lesson 3.1 aim: properties of parallel lines -...

Warm Up

Lesson 3.1 AIM: Properties of Parallel Lines

DO NOW: Name an angle congruent to angle 2.

Angles 2, 3, 6, and 7 are

congruent.

A transversal is… A line that intersects two or more lines at different

points.

Which angles are alternate interior angles?

Angles 3 and 6.

Angle 4 and 5.

Alternate Interior Angles

are congruent.

Which angles are same side interior angles?

Angles 3 and 5.

Angle 4 and 6.

Same Side Interior Angles

are supplementary.

Which angles are corresponding angles?

Angles 1 and 5.

Angle 2 and 6.

Angles 3 and 7.

Angles 4 and 8.

Corresponding Angles

are congruent.

Which angles are alternate exterior angles?

Angles 1 and 8.

Angle 2 and 7.

Alternate Exterior Angles

are congruent.

What is the measure of angle 5?

Angle 5 = 135º


are congruent.

What is the measure of angle PQR?

Angle PQR = 120º

Alternate Interior Angles

are congruent.


Angle 9 = 90º

Alternate Exterior Angles

are congruent.


Angle 6 = 50º

Same Side Interior Angles

are supplementary.

Finding Measures of Angles What are the measures of the missing angles?

3 105m 4 105 180m

4 75m

Find the value of x. Relationship:


125 = x + 15

-15 -15

110 = x

Finding an Angle Measure What is the value of y?

( 40) 80 180y

120 180y 60y

Summary Question

Practice!!

Pgs. 118-120 #1-15 odd #27 and #29

Turn in #27 and 29!

Warm Up

18

Lesson 2 - 2

Proving Lines

Parallel

19

Proving Lines Parallel - Postulates & Theorems

If two lines are cut by a transversal and corresponding

angles are congruent, then the lines are parallel.

DC

BA

20

Proving Lines Parallel - Postulates &Theorems

If two lines are cut by a transversal and alternate

interior angles are congruent, then the lines are parallel.

DC

BA

21


If two lines are cut by a transversal and consecutive

interior angles are supplementary, then the lines are

parallel.

DC

BA

22


If two lines are cut by a transversal and consecutive

exterior angles are supplementary, then the lines are

parallel.

DC

BA

23

Examples: Proving Lines Parallel

Find the value of x which will make lines a and lines b parallel.

4. 20° Answers: 1.

20°

2.

50°

3.

90°

2.

80

2x

b

a1.

60

3x

b

a

3.

70

(x-20)

b

a 4.

60

3x

b

a

24

Ways to Prove Two Lines Parallel

Show that corresponding angles are equal.

Show that alternative interior angles are equal.

Show that consecutive interior angles are supplementary.

Show that consecutive exterior angles are supplementary.

In a plane, show that the lines are perpendicular to the same line.

Practice

Pgs. 125-127 # 2-16 even

#18-24 all

#25

Section 3-3 Parallel Line and the Triangle Angle-Sum Theorem

Purpose:

Students will classify triangles, find the measures of their

angles, and use exterior angles of triangles to solve problems.

Finding Angle Measures in Triangles

Theorem 3-7: Triangle Angle-Sum Theorem:

The sum of the measures of the angles of a triangle is 180.

mAmBmC 180

A

C

B

50° 35°


A

C

B

50° 35°

Find the measure of

the three angles.

Justify your work.


F

G

H J

65°

39° 21°

x° y° z°

Find the values of x, y, and z.

Justify your work.


A

B

D C

62° 49°

x° y°

z°

Find the values of x, y, and z.

Classifications of Triangles

Equiangular

All 's are

Acute

All 's acute

Right

One right Obtuse

One obtuse

Equilateral

All sides

Isosceles

At least 2 sides

Scalene

No sides

Classifications of Triangles

37°

37°

Classify the following triangles by their angles and sides.

Draw an equilateral right triangle.

NOT POSSIBLE!!!

1

2

3

Using Exterior Angles of Triangles

Exterior

Angle

Exterior Angle of a Polygon: An angle formed by a side and an

extension of an adjacent side.

Remote Interior Angles: The two nonadjacent interior angles

corresponding to each exterior angle of a triangle.

Remote

Interior

Angles


Theorem 3-8: Triangle Exterior Angle Theorem:

The measure of each exterior angle of a triangle equals the sum

of the measures of its two remote interior angles.

m1m2m3

1

2

3


x°

100°

50°

Find the measure of

the x. Justify your

work.


53° w° 86°

x° y°

z°

Find the values of w, x, y, and z. 97°

Little More Practice

(80 – 2x)°

(70 – 3x)°

Find the value of x.

Practice:

Pg. 134-1371-51 odd

#51 can be turned in for 5 extra credit points!!

Warm Up

3-4 The Polygon Angle-Sum Theorem

Theorem 3-9 Polygon Angle-Sum Theorem

The sum of the measures of the interior angles of an n-gon is (n – 2) * 180.

Finding a Polygon Angle Sum What is the sum of the interior angle measures of a

heptagon?

Sum = (n – 2) * 180

= (7 – 2) * 180

= 5 * 180

= 900°

The sum of the interior angle measures of a heptagon is 900°.

An equilateral polygon is a polygon with all sides congruent.

An equiangular polygon is a polygon with all angles congruent.

A regular polygon is a polygon that is both equilateral and equiangular.

Equilateral Polygon Equiangular

Polygon

Regular

Polygon

Corollary to the Polygon Angle-Sum Theorem

The measure of each interior angle of a regular n-gon is ( 2) 180

.n

n

Using the Polygon Angle-Sum Theorem The common housefly, Musca domestica, has eyes that

consist of approximately 4000 facets. Each facet is a regular hexagon. What is the measure of each interior angle in one hexagonal facet?

( 2) 180n

n

(6 2) 180

6

4 180

6

120

Using the Polygon-Angle Theorem

What is the measure of angle Y in pentagon TODAY?

Using the Polygon-Angle Theorem (5 2) 180m T m O m D m A m Y

110 90 120 150 3 180m Y

470 540m Y

70m Y

Polygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

1 2 3 4 5 360m m m m m

Finding an Exterior Angle Measure

What is the measure of angle 1 in the regular octagon?

3601

8m

1 45m

Practice!!!!! Homework – Textbook p. 147-149

1-25 odd

28-31 all

32-55 odd

Warm-Up

Parallel and Perpendicular Lines

Parallel Lines Two lines with the same slope are said to be parallel lines. If

you graph them they will never intersect.

We can decide algebraically if two lines are parallel by finding

the slope of each line and seeing if the slopes are equal to each

other.

We can find the equation of a line parallel to a given line and

going through a given point by: a.)

first finding the slope m of the given line; b.) finding

the equation of the line through the given point with slope

m.

Testing if Lines are Parallel Are the lines parallel? 12 3 9 and -8 2 14x y x y

Find the slope of 12 3 9

3 12 9

4 3

x y

y x

y x

The slope m = -4


2 8 14

4 7

x y

y x

y x

The slope m = -4

Since the slopes are equal the lines are parallel.

Graphs of Parallel Lines

The red line is the graph of

y = – 4x – 3

and the blue line is the graph of

y = – 4x – 7

Practice Testing if Lines are Parallel Are the lines 6 3 5 and 2 4 4x y y x parallel? (click mouse for answer)

6 3 5

3 6 5

523

2

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are different

the lines are not parallel.

Are the lines 2 4 and 2 4 12x y x y parallel? (click mouse for answer)

2 4

2 4

1 22

12

x y

y x

y x

m

2 4 12

4 2 12

1 32

12

x y

y x

y x

m

Since the slopes are equal

the lines are parallel.

Constructing Parallel Lines Find the equation of a line going through the point (3, -5) and

parallel to 2 83

y x

Using the point-slope equation where the slope m = -2/3

and

the point is (3, -5) we get 25 33

25 23

2 33

y x

y x

y x

Practice Constructing Parallel Lines Find the equation of the line going through the point (4,1) and

parallel to (click mouse for answer) 3 7y x

1 3 4

1 3 12

3 13

y x

y x

y x

Find the equation of the line going through the point (-2,7) and

parallel to (click mouse for answer) 2 8x y

7 2 2

7 2 2

7 2 4

2 3

y x

y x

y x

y x

Perpendicular Lines Perpendicular lines are lines that intersect in a right angle.

We can decide algebraically if two lines are perpendicular by finding the

slope of each line and seeing if the slopes are negative reciprocals of each

other. This is equivalent to multiplying the two slopes together and seeing if

their product is –1.

We can find the equation of a line perpendicular to a given line and going

through a given point by:

a.) first finding the slope m of the given line;

b.) finding the equation of the line through the given point with

slope = –1 /m.

Testing if Lines Are Perpendicular

1Are the lines 2 5 and 4 perpendicular?

2x y y x


2 5

x y m

y x

1 1Find the slope of 4

2 2y x m

Since the slopes are negative reciprocals of each other the lines

are perpendicular. 12 1

2

Graphs of Perpendicular Lines

The red line is the graph of

y = – 2x + 5

and the blue line is the

graph of

y = – 1/2 x +4

Practice Testing if Lines Are Perpendicular Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x

6 3 5

3 6 5

523

2

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are not

negative reciprocals of

each other (their product

is not -1) the lines are

not perpendicular

Are the lines 2 4 and 4 2 6 perpendicular?x y x y

2 4

2 4

1 22

12

x y

y x

y x

m

4 2 6

2 4 6

2 3

2

x y

y x

y x

m

Since the slopes are

negative reciprocals of

each other (their

product is -1) the lines

are perpendicular.

Constructing Perpendicular Lines

Find the equation of a line going through the point (3, -5) and

perpendicular to 2 83

y x

The slope of the perpendicular line will be m = 3/2 Using

the point-slope equation where the slope m = 3/2 and

the point is (3, -5) we get 35 3

2

3 952 2

3 192 2

y x

y x

y x

Practice Constructing Perpendicular Lines

Find the equation of the line going through the point (4,1) and

perpendicular to (click mouse for answer) 3 7y x

11 43

1 413 3

1 13 3

y x

y x

y x

Find the equation of the line going through the point (-2,7) and

perpendicular to (click mouse for answer) 2 8x y

17 22

17 22

17 12

1 82

y x

y x

y x

y x

Practice!!

Pg. 1-23 odd

25-30 all

#35, 38, & 41

lesson 3.1 aim: properties of parallel lines -...

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