Angle Pair Relationships

Angle Pair Relationship Essential Questions

How are special angle pairs identified?

Straight Angles

___________ are two rays that are part of a the same line and have

only their endpoints in common.

Opposite rays

XY Z

XY and XZ are ____________.opposite rays

The figure formed by opposite rays is also referred to as a

____________. A straight angle measures 180 degrees.straight angle

Angles – sides and vertex

There is another case where two rays can have a common endpoint.

R

S

T

This figure is called an _____.angle

Some parts of angles have special names.

The common endpoint is called the ______,vertex

vertex

and the two rays that make up the sides of

the angle are called the sides of the angle.

side

Naming Angles

R

S

T

vertex

side

There are several ways to name this angle.

1) Use the vertex and a point from each side.

SRT or TRS

The vertex letter is always in the middle.

2) Use the vertex only.

R

If there is only one angle at a vertex, then the

angle can be named with that vertex.

3) Use a number.

1

1

Angles

Definition

of Angle

An angle is a figure formed by two noncollinear rays that

have a common endpoint.

E

D

F

2

Symbols: DEF

2

E

FED

Angles

B

A

1

C

1) Name the angle in four ways.

ABC

1

B

CBA

2) Identify the vertex and sides of this angle.

Point B

BA and BC

vertex:

sides:

Angles

W

Y

X1) Name all angles having W as their vertex.

1

2

Z

1

2

2) What are other names for ?1

XWY or YWX

3) Is there an angle that can be named ? W

No!

XWZ

Once the measure of an angle is known, the angle can be classified

as one of three types of angles. These types are defined in relation

to a right angle.

Types of Angles

A

right angle

m A = 90

acute angle

0 < m A < 90

A

obtuse angle

90 < m A < 180

A

Angle Measure

Classify each angle as acute, obtuse, or right.

110°

90°40°

50°

130° 75°

Obtuse

Obtuse

Acute

Acute Acute

Right

Angle Measure

Adjacent Angles

When you “split” an angle, you create two angles.

A

C

B1

2

The two angles are called

_____________adjacent angles

1 and 2 are examples of adjacent angles.

They share a common ray.

Name the ray that 1 and 2 have in common. ____BD

adjacent = next to, joining.

Adjacent Angles

Definition of

Adjacent

Angles

Adjacent angles are angles that:

J

N

R1

21 and 2 are adjacent

with the same vertex R and

common side RM

A) share a common side

B) have the same vertex, and

C) have no interior points in common

Adjacent Angles

Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but

_____________no common side1 2

B

12

G

Yes. They have the same vertex G and a

common side with no interior points in

common.

N

1

2J

L

No. They do not have a common vertex or

____________a common side

The side of 1 is ____LN

JNThe side of 2 is ____

Adjacent Angles and Linear Pairs of Angles

Determine whether 1 and 2 are adjacent angles.

No.

21

Yes.

1 2

X D Z

In this example, the noncommon sides of the adjacent angles form a

___________.straight line

These angles are called a _________linear pair

Linear Pairs of Angles

Definition of

Linear Pairs

Two angles form a linear pair if and only if (iff):

1 and 2 are a linear pair.

A) they are adjacent and

B) their noncommon sides are opposite rays

A DB

1 2

AD form and BDBA

180 2 1

Linear Pairs of Angles

In the figure, and are opposite rays.CM CE

1

2

M

43 E

H

T

A

C

1) Name the angle that forms a

linear pair with 1.

ACE

ACE and 1 have a common side

the same vertex C, and opposite rays

and

CA

CM CE

2) Do 3 and TCM form a linear pair? Justify your answer.

No. Their noncommon sides are not opposite rays.

Definition of

Complementary

Angles

30°

A

BC

60°D

E

F

Two angles are complementary if and only if (iff) The sum of their degree measure is 90.

mABC + mDEF = 30 + 60 = 90

Complementary and Supplementary Angles

30°

A

BC

60°D

E

F

If two angles are complementary, each angle is a

complement of the other.

ABC is the complement of DEF and DEF is the

complement of ABC.

Complementary angles DO NOT need to have a common side

or even the same vertex.

Complementary and Supplementary Angles

15°H

75° I

Some examples of complementary angles are shown below.

mH + mI = 90

mPHQ + mQHS = 9050°

H

40°Q

P

S

30°

60°T

UV

WZ

mTZU + mVZW = 90

Complementary and Supplementary Angles

Definition of

Supplementary

Angles

If the sum of the measure of two angles is 180, they form a

special pair of angles called supplementary angles.

Two angles are supplementary if and only if (iff) the sum of their degree measure is 180.

50°

AB

C

130°

D

E F

mABC + mDEF = 50 + 130 = 180

Complementary and Supplementary Angles

105°H

75° I

Some examples of supplementary angles are shown below.

mH + mI = 180

mPHQ + mQHS = 18050°H

130°

Q

P S

mTZU + mUZV = 180

60°120°

T

UV

W

Z

60° and

mTZU + mVZW = 180

Complementary and Supplementary Angles

Congruent Angles

Recall that congruent segments have the same ________.measure

_______________ also have the same measure.Congruent angles

Definition of

Congruent

Angles

Two angles are congruent iff, they have the same

______________.degree measure

50°B

50°

V

B V iff

mB = mV

Congruent Angles

1 2

To show that 1 is congruent to 2, we use ____.arcs

ZX

To show that there is a second set of congruent angles, X and Z,

we use double arcs.

X Z

mX = mZ

This “arc” notation states that:

Congruent Angles

When two lines intersect, ____ angles are formed.four

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34

There are two pair of nonadjacent angles.

These pairs are called _____________.vertical angles

Vertical Angles

Definition of

Vertical

Angles

Two angles are vertical iff they are two

nonadjacent angles formed by a pair of

intersecting lines.

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34

Vertical angles:

1 and 3

2 and 4

Vertical Angles

Theorem 3-1

Vertical Angle

Theorem

Vertical angles are congruent.

1

4

3

2mn

1 3

2 4

Vertical Angles

Find the value of x in the figure:

The angles are vertical angles.

So, the value of x is 130°.130°

x°

Vertical Angles

Find the value of x in the figure:

The angles are vertical angles.

(x – 10) = 125.(x – 10)°

125°

x – 10 = 125.

x = 135.

Vertical Angles

Suppose A B and mA = 52.

Find the measure of an angle that is supplementary to B.

A

52°

B

52° 1

B + 1 = 180

1 = 180 – B

1 = 180 – 52

1 = 128°

Congruent Angles

1) If m1 = 2x + 3 and the

m3 = 3x + 2, then find the

m3

2) If mABD = 4x + 5 and the mDBC = 2x +

1, then find the mEBC

3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4

4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1

x = 17; 3 = 37°

x = 29; EBC = 121°

x = 16; 4 = 39°

x = 18; 1 = 43°

A B C

D

E

G

H

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Congruent Angles