Angle RelationshipsDay 1

Angles

•An angle consists of two different rays (sides) that share a common endpoint (vertex).B

A

C

Sides

Vertex

Naming AnglesThere are several ways to name an 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

3) Use a number.

1

R

S

T

vertex

side

side

1

This only works if there is only one angle at a vertex.

Naming 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:

You Try

W

Y

X

1) Name all angles having W as their vertex.

1

2

Z

1

22) What are other names for ? 1

XWY or YWX

3) Is there an angle that can be named ?

W

No!

XWZ ZWX

ZWY YWZ

XWY YWX

Vocabulary

congruentvertical anglesadjacent anglescomplementary anglessupplementary 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

This “arc” notation says that:

Congruent Angles

When two angles are congruent they have the SAME measure.

This “arc” notation says that:

1 2

When 2 lines intersect, they make vertical angles.

2

4

31

Vertical angles are opposite one another and are

congruent.

2

4

31

1 3

2 4

Find the value of x in the figure:

The angles are vertical angles.

So, the value of x is 130°.130°

x°

Example:

Find the value of x in the figure:

The angles are vertical angles.

(x – 10) = 125(x – 10)°

125°x – 10 = 125

x = 135°

You Try:

+ 10 +10

Adjacent Angles Adjacent angles are angles that:

M

J

N

R1

2 1 and 2 are adjacent with the same vertex R and a common side

RM

A) share a common side, and

B) have the same vertex

Adjacent angles are “side-by-side”

ExamplesDetermine whether 1 and 2 are adjacent angles.

No. They have no common side. 1 2

B

12

GYes. They are “side-by-side”.

N

1

2J

LNo. They do not have a common vertex or a common side.

The side of 1 isLN

JNThe side of 2 is

Complementary Angles

Complementary angles are two angles that form a right angle and whose measures have a sum of 90 degrees.

Complementary angles can be adjacent or nonadjacent 200

700

Remember: The box in the corner means it’s a right angle.

Examples

65º

25º

These are examples of complementary angles.

60º30º

65° + 25° = 90° 30° + 60° = 90°

xH

75°I

The angles below are complementary angles. Find the missing angle measure.

mH + mI = 90°x + 75 = 90 -75 -75

x = 15°

mPHQ + mQHS = 90°x + 50 = 90 -50 -50

x = 40°

50°H

xQ

P

S

Examples

Supplementary Angles

Supplementary angles are two angles that form a straight line and whose measures have a sum of 180 degrees.

Supplementary angles can be adjacent or nonadjacent.

520 1280

ExamplesThese are examples of supplementary

angles.

120° + 60° = 180°

135° + 45° = 180°

60º120º 45º135º

xH

75° I mH + mI = 180°x + 75 = 180 -75 -75x = 105°

mPHQ + mQHS = 180°130 + x = 180-130 -130

x = 50°

x

H

130°

Q

P S

ExamplesThe angles below are supplementary angles.

Find the missing angle measure.

Find each unknown angle measure.

x + y + 80° = 180°–80° –80°

x + y = 100°

The sum of the measures is 180°.

x and y are congruent.

x = 50° and y = 50° Each angle measures half of 100°.

x 80°

K

ML

NJ

y

Example:

Find each unknown angle measure.

x + y + 50° = 180°–50° –50°

x + y = 130°

The sum of the measures is 180°.

ABC and DBE are congruent.

x = 65° and y = 65° Each angle measures half of 130°.

x50°

B

DC

EA

y

You Try:

You Try:

20°

C

J

D

EF

G

H

= 70°

= 90°

70°=

= 20°

90°=Box in the corner indicates a right angle.

Find the missing angle measures given that m<u = 20°.

v

wx

y

z

<u and <v are complementary angles,

sov + 20 = 90 -20 -20

v = 70°

<w and <ECG are supplementary angles, so

w + 90 = 180 -90 -90

w = 90°

<v and <y are vertical angles

so y = 70°

<u and <x are vertical angles

so x = 20°

u

<w and <z are vertical angles

so z = 90°

Practice:Angle Relationships Packet