geometry chapter 3 notes - canyon crest academy...

1 Geometry Chapter 3 Notes

Notes #13: Section 3.1 and Algebra Review

A. Definitions:

Parallel Lines:

Draw:

Example:

Transversal: a line that intersects two or more coplanar lines in different points

Draw:

Example:

Parallel Planes:

Draw:

Example:

Practice:

1.) Name a plane parallel to ABGH.

2.) Name three lines parallel to CF suur

3.) Classify the planes as intersecting or parallel: ADEH and BCFG

4.) Draw and label the figure described: AB suur

and XY suur

are coplanar and parallel. h is a transversal that intersects them at points C and Z, respectively.

B. Special Angles: The angles formed by lines and their transversals are special: Alternate Interior Angles: ( ) Sameside Interior Angles: ( )

2 Corresponding Angles: ( ) Alternate Exterior Angles: ( )

Sameside Exterior Angles: ( )

Vertical Angles: (reminder) ( )

Practice: Classify each pair of angles as alt. int., ss int., corr, alt. ext., ss ext, or vertical. 5.) 1, 5 ∠ ∠ 6.) 2, 8 ∠ ∠

7.) 4, 2 ∠ ∠ 8.) 3, 8 ∠ ∠

9.) 1, 7 ∠ ∠ 10.) 4, 7 ∠ ∠

8 7 6 5 4 3

2 1

C. Identifying Lines and Transversals: Name the two lines and the transversal that form each pair of angles. What type of special angles are they? (Hint: Trace the angles in two different colors – where they overlap is the transversal, the leftovers are the two lines.)

11.) 4, 2 ∠ ∠ lines: ____, _____ transversal: ______ type: _______________

12.) , B BAD ∠ ∠ lines: ____, _____ transversal: ______ type: _______________

13.) , 5 BAD ∠ ∠ lines: ____, _____ transversal: ______ type: _______________

5 4 3

2 1

B

A

C

D E

14.) , 5 BCD ∠ ∠ lines: ____, _____ transversal: ______ type: _______________

3 D. Algebra Practice: Solving Linear Systems by Addition/Subtraction/Elimination

Rearrange each equation so that the variable expressions are on the left side of the equals sign and the constant is on the right side of the equal sign (called Standard Form) Multiply whole equations so that one variable expression is equal but has the opposite sign. Add the equations together; watch one variable cancel out Solve for BOTH variables Write your answer as a point ( x, y ) (in alphabetical order)

15.) 2x – 3y = 8 16.) x = 4y 3 4x + 3y = 2 3x – 2y = 5

17.) 3x – 5y = 11 18.) 2x + 3y – 10 = x + y 14 2x – 4y = 9 x – 2y + 5 = 2x – y + 6

Solve for x and y: 19.)

2x y 70

3x 35 x 15

4 Notes #14: Sections 3.1 and 3.2

A. Relationships formed by parallel lines

Alternate Interior Angles Theorem

If two ________________ lines are cut by a __________, then alternate interior angles are _________________.

(Its converse): If two lines cut by a ________________ form congruent alternate interior angles, then the lines are ________________.

Corresponding Angles Postulate

If two ________________ lines are cut by a __________, then corresponding angles are _________________.

(Its converse): If two lines cut by a ________________ form congruent corresponding angles, then the lines are ________________.

SameSide Interior Angles Theorem

If two ________________ lines are cut by a __________, then sameside interior angles are _________________.

(Its converse): If two lines cut by a ________________ form supplementary sameside interior angles, then the lines are ________________.

Alternate Exterior Angles Theorem

If two ________________ lines are cut by a __________, then alternate exterior angles are _________________.

(Its converse): If two lines cut by a ________________ form congruent alternate exterior angles, then the lines are ________________.

SameSide Exterior Angles Theorem

If two ________________ lines are cut by a __________, then sameside exterior angles are _________________.

(Its converse): If two lines cut by a ________________ form supplementary sameside exterior angles, then the lines are ________________.

5 Complete the sentences and solve for x.

1.) The labeled angles are ____________ angles and their measures are ___________ because of the _________________________________________

3x + 10

100

2.) The labeled angles are ____________ angles and their measures are ______________________ because of the _____________________________

120 3x

3.) The labeled angles are ____________ angles and their measures are ___________________ because of the _____________________________

2x + 52

4x 8

4.) The labeled angles are ____________ angles and their measures are ___________________ because of the _____________________________

2x + 21 7x 4

B. Identifying Parallel Lines: Use the given information to name the lines that must be parallel. (Trace angles and look for special pairs of angles and special relationships.)

10

9

8 7 6

5 4 3

2 1

W

S T

U

V

5.) 1 4 ∠ ≅ ∠ Type of angle pair: Relationship:

Parallel lines?:

6.) 1 2 3 180 m m m ∠ + ∠ + ∠ = o

Type of angle pair: Relationship:

Parallel lines?:


Parallel lines?:


Parallel lines?:


Parallel lines?:

6

C. Special Pairs of Angles

Solve for all variables. All measurements are in degrees. (Hint: extend the parallel lines and look for special pairs of angles)

10.)

120

x y

11.)

4x

120

80

2y

12.)

e

d c

b a

27

60

56

13.)

130

2x 5y 2x 3y

110

D. Proofs with Parallel Lines:

14.) Prove the alternate exterior angles theorem:

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

Given: k l P Prove: 1 3 ∠ ≅ ∠

l

k

3 2

1

Statements Reasons 1.)

2.)

3.) 3 ____ ∠ ≅ ∠

4.)

1.)

2.) Corresponding Angle Postulate

3.)

4.)

7

15.) Prove the converse of the alternate exterior angles theorem:

If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.

Given: 1 3 ∠ ≅ ∠ Prove: m n P

n

m

3

2 1


2.) 1 _____ ∠ ≅

3.)

4.)

1.)

2.)

3.) Substitution

4.) If two lines and a transversal form ____________ __________________ angles, then the two lines are ________________.

Notes #15: Sections 3.3 and 3.4 A. Parallel and Perpendicular Lines

Prove the converse of the sameside interior angles theorem:

If two lines and a transversal form sameside interior angles that are supplementary, then the two lines are parallel.

Given: 2 ∠ and 3 ∠ are supplementary Prove: m n P

n

m

3

2 1


2.) 2 3 _____ m m ∠ + ∠ =

3.) 2 ____ _____ m m ∠ + ∠ =

4.)

5.) 2 2 m m ∠ = ∠

6.)

7.)

1.)

2.) Definition of __________________ angles

3.)

4.) Substitution

5.)

6.) Subtraction

7.) If two lines and a transversal form ____________ __________________ angles, then the two lines are ________________.

8

If two lines are parallel to the same line, then they are ___________ to each other.

In a plane, if two lines are perpendicular to the same line, then they are _____________ to each other.

In a plane, if a line is perpendicular to one of two parallel lines, then it is also _____________ to the other.

Examples: For #13, consider coplanar lines j, k, l, and m. Given each of the following statements, what more, if anything, can you conclude about the lines?

1.) , j k k m P P 2.) , j k l k ⊥ P 3.) , , j k k l l m ⊥ ⊥ P

B. Classifying Triangles Triangles are described based on the lengths of their sides and the measures of their angles

Sides

Scalene Isosceles Equilateral

9 Angles Acute Obtuse Right

Equiangular

Examples: For #57, classify each triangle (drawn to scale) by its angles and sides. 5.) 6.) 7.)

For #810, draw a triangle, if possible, to fit each description. 8.) obtuse scalene 9.) acute isosceles 10.) right equilateral

11.) The perimeter of ∆ABC is 32m. AB = 4x – 2, BC = 3x + 1, AC = 2x + 6. Write an equation and solve for x. Then, classify the triangle as scalene, isosceles, or equilateral. (Hint: draw a picture first and label what you know)

C. The angles of a triangle:

** The sum of the interior angles of a triangle is always ______**

10

13.) Find the missing values and classify LMN V : 12.)

3

2

1

1 2 3 _____ m m m ∠ + ∠ + ∠ = x

4x + 11

2x + 8

M

L N

LMN V is __________ and __________

____ ____ ____ ____

x m L m M m N

= ∠ = ∠ = ∠ =

14.) Find the values of each variable and the measure of each angle. Then classify each triangle by its angles. (All measurements shown are in degrees)

134

61 w x

y

z D

C

B A ____ ____ ____ ____

w x y z

= = = =

is _________ is _________ is _________

ABC ABD DBC

V V V

** The measure of an exterior angle of a triangle equals the sum of the measures of the two _________________ __________________ angles.**

Explore:

85

40

1 2

3 4

5 6

7

Diagram for #15

15.) Complete:

1 2 _____ 2 7 _____ 7 3 _____ 4 3 5 6 _____ 2 3 7 _____

m m m m m m m m m m m m m

∠ + ∠ = ∠ + ∠ = ∠ + ∠ = ∠ + ∠ + ∠ + ∠ = ∠ + ∠ + ∠ =

11

16.) Solve for x and y:

y

95

50 x

17.) 2 103 , 3 156 , 1 _____ m m m ∠ = ∠ = ∠ = o o

3

2

1

Notes #16: Section 3.5 and Algebra Review

A. Polygons: ( _________ sided figures)

Convex Polygon Concave Polygon

Explore: Triangle ____ sides

Sum of Interior Angles = ______ Sum of Exterior Angles = ______

Quadrilateral____ sides


Pentagon ____ sides


Hexagon ____ sides


Other Common Polygons: Octagon: ______ sides Nonagon: ______ sides Decagon: ________ sides Dodecagon: ________ sides

18gon: ______ sides 20gon: ______ sides ngon: ______ sides

12 Patterns for polygonal angle sums:

Sum of Interior Angles Sum of Exterior Angles

each interior angle + each exterior angle = ___________

Find the sum of the measures of the angles of each polygon: (interior angles)

1.) decagon 2.) octagon 3.) 22gon

Find the missing angle measures: (all measures shown are in degrees) 4.)

130

a

114 28

5.) b 131

107

160

123

85

B. Regular Polygons (where n is the number of sides in the polygon)

Regular Polygons: all sides ________________

all angles ________________

Interior Angles

Sum of Interior Angles:

OR

Each Interior Angle:

Exterior Angles

Sum of Exterior Angles:

Each Exterior Angle:

Extra Trick:

(each interior angle) + (each exterior angle) = ________

13 Complete the chart using these relationships. (Pictures may help!)

6. 7. 8. 9. 10. 11. # of sides (n) 6 8 Sum of Exterior Angles Each Exterior Angle 72˚ Each Interior Angle 90˚ Sum of Interior Angles 900˚ 2880˚

6.) 7.) 8.)

9.) 10.) 11.)

D. Word Problems: Define two variables and write two equations to solve.

12.) The sum of two numbers is 18 and their difference is 6. Find each of the numbers.

13.) The sum of two numbers is one more than twice the smaller number. Their difference is seven less than twice the larger number. Find the numbers.

14

D. Algebra Practice: Solving linear systems with fractions and/or decimals.

To clear decimals: multiply both sides of the equation by a multiple of 10; scoot the decimal over

To clear fractions: multiply both sides of the equation by the common denominator; cross cancel

14.) 0.2 0.5 1.4 1 1 1 2 3 3

m n

m n

− = −

+ = − 15.)

0.2 1.2 8.8 1 1 1 4 6 6

m n

m n

+ =

− =

Notes #17: Section 3.6

A. Slope:

Slope is used to describe the ________________ and _________________ of lines.

Sketch a line with: a) positive slope b) negative slope c) zero slope d) undefined slope

15 A line is shown. Use two marked points and count “rise over run” to find the slope of the line. 1.) 2.) 3.)

Slope = Slope = Slope =

Without using a graph and given two points: ( ) 1 1 , x y and ( ) 2 2 , x y

Slope = m = 2 1

2 1

y y x x

− −

0 0

0

n

n undefined

=

=

For #45, find the slope of the line passing through the two given points:

4.) (4, 1), (3, 2) 5.) (6, 3) and (2, 1) 6.) A line with slope

7 3 passes through the

points (1, 2) and (2, y). Find y.

B. Graphing Lines

There are many ways to graph a line. You need to know how to graph a line: (i) given a point and a slope, (ii) by finding the xintercept and yintercept, and (iii) by finding the yintercept and the slope of the line.

16 (i) Graphing lines using a point and a slope A point P on a line and the slope of the line are given. Sketch the line and find the coordinates of two other points on the line

7.) 8.) 9.)

P (2, 1); slope = 4 5

1 st point:

2 nd point:

P (0, 3); slope = 2

1 st point:

2 nd point:

P (2, 0); slope = 2 3

1 st point:

2 nd point:

(ii). Graphing Lines using the x and y intercepts. The intercepts are the point(s) where a line intersects the axes of the coordinate plane. Find the x and y intercepts (by setting the opposite variable to zero) Write these answers as two different points Graph and connect these points to graph the line Label the graphed line with the original equation

Most common error: • Forgetting that the intercepts are two different points and graphing as just one

10.) x + 2y = 4 xintercept yintercept (set y = 0) (set x = 0)

xint: ( , 0) yint: (0, )

11.) 3x – y = 3

xint: ( , ) yint: ( , )

10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

17

(iii) Graphing Lines using the slope and yintercept: Get y alone so the equation is in y = mx + b form (m = _________, b = _________) Graph b first. This point goes on the ____ axis. Use slope and count rise over run to the next point(s). When you have at least three points, then connect the points to make a line. Label your graphed line with the original equation

Most common errors: • Graphing b on the xaxis instead of the yaxis • Graphing the slope in the wrong direction (e.g. forgetting a negative)

12.) 1 5 2

y x = − −

(↑ I’m already in slopeintercept form!)

m = ___ (ß graph me second! Watch the negative!)

b = ___ (ß graph me first! I go on the yaxis!)

13.) x – 2y =2 (↑ Get me in slopeintercept form first)

m = ______

b = ______

14.) x + 3y = 6 (↑ Get me in slopeintercept form first)

m = ______

b = ______

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

Graph for #13 AND #14 (be sure to label your lines!)

18 Special Cases: Graphing Horizontal and Vertical Lines

15.) x = 4 (This equation describes the line for which ALL points have an x coordinate of 4. There are no restrictions on the value of y).

16.) y = 2 (This equation describes the line for which ALL points have an y coordinate of 2. There are no restrictions on the value of x).

17) x = 1

Use the pattern you found above to complete these sentences: • Any line in the form x = _____ is a ________________ line because it intersects the ___ _________

• Any line in the form y = _____ is a ________________ line because it intersects the ___ _________

Use this pattern to graph these lines without a table of solutions. 18.) y = 3 19.) x = 2 20.) y = 4

19 Notes #18: Writing Linear Equations

A. Converting equations of lines: Lines can be written in either SlopeIntercept form (y = mx + b) or Standard Form (Ax + By = C). You need to know how to convert from one to the other.

Converting to SlopeIntercept Form

Goal: y =mx + b (where m and b are integers or fractions)

• Get y alone • Reduce all fractions

Converting to Standard Form

Goal: Ax + By = C (where A, B, and C are integers and

where A is positive)

• Get x and y terms on the left side and the constant term on the right side of the equation

• Multiply ALL terms by the common denominator to eliminate the fractions

• If necessary, change ALL signs so that the x term is positive

1.) Convert to slopeintercept form:

4x – 12y = 8

2.) Convert to standard form:

2 5 3

y x = −

B. Writing linear equations given the slope and yintercept Find the slope (m) and yintercept (b) [If the given information is a graph, then you will have to count by hand to find these values.] Fill in m and b so you have an equation of the line in y = mx + b form.

y = ________ x + ____________ (↑ Put m here!) (↑ Put b here!)

3.) Find the equation of the line with slope of 5 and yintercept of 2. Write in standard form.

4.) Find the equation of the given line in slopeintercept form.

5.) Write the equation of a line that has the same slope

as 4 3 5

y x = − and has a yintercept

of 1. Write in standard form.

20 C. Writing linear equations given the slope and a point

• plug slope = m into y = mx + b • name your point (x, y) and plug these values in for x and y • solve for b • plug m and b back into y = mx + b • convert to standard form, if necessary

** Remember to leave x and y as variables! ** 6.) Find the equation of the line with slope of 2 and going through (1, 3) in slopeintercept form.

7.) Find the equation of the

line with slope of 1 3 and

going through (6, 2) in standard form.

8.) Find the equation of the line in slopeintercept form with

slope 2 5

− and passing through the

point (3, 7).

D. Writing linear equations given two points • find the slope • pick one of your points to be x and y • plug m, x, y into y = mx + b • solve for b; plug m and b into y = mx + b • convert to standard form, if necessary

** Remember to leave x and y as variables! ** 9.) Find the equation of the line going through (3, 1) and (4, 8) in slopeintercept form.

10.) Find the equation of the line with xintercept 3 and yintercept 2 in standard form.

11.) Find the equation of the line going through (5, 2) and (1, 3) in standard form.

12.) Find the equation of the line with xintercept 5 and yintercept 4 in slopeintercept form.

21 Notes #19: Section 3.7 A. Review Writing Linear Equations: 1.) Find the equation of the line with slope of 3 4 and going through (1, 5) in slopeintercept

form.

2.) Find the equation of the line going through (1, 0) and (4, 2) in slopeintercept form.

3.) Find the equation of the line with xintercept 1 and yintercept 2 in standard form.

4.) Find the equation of the line going through (4, 3) with xintercept 6 in standard form.

B. Parallel and Perpendicular Lines For #56, a pair of parallel lines and a pair of perpendicular lines are graphed below. Use the graphs to find the slope of each of the four lines and to complete the sentences.

5.)

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

Slope of 1 l : Slope of 2 l :

Parallel lines have _________ slopes.

22

6.)

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

Slope of 3 l : Slope of 4 l :

Perpendicular lines have ____________, _____________ slopes.

The slope of a line is given. Find the slope of a line parallel to it and the slope of a line perpendicular to it:

7.) m = 2 3

− 8.) m = 7 9.) m = 0

Are the lines with these slopes parallel, perpendicular, or neither?

10.) 11.) 4, 4 12.) 1, 1

13.) Find the slope of a line parallel and perpendicular to AB suur

where A(3, 1) and B (2, 4)

, 2 4 3 6

23 For #1416, state whether the given pair of lines is parallel, perpendicular, or neither:

14.) 3 1 4

8 6 12

y x

x y

= − +

− =

15.) 1 5 2

2 4 9

y x

x y

= − +

+ =

16.) 5 4 5 4

y x x y

= − + = − +

C. Writing linear equations given a point and another line (parallel or perpendicular to your line) • find m from the given line • if the line is parallel, this is your m;

if the line is perpendicular, find its ______________ _______________ • plug m, x, y into y = mx + b • solve for b; plug m and b into y = mx + b • convert to standard form, if necessary

** Remember to leave x and y as variables! ** 17.) Find the equation of the line going through (1, 2) and parallel to y = 3x + 4 in slopeintercept form.

18.) Find the equation of the line going through (3, 2) and perpendicular to x – 4y = 3 in standard form.

19.) Find the equation of the line going through (1, 5) and perpendicular to y = 3x + 4 in slope intercept form.

20.) Find the equation of the line going through (9, 3) and parallel to 2x – 3y = 3 in standard form.

24 Notes #20: Review You can now solve linear systems (a set of 2 lines) using algebra (substitution/elimination) AND using coordinate Geometry (graphing). You should get the same answer for both methods.

solve the equations using substitution or elimination; write your answer as a point ( , ) graph the two lines using either the intercept method OR slopeintercept method confirm that the two lines intersect (meet) at your solution point

1.) y = x – 2 x + y = 4

1 st Method: substitution or elimination

solution: ( , )

2 nd Method: graphing (graph both lines on the coordinate plane below)

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8 7

6 5

4 3 2

1

1 2 3

4 5

6 7

8 9

10 y

2.) 2x – y = 3 x + 2y = 6

1 st Method: substitution or elimination

solution: ( , )

2 nd Method: graphing (graph both lines on the coordinate plane below)

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10 9 8 7

6 5 4

3 2

1

1 2 3

4 5

6 7

8 9

10 y

25 Chapter 3 Study Guide

For #14, identify whether the angles are vertical angles, same side interior angles, corresponding angles, alternate interior angles, sameside exterior angles, or alternate exterior angles. 1.) 2 and 6 ∠ ∠

3.) 5 and 2 ∠ ∠

2.) 1 and 6 ∠ ∠

4.) 4 and 7 ∠ ∠

7

6 5

4 3

2 1

For #56, find the slope of the line passing through the two points 5.) (3, 2) and (4, 1) 6.) (9, 2) and (2, 2) 7.) The slope of line l is

given. Find the slope of the line parallel to it and the slope of the line perpendicular to it: a) 2 b) 3/2

For #810, name the two lines and transversal that form each pair of angles:

8.) 1, 3 ∠ ∠ lines: ____, ____ trans: ______

9.) , BAD CDA ∠ ∠ lines: ____, ____ trans: ______

10.) , 5 BAD ∠ ∠ lines: ____, ____ trans: ______

5 4 3

2 1

B

A

C

D E

In the diagrams, the lines shown are parallel. Write an equation and solve for x and y. (The answer to #12 is two fractions.) Justify your work.

2x+40

x+80 5y+20

11.)

2x+10 2y+20

y+65

12.)

Find the values of x and y. 13.)

40

20

y

x

14.)

5x

15x20 5y

26 Define your variables, write an equation, and solve: 15.) The sum of two numbers is 10. The difference of the first number and twice the second number is 1. Find the numbers.

In each exercise, some information is given. Use this information to name the segments that must be parallel. If there are no such segments, write none. 16.) 3 10 ∠ ≅ ∠

18.) 2 3 ∠ ≅ ∠

17.) 7 10 ∠ ≅ ∠

19.) 9 5 ∠ ≅ ∠ 11

10 9

8

7

6 5

4

3 2 1 A B

F C

E D


6x 2y 140

x + y

130


1 1 1 2 4 2 1 8 3 2

x y

x y

+ = −

− = −

22.) Prove the converse of the alt ext angles theorem:

If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.

Given: 1 3 ∠ ≅ ∠ Prove: m n P

n

m

3

2 1


2.)

3.)

4.)

1.)

2.)

3.)

4.)

27 23.) Given: l m P

Prove: 2 4 180 m m ∠ + ∠ = o 4 3

2 1

m

l


2.)

3.)

4.)

1.)

2.)

3.)

4.)

For #2526, classify the triangles based on their sides and their angles: 25.) 23.)

2x 3x + 5

8x 20

26.) 24.)

60 60

24.) Given: AB = CD AE = FD

Prove: EB = CF

A B

C D

E

F


2.)

3.)

4.) AE = FD

5.)

1.)

2.)

3.)

4.)

5.)

28

Use the diagram for reference. Show all equations and work.

27.) If m 6 42 and m 8 61, then m 10 ____

∠ = ∠ = ∠ =

11

10

9

8

7 6

28.) If m 6 7 , m 7 2 5, and m 11 6 + 35, then = ___.

x x x

x

∠ = ∠ = + ∠ = 29.)

If m 8 7 2, m 7 4 7, and m 9 10 + 3, then = ___.

x x x

x

∠ = − ∠ = − ∠ =

For #3031, a, b, c, and d are distinct coplanar lines. How are a and d related? 30.) , , a b b c c d ⊥ P P 31.) , , a b b c c d ⊥ ⊥ P

For #3233, find the measure of an interior angle and an exterior angle of each regular polygon. 32.) an octagon 33.) a pentagon

Complete the table for regular polygons 34.) (a) (b) (c)

Work: (a) (b) (c)

Number of Sides 6 Sum of exterior angles Measure of each exterior angle 20 Measure of each interior angle 162 Sum of interior angles

29 Graphing Linear Equations: 35.) Graph each line using the slope and y intercept:

a) y = 3 2 x + 1 b) 2x + y = 4 c) 3x – 2y = 8

109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8 7

6 5

4

3 2

1

1 2 3

4 5

6 7

8 9

10 y

36.) Graph the lines using the x and y intercepts:

a) 2x + 3y = 6 b) 3x – 5y = 15 c) 2x – 3y = 10

10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x

10

9 8

7

6 5

4

3 2

1

1 2

3

4 5

6

7

8 9

10 y

37. Find the x and y intercept of each line:

a) x + 2y = 6 b) 3x – 4y = 8 c) y = 1 2

− x + 3

38. Find the slope and yintercept of each line: a) y = 3x – 5 b) y = 3 c) 3x – 2y = 4

39. Find the intersection of the two lines using the substitution or elimination method. x + 2y = 8 and 2x + 3y = 10

40. Explain why these two lines will not intersect y = 2x – 1 and 8x – 4y = 16

Writing Linear Equations: Write an equation of the line with: 41. yintercept 2 and slope 4 in standard form 42. xintercept 4 and yintercept 2 in slope

intercept form

43. through (1, 2) with slope 3 in slope intercept form

44. through (6, 2) and parallel to x – 2y = 5 in standard form

30

45. through (2, 1) and perpendicular to x + 3y = 7 in standard form

47. through (3, 2) and (7, 2) in slopeintercept form

49. xintercept 3 and yintercept 5 in standard form

46. through (3, 2) and (4, 7) in slopeintercept form

48. through (4, 3) and with xintercept 2 in standard form

50. through (3, 3) and perpendicular to 2x – y = 1 in slopeintercept form

For #5153, are the given lines parallel, perpendicular, or neither?

51.) 1 2 3

2 6 10

y x

x y

= − +

− = 52.)

1 2 3

2 6 10

y x

x y

= − +

+ = 53.)

1 2 3

6 2 10

y x

x y

= − +

− =

geometry chapter 3 notes - canyon crest academy...

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