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Angle Pair Relationships Learner Objective: As a result in learning, students should be able to classify triangles by side lengths and angles and determine the measurement of angles given measurement of another angle. CCES: 7.G.2, 7.G.5, MP.4, MP.5, MP6. Think about it….

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Angle Pair RelationshipsLearner Objective:As a result in learning, students should be able to classify triangles by side lengths and anglesand determine the measurement of angles given measurement of another angle.CCES: 7.G.2, 7.G.5, MP.4, MP.5, MP6

• If a triangle has all equal sides and has a perimeter of 42cm, what is the length of each side?

• If a triangle has 2 equal side lengths and has a perimeter of 94 in, what are possible side lengths for the triangle?

• Draw a quadrilateral with one set of parallel sides and no right angles

• Can a triangle have more than one obtuse angle? Explain your reasoning

How are special angle pairs identified?

Y

XY and XZ are ____________.

X

Straight Angles

Opposite rays

___________ are two rays that are part of a the same line and have only their endpoints in common.

opposite rays

The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees.

straight angle

vertex

T

Angles – sides and vertex

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

angle

This figure is called an _____.

Some parts of angles have special names.

side

vertex

The common endpoint is called the ______,

and the two rays that make up the sides ofthe angle are called the sides of the angle.

side

R

vertex

SRT

R

TRS

1

T

Naming Angles

There are several ways to name this angle.

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

or

side

The vertex letter is always in the middle.

2) Use the vertex only.

1

R

side

If there is only one angle at a vertex, then theangle can be named with that vertex.

3) Use a number.

2

F

DEF

2

E

FED

E

Angles

Symbols:

C

A

1

B

ABC

1

B

CBA

BA and

BC

1) Name the angle in four ways.

2) Identify the vertex and sides of this angle.

vertex:

Point B

sides:

3) Is there an angle that can be named ?

1

XWZ

YWX

XWY or

1

2

W

Angles

1) Name all angles having W as their vertex.

X

W

1

2

Y

Z

No!

A

A

obtuse angle 90 < m A < 180

acute angle 0 < m A < 90

right angle m A = 90

Angle Measure

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.

40°

110°

90°

50°

75°

130°

Angle Measure

Classify each angle as acute, obtuse, or right.

Acute

Obtuse

Right

Obtuse

Acute

Acute

B

D

C

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

The two angles are called

_____________

2

1

1 and 2 are examples of adjacent angles. They share a common ray.

Name the ray that 1 and 2 have in common. ____

J

2

common side

R

M

1

with the same vertex R and

N

A) share a common side

B) have the same vertex, and

C) have no interior points in common

B

2

1

1

2

G

N

L

1

J

2

Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but

_____________

no common side

Yes. They have the same vertex G and a common side with no interior points in common.

No. They do not have a common vertex or ____________

a common side

The side of 1 is ____

The side of 2 is ____

E

D

A

60°

30°

F

B

C

Two angles are complementary if and only if (iff)

The sum of their degree measure is 90.

mABC + mDEF = 30 + 60 = 90

D

A

60°

30°

F

B

C

Complementary and Supplementary Angles

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.

I

75°

15°

H

P

Q

40°

50°

H

S

U

V

60°

T

30°

Z

W

Some examples of complementary angles are shown below.

mH + mI = 90

mPHQ + mQHS = 90

mTZU + mVZW = 90

D

C

130°

50°

E

B

F

A

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.

mABC + mDEF = 50 + 130 = 180

I

75°

105°

H

Q

130°

50°

H

S

P

U

V

60°

120°

60°

Z

W

T

Some examples of supplementary angles are shown below.

mH + mI = 180

mPHQ + mQHS = 180

mTZU + mUZV = 180

and

mTZU + mVZW = 180

measure

Recall that congruent segments have the same ________.

Congruent angles

_______________ also have the same measure.

50°

50°

B

V

Congruent Angles

Two angles are congruent iff, they have the same

______________.

degree measure

B  V iff

mB = mV

2

X

Z

Congruent Angles

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

arcs

To show that there is a second set of congruent angles, X and Z, we use double arcs.

This “arc” notation states that:

X  Z

mX = mZ

When two lines intersect, ____ angles are formed.

four

There are two pair of nonadjacent angles.

vertical angles

These pairs are called _____________.

1

4

2

3

Two angles are vertical iff they are two

nonadjacent angles formed by a pair of

intersecting lines.

Vertical angles:

1 and 3

1

4

2

2 and 4

3

Vertical angles are congruent.

n

m

2

1  3

3

1

2  4

4

130°

Vertical Angles

Find the value of x in the figure:

The angles are vertical angles.

So, the value of x is 130°.

Find the value of x in the figure:

The angles are vertical angles.

(x – 10) = 125.

(x – 10)°

x – 10 = 125.

125°

x = 135.

52°

A

B

Congruent Angles

Suppose A  B and mA = 52.

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

1

B + 1 = 180

1 = 180 – B

1 = 180 – 52

1 = 128°

D

1

2

A

C

4

B

3

E

H

Congruent Angles

1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3

x = 17; 3 = 37°

2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC

x = 29; EBC = 121°

3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4

x = 16; 4 = 39°

4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1

x = 18; 1 = 43°