1.7 angles and perpendicular lines

AnglesAngles

§ 1.6 Angles

§ 1.6 Adjacent Angles and Linear Pairs of Angles

§ 1.6 The Angle Addition Postulate

§ 1.6 Angle Measure

§ 1.6 Congruent Angles

§ 1.6 Complementary and Supplementary Angles

§ 1.6 Perpendicular Lines

Angles

You will learn to name and identify parts of an angle.

1) Opposite Rays2) Straight Angle3) Angle4) Vertex5) Sides6) Interior7) Exterior

Angles

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

XY Z

XY and XZ are ____________.opposite rays

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

Angles

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 ofthe angle are called the sides of the angle.

side

side

Angles

R

S

T

vertex

side

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 theangle can be named with that vertex.

3) Use a number.

1

1

Angles

Definitionof 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

Angles

An angle separates a plane into three parts:

1) the ______

2) the ______

3) the _________

interior

exterior

angle itself

exterior

interior

W

Y

Z

A

B

In the figure shown, point B and all other points in the blue region are in the interiorof the angle.

Point A and all other points in the greenregion are in the exterior of the angle.

Points Y, W, and Z are on the angle.

Angles

B

Is point B in the interior of the angle, exterior of the angle, or on the angle?

Exterior

G

Is point G in the interior of the angle, exterior of the angle, or on the angle?

On the angle

Is point P in the interior of the angle, exterior of the angle, or on the angle?

Interior

P

§1.6 Angle Measure

You will learn to measure, draw, and classify angles.

1) Degrees2) Protractor3) Right Angle4) Acute Angle5) Obtuse Angle

In geometry, angles are measured in units called _______.degrees

The symbol for degree is °.

Q

P

R

75°

In the figure to the right, the angle is 75 degrees.

In notation, there is no degree symbol with 75because the measure of an angle is a real number with no unit of measure.

m PQR = 75

§1.6 Angle Measure

Postulate 3-1Angles

Measure Postulate

For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle.

B

A

C

n°

0 180

m ABC = nand 0 < n < 180

§1.6 Angle Measure

You can use a _________ to measure angles and sketch angles of givenmeasure.

protractor

Q

R S

Use a protractor to measure SRQ.

1) Place the center point of the protractor on vertex R. Align the straightedge with side RS.

2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale.

§1.6 Angle Measure

J

H

G

SQ R

m SRQ =

Find the measurement of:

m SRJ =

m SRG =

m QRG =

m GRJ =

180

45

150

70

180 – 150= 30

150 – 45= 105

m SRH

§1.6 Angle Measure

Use a protractor to draw an angle having a measure of 135.

1) Draw AB

2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.

3) Locate and draw point C at the mark labeled 135. Draw AC.

C

A B

§1.6 Angle Measure

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

§1.6 Angle Measure

Classify each angle as acute, obtuse, or right.

110°

90°40°

50°

130° 75°

Obtuse

Obtuse

Acute

Acute Acute

Right

§1.6 Angle Measure

5x - 7

B

The measure of B is 138.Solve for x.

9y + 4H

The measure of H is 67.Solve for y.

B = 5x – 7 and B = 138

Given: (What do you know?)

5x – 7 = 138

5x = 145

x = 295(29) -7 = ?

145 -7 = ?

138 = 138

Check!

H = 9y + 4 and H = 67

Given: (What do you know?)

9y + 4 = 67

9y = 63

y = 79(7) + 4 = ?

63 + 4 = ?

67 = 67

Check!

§1.6 Angle Measure

? ? ?

ba

Is m a larger than m b ?

60° 60°

§1.6 The Angle Addition Postulate

You will learn to find the measure of an angle and the bisectorof an angle.

NOTHING NEW!

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

2) Draw and label a point X in the interior of the angle. Then draw SX.

X

3) For each angle, find mRSX, mXST, and RST.

30°

45°

75°


R

TS

X

30°

45°

75°

= mRST = 75

mXST = 30

+ mRSX = 45

1) How does the sum of mRSX and mXST compare to mRST ?

2) Make a conjecture about the relationship between the two smaller angles and the larger angle.

Their sum is equal to the measure of RST .

The sum of the measures of the twosmaller angles is equal to the measureof the larger angle.


Postulate 3-3Angle

Addition Postulate

For any angle PQR, if A is in the interior of PQR, thenmPQA + mAQR = mPQR.

2

1

A

R

P

Q m1 + m2 = mPQR.

There are two equations that can be derived using Postulate 3 – 3.

m1 = mPQR – m2

m2 = mPQR – m1

These equations are true no matter where A is locatedin the interior of PQR.


2

1Y

Z

X

W

Find m2 if mXYZ = 86 and m1 = 22.

m2 = mXYZ – m1

m2 = 86 – 22

m2 = 64

m2 + m1 = mXYZ Postulate 3 – 3.


2x°

(5x – 6)°

B

DC

A

Find mABC and mCBD if mABD = 120.

mABC + mCBD = mABD Angle Addition Postulate

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126

x = 18

Add 6 to both sides

Divide each side by 7

mABC = 2x

mABC = 2(18)

mABC = 36

mCBD = 5x – 6

mCBD = 5(18) – 6

mCBD = 90 – 6

mCBD = 84

36 + 84 = 120


Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.ray

This ray is called an ____________ .angle bisector


Definition ofan Angle Bisector

The bisector of an angle is the ray with its endpoint at thevertex of the angle, extending into the interior of the angle.

The bisector separates the angle into two angles of equalmeasure.

2

1

A

R

P

Q

m1 = m2

QA is the bisector of PQR.


If bisects CAN and mCAN = 130, find 1 and 2.AT

ATSince bisects CAN, 1 = 2.

1 + 2 = CAN Angle Addition Postulate

1 + 2 = 130 Replace CAN with 130

1 + 1 = 130 Replace 2 with 1

2(1) = 130 Combine like terms

(1) = 65 Divide each side by 2

Since 1 = 2, 2 = 65

1

2

AC

N

T


Adjacent Angles and Linear Pairs of Angles

You will learn to identify and use adjacent angles and linear pairs of angles.

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

D

A

C

B1

2The 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.


Definition ofAdjacentAngles

Adjacent angles are angles that:

M

J

N

R1

2

1 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


Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but _____________no common side

1 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 ____


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


Definition ofLinear 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

C

A DB

1 2

AD form and BDBA

180 2 1


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.

§1.6 Complementary and Supplementary Angles

You will learn to identify and use Complementary and Supplementary angles

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


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.


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


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


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


§1.6 Congruent Angles

You will learn to identify and use congruent andvertical angles.

Recall that congruent segments have the same ________.measure

_______________ also have the same measure.Congruent angles

Definition of

CongruentAngles

Two angles are congruent iff, they have the same

______________.degree measure

50°B

50°

V

B V iff

mB = mV


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:


When two lines intersect, ____ angles are formed.four

12

34

There are two pair of nonadjacent angles.

These pairs are called _____________.vertical angles


Definition of

VerticalAngles

Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines.

12

34

Vertical angles:

1 and 3

2 and 4


1) On a sheet of paper, construct two intersecting lines that are not perpendicular.

2) With a protractor, measure each angle formed.

12

34

3) Make a conjecture about vertical angles.

Consider:

A. 1 is supplementary to 4.

m1 + m4 = 180

B. 3 is supplementary to 4.

m3 + m4 = 180

Therefore, it can be shown that 1 3

Likewise, it can be shown that 2 4


1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3

12

34

2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4

3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3


x = 4; 3 = 19°

x = 56; 4 = 65°

x = 9; 3 = 127°

x = 10; 4 = 97°


Theorem 3-1

Vertical AngleTheorem

Vertical angles are congruent.

1

4

3

2mn

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°


Find the value of x in the figure:

The angles are vertical angles.

(x – 10) = 125.(x – 10)°

125°

x – 10 = 125.

x = 135.


Suppose two angles are congruent.What do you think is true about their complements?

1 2

1 + x = 90 2 + y = 90

x = 90 - 1 y = 90 - 2

x = y

x = 90 - 1 y = 90 - 1

Because 1 2, a “substitution” is made.

x is the complement of 1

y is the complement of 2

If two angles are congruent, their complements are congruent.

x y


Theorem 3-2

If two angles are congruent, then their complements are_________.

The measure of angles complementary to A and Bis 30.

A B60° 60°

A B

Theorem 3-3

If two angles are congruent, then their supplements are_________.

The measure of angles supplementary to 1 and 4is 110.

70° 70°4 3 2 1

110° 110°

4 1

congruent

congruent


Theorem 3-4

If two angles are complementary to the same angle,then they are _________.

3 is complementary to 4

3 5

Theorem 3-5

If two angles are supplementary to the same angle,then they are _________.

congruent

congruent

45 is complementary to 4

5 3

3 1

2

1 is supplementary to 2


1 3


Suppose A B and mA = 52.

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

A52°

B52° 1

B + 1 = 180

1 = 180 – B

1 = 180 – 52

1 = 128°


If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25,

What are m1 and m2 ?

m1 + m3 = 90 Definition of complementary angles.

m1 = 90 - m3 Subtract m3 from both sides.

m1 = 90 - 25 Substitute 25 in for m3.

m1 = 65 Simplify the right side.

m2 + m3 = 90 Definition of complementary angles.

m2 = 90 - m3 Subtract m3 from both sides.

m2 = 90 - 25 Substitute 25 in for m3.

m2 = 65 Simplify the right side.

You solve for m2


1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3

2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC


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

12

34


Suppose you draw two angles that are congruent and supplementary.What is true about the angles?

Theorem 3-6

If two angles are congruent and supplementary then each is a __________.


1 2

Theorem 3-7

All right angles are _________.

right angle

congruent

1 and 2 = 90

C

BA

A B C


A

D

C

B

E1

23

4

If 1 is supplementary to 4, 3 is supplementary to 4, andm 1 = 64, what are m 3 and m 4?

1 3 They are vertical angles.

m 1 = m3

m 3 = 64


m3 + m4 = 180 Definition of supplementary.

64 + m4 = 180

m4 = 180 – 64

m4 = 116

Given


§1.6 Perpendicular Lines

You will learn to identify, use properties of, and construct perpendicular lines and segments.


Lines that intersect at an angle of 90 degrees are _________________.perpendicular lines

In the figure below, lines are perpendicular.CDAB and

A

DC

B

1 2

3 4


Definition ofPerpendicular

Lines

Perpendicular lines are lines that intersect to form aright angle.

m

nnm

1

3 4

2


m

l

In the figure below, l m. The following statements are true.

1) 1 is a right angle.

2) 1 3.

3) 1 and 4 form a linear pair.

4) 1 and 4 are supplementary.



Definition of Perpendicular Lines

Vertical angles are congruent

Definition of Linear Pair

Linear pairs are supplementary

m4 + 90 = 180, m4 = 90

Vertical angles are congruent


Theorem 3-8 1

3 4

2

a

b

If two lines are perpendicular, then they form four rightangles.

ba 901 m

902 m903 m904 m


false. or true is following the

of each whetherDetermine .QSNP and MN OP figure, the In

1) PRN is an acute angle.

False.

angle. right a is PRN

,MNOP Since

2) 4 8

True

congruent.

are angles vertical and

angles, vertical are 8 and 4 N

R

P7

1

2

5

6

8

4

3

M

O

Q


Theorem 3-9

If a line m is in a plane and T is a point in m, then thereexists exactly ___ line in that plane that is perpendicular tom at T.

one

m

T

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1.7 angles and perpendicular lines

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