁

< 𝑾𝑿𝒁

The Protractor Postulate:

The measure of < 𝑨𝑶𝑩 is equal to the absolute value of the difference between

the real numbers for 𝑶𝑨 and 𝑶𝑩.

Notation:

𝒎 < 𝑨𝑶𝑩

Example:

𝒎 < 𝑨𝑶𝑩 = 𝟏𝟖𝟎° − 𝟒𝟎°= 𝟏𝟒𝟎°

Angles can be classified into one of the four following names:

Acute: Less than

𝟗𝟎° in measure

(𝟎° < 𝒎 < 𝑨 < 𝟗𝟎°)

Right : Always 90° in measure

(𝒎 < 𝑨 = 𝟗𝟎°)

Obtuse: Greater than

90°, but less than 180°in measure

(𝟗𝟎° < 𝒎 < 𝑨 < 𝟏𝟖𝟎°)

Straight: Always 180° in

measure (𝒎 < 𝑨 = 𝟏𝟖𝟎°)

Use the diagram (pg. 24) to find the measure of the indicated angle. Then classify the angle.

a.) < 𝐾𝐻𝐽

b.) < 𝐺𝐻𝐾

c.) < 𝐺𝐻𝐽

d.) < 𝐺𝐻𝐿

𝒎 < 𝑲𝑯𝑱 = 𝟓𝟓° ;

Acute Angle

𝒎 < 𝑮𝑯𝑲 = 𝟏𝟐𝟓° ;

Obtuse Angle

𝒎 < 𝑮𝑯𝑱 = 𝟏𝟖𝟎° ;

Straight Angle

𝒎 < 𝑮𝑯𝑳 = 𝟗𝟎° ;

Right Angle

The Angle Addition Postulate:

If 𝑷 is the interior of < 𝑹𝑺𝑻, then 𝒎 < 𝑹𝑺𝑻 = 𝒎 < 𝑹𝑺𝑷 +𝒎 < 𝑷𝑺𝑻

Given that 𝒎 < 𝑳𝑲𝑵 = 𝟏𝟒𝟓°, find 𝒎 < 𝑳𝑲𝑴 and 𝒎 < 𝑴𝑲𝑵.

Solution:

Because M is in the interior of < 𝑳𝑲𝑵, we

can use the Angle Addition Postulate to

first solve for x:

𝒎 < 𝑳𝑲𝑴+𝒎 < 𝑴𝑲𝑵 = 𝒎 < 𝑳𝑲𝑵

𝟐𝒙 + 𝟏𝟎 + 𝟒𝒙 − 𝟑 = 𝟏𝟒𝟓

𝟔𝒙 + 𝟕 = 𝟏𝟒𝟓

𝟔𝒙 = 𝟏𝟑𝟖

𝒙 = 𝟐𝟑

Next, plug in 𝑥 = 23 for 𝒎 < 𝑳𝑲𝑴 and 𝒎 < 𝑴𝑲𝑵.

For 𝒎 < 𝑳𝑲𝑴:

𝒎 < 𝑳𝑲𝑴 = 𝟐𝒙 + 𝟏𝟎

= 𝟐 𝟐𝟑 + 𝟏𝟎

= 𝟓𝟔

For 𝒎 < 𝑴𝑲𝑵:

𝒎 < 𝑴𝑲𝑵 = 𝟒𝒙 − 𝟑

= 𝟒 𝟐𝟑 − 𝟑

= 𝟖𝟗

Thus, 𝒎 < 𝑳𝑲𝑴 = 𝟓𝟔° and 𝒎 < 𝑴𝑲𝑵 = 𝟖𝟗°

Congruent Angles: Angles that have the same measure

Symbol: ≅ means “is Congruent to”

Identify the congruent angles in the diagram on page 27 in your textbook.

There are two pairs of congruent angles:

< 𝑫𝑬𝑭 ≅< 𝑱𝑲𝑳

And

< 𝑫𝑬𝑮 ≅< 𝑳𝑲𝑮

Angle bisector: A ray that divides the angle into two angles that are congruent.

B

C

A

D

In the diagram, 𝑩𝑫 bisects < 𝑨𝑩𝑪.

So, 𝒎 < 𝑨𝑩𝑫 = 𝒎 < 𝑫𝑩𝑪

And < 𝑨𝑩𝑫 ≅< 𝑫𝑩𝑪

In the diagram at the right, 𝑌𝑊 bisects < 𝑋𝑌𝑍, and 𝑚 < 𝑋𝑌𝑊 = 18°, Find 𝑚 < 𝑋𝑌𝑍

YZ

X

W

Solution:

Because 𝑌𝑊 bisect < 𝑋𝑌𝑍, then

𝑚 < 𝑋𝑌𝑊 = 𝑚 < 𝑊𝑌𝑍 = 18°

And with the Angle Addition Postulate

𝒎 < 𝑿𝒀𝑾+𝒎 < 𝑾𝒀𝒁 = 𝒎 < 𝑿𝒀𝒁

𝟏𝟖 + 𝟏𝟖 = 𝒎 < 𝑿𝒀𝒁

𝒎 < 𝑿𝒀𝒁 = 𝟑𝟔°

Pg. 28-30

#’s 3, 6, 11-14, 15-20, 23, 25, 26, 29-31, 33-38, 40, 41

44-47 (EC)