Geometry Chapter 3 2013-2014 Parallel lines

Proving Parallel lines

Triangles

Angles of Polygons

1

2

Name: __________________________________________ Geometry Assignments – Chapter 3

Parallel Lines and Planes Date Due

Section

Topics

Assignment

Written Exercises

3.1

3.2

Parallel Lines/Planes Skew Lines Transversal Alternate Interior <’s Corresponding <’s Properties of Parallel lines

and their angles

Pg 76-77 #2-16 even, 23-28 all Pg 80 -81 (bottom of page): #2-6even, 7-12, 14,16,18

3.4

3.5

Types of Triangles by sides Types of Triangles by

angles Exterior <’s Remote Interior <’s Polygons (convex) Diagonal Regular Polygon <’s in a Polygon

Pg 97-99 #5-15 odd, 29 Pg 104 #1-6,8,9, 16, 17, 22, 25

3.3

Proving Lines Parallel

Pg 87 #2-16even, 17,18,20,27,28,31 Worksheet

CHPT 3 Review

Suggested Chapter 3 Review Questions from your Textbook

Pg. 89 (Self-Test 1) Pg.110 (Self-Test 2) #1-11 Pg 111-112 (Chapter Review) #1-19, not 16 Pg 112-113 (Chapter Test) #7-13 All answers for above questions are

available, simply ask for them! STUDY MATH BY DOING THE

MATH!!! Use Suggested Practice as a Guide,

Ask for help!!!!

3

4

5

Parallel Lines with Algebra WS DATE: _________________

1. In the figure below, if AB CD , find :

a. 5m when 3 80m

b. 2m when 6 150m

c. 4m when 5 60m

d. 7m when 1 75m

e. 8m when 3 65m

Classify each pair of angles as alternate interior angles, corresponding angles, alternate

exterior angles, same side interior angles, or none of these.

2. 5 7

4. 2 4

6. 6 7

8. 1 3

10. 2 3

and

and

and

and

and

3. 1 5

5. 6 3

7. 2 7

9. 6 8

11. 2 5

and

and

and

and

and

In questions #12-17, AB CD and these lines are cut by transversal EF at G and H,

respectively.

12. If 3m BGH x and 60m GHC , find x

13. If 2m EGA x and 5 54m GHC x , find the

value of x

14. If 90m AGH x and 3 10m DHG x , find x.

15. If m AGH is twice the measure of GHC , find

The m GHC .

16. If the ratio of the measures of BGH to GHD is

2:3, find m BGH and m GHD .

17. If 3 40m AGH x and 20m CHG x , find:

a. The value of x. b. m AGH c. m CHG d. m BGH

6

18. If AB CD , 5 40 , 4 30m and m , find the measures of the remaining

angles in the diagram given below.

Questions #19-20, as shown in the figure AB CD .

19. If 3 2 40m x and

7 3 20m x , find 3m .

20. If 4 4 10m x and 5 2 20m x ,

find the measure of the smaller of the two angles.

7

Algebra in Geometry (|| Lines Vocab) Date ____________

YOU MUST SHOW ALL WORK!

1. Two corresponding angles formed by parallel lines and a transversal are

2 8x and 3 20x . Find the two possible solutions for the angle measures.

2. The measures of two angles formed by parallel lines and a transversal

such that the angles are same-side interior angles have measures of

2 80x x and 4 50x . If both angles are greater than 50, find the

measures of the angles.

3. Alternate interior angles formed by parallel lines and a transversal are

represented by 23 40x and 2 24x . Find the measures of the angles.

8

4. Two alternate exterior angles formed by parallel lines and a transversal

are represented by 25 7 10x x and 2 10 2x x . Solve for x.

5. Corresponding angles formed by parallel lines and a transversal are

represented by 22 4 1x x and 24 3 4x x . Solve for x.

6. Same-side interior angles formed by parallel lines have measures of

2 20 100x x and 2 9 59x x . If x is a positive integer, find the measures

of the angles.

9

A

B

C

Triangles and Angles Notes Date ____________

Use your textbook [pages 93 – 95] to do the following.

____________________________________________________________________

1. A triangle is formed by three segments joining noncollinear points.

Use the diagram to answer the following:

a) Name the triangle: ________________

b) Name the vertices of the triangle: ________________________

c) Name the sides of the triangle: ___________________________

d) Name the angles of the triangle: _________________________

_____________________________________________________________________

2. Every triangle has two names for it. A triangle may be named by the

characteristics of the sides of the triangle.

a) A triangle with no congruent sides is called: _________________

Draw a diagram:

b) A triangle with two congruent sides is called: ________________

Draw a diagram:

c) A triangle with all three sides congruent is called: _____________

Draw a diagram:

3. A triangle may also be named by its angles.

a) A triangle with all acute angles is called an _______________ triangle.

Draw a diagram:

10

b) A triangle with one obtuse angle is called an _______________ triangle.

Draw a diagram:

c) A triangle with one right angle is called a ________________ triangle.

Draw a diagram:

d) A triangle with all angles congruent is called an __________________

triangle.

Draw a diagram:

______________________________________________________________________

Theorem:

The sum of the measure of the angles of a triangle is __________________

degrees.

Theorem:

The measure of an exterior angle of a triangle is equal to the sum of the

measures of the two remote interior angles of the triangle.

Diagram:

_______________________________________________________________________

Corollaries:

1) If two angles of one triangle are congruent to two angles of another

triangle, then the 3rd pair of angles are congruent.

2) Each angle of an equiangular triangle has measure of ___________

degrees.

3) In a triangle there can be at most one ______angle or one ______angle.

4) The acute angles of a right triangle are _________________ angles.

11

Exterior Angle Theorem Practice DATE: ____________

Questions #1-4, complete.

1. 40 , 30 , 1 ______

2. 45 , 35 , 1 ______

3. 50 , 1 85 , ______

4. 50 , 1 75 , ______

m A m B m

m A m B m

m A m m B

m B m m A

Questions #5 – 8, state two angles whose measures have a sum equal to

the measure of the given angle.

5. 9

6. 2

7. 3

8. 4

Questions #9 – 12, complete.

9. 2 45 , 3 60 , 5 _______

10. 3 70 , 4 80 , 1 _______

11. 1 140 , 3 55 , 4 _______

9. 3 84 , 5 110 , 2 _______

m m m

m m m

m m m

m m m

Questions #13-16, find the measures of 6, 7, 8and .

13. 6 5 17, 7 2 , 8 35

14. 6 4 80, 7 , 8 10

15. 6 3 7, 7 40 , 8 32

1 116. 6 15, 7 21 , 8 50

2 2

m x m x m

m x m x m x

m x m x m x

m x m x m x

12

Triangle Angle Sum WS DATE: ___________

1. If the measures of the angles of a triangle are represented by 30 ,x

4 30 , 10 30x and x , the triangle must be

a. obtuse b. right c. scalene d. isosceles

2. In an isosceles triangle, the measure of a base angle is 65 . Find the

number of degrees in the measure of the vertex angle.

3. If the measures of three angles of a triangle are represented by

, 2 20x x , and 3 10x , then the triangle is

a. equilateral b. right c. obtuse d. isosceles

4. The measure of a base angle of an isosceles triangle is 4 times the

measure of the vertex angle. The number of degrees in the vertex angle is

a. 20 b. 30 c. 135 d. 36

5. If two angles of a triangle measure 48 42and , the triangle is

a. acute b. isosceles c. obtuse d. right

6. The measures of the angles of a triangle are in the ratio 1:3:5, the

number of degrees in the measure of the smallest angle is

a. 10 b. 20 c. 60 d. 180

7. If the measures of the angles of a triangle are represented by

, 3 6 , 2 6x x and x , find the value of x.

13

8. In the accompanying diagram of ABC , AC is extended to D,

, , ,DEF BEC AFB 50m B , 25m BEF and 65m ACB . What is m D?

a. 55 b. 45 c. 40 d. 50

9. One angle of a triangle measures 30 . If the measures of the other two

angles are in the ratio 3:7, the measure of the largest angle of the triangle

is ___________________.

10. In QRS , , 8 40m Q x m R x , and 2m S x . What type of

triangle is QRS ?

a. obtuse b. right c. acute d. isosceles

11. If the base angle of an isosceles triangle measures 50 , what is the

number of degrees in the vertex angle?

12. The measures of the angles of a triangle are represented by

4 , 40 , 2x x and x . Find the value of x.

13. If the angles of a triangle are represented by , 3 20 , 6x x and x ,

the triangle must be

a. obtuse b. acute c. right d. isosceles

14

Polygons & Angles Notes Date: _________

Polygon:

Convex vs. Concave

Regular Polygon:

________________________________________________________________________

Fill in the table below.

Note: To find the sum of the interior angles, draw all diagonals from one

vertex as shown below.

Number of

Sides

Name of

Polygon

Sum of the

measures of

the interior

angles

Sum of the

measures of

the exterior

angles

Number of

diagonals

from 1 vertex

of polygon

3

4

5

6

7

8

9

10

n

15

Complete the 5 questions below with your partner.

1. If the sum of the measures of the interior angles of a polygon is 1800,

how many sides does the polygon have?

2. How would you find the measure of each interior angle of a regular

polygon with n sides?

3. How would you find the measure of each exterior angle of a regular

polygon with n sides?

4. A certain regular polygon has 30 sides. Find the measure of each

interior and each exterior angle of the polygon.

5. In a certain regular polygon, the measure of each interior angle is

equal to the measure of each exterior angle. How many sides does the

polygon have?

16

Polygon Angle Formulas

Sum of the measures of the interior

angles of a polygon with n sides

Sum of the measures of the exterior

angles of a polygon with n sides

Each interior angle of a regular

polygon with n sides

Each exterior angle of a regular

polygon with n sides

Show any calculations or diagrams used in determining your answers for

each question.

Find the sum of the interior angles of a polygon with the given number of

sides.

1. 14 sides 2. 18 sides

_______________________________________________________________________

Find the sum of the measures of the exterior angles of the given polygons.

3. nonagon 4. 22-gon

_______________________________________________________________________

Find the measure of one interior angle of a regular polygon with the given

number of sides.

5. 10 sides 6. 20 sides

________________________________________________________________________

Find the measure of one exterior angle of each regular polygon given

below.

7. dodecagon 8. 24 gon

17

Find the number of sides of a polygon whose interior angles have the

given number as the sum of their measures.

9. 1800 degrees 10. 2340 degrees

__________________________________________________________________

Find the measure of one exterior angle of a regular polygon if the sum of

the measures of the interior angles is the given number.

11. 1260 degrees 12. 3240 degrees

__________________________________________________________________

13. Find the measure of the sixth interior angle of a hexagon if the

measures of the other five angles are 70, 90, 140, 150, and 160.

___________________________________________________________________

14. The measure of an interior angle of a regular polygon is x+17 and the

measure of an exterior angle is 3x-9. How many sides does the polygon

have?

18

Algebra in Geometry Date ____________

SHOW ALL WORK ON SEPARATE PAPER!

1. The angles in an obtuse triangle are 2 45x , 2 95x , and 3 46x . Find the

measure of the angles of the triangle.

2. Base angles of an isosceles triangle are 27 10 6x x and 2 8 50x x . If x

is a positive integer, find the measure of the VERTEX angle of the triangle.

3. The angles of a triangle are represented by 22 30x , 23 7 55x x , and 25 4 98x x . Solve for x.

4. Two acute angles in a right triangle are 2 3 13x x and 15 180x . Find

the measures of the angles.

5. The measures of the angles in a hexagon are represented by 22 15 50x x

, 20 120x , 2 10 200x x , 22 20x x , 25 5 500x x , and 22 10 250x x . Solve

x.

6. Two acute angles in a right triangle are 26 30 20x x and 22 7 55x x . If

x is a positive integer, find the measures of the angles in the triangle.

7. The measures of two same-side interior angles formed by parallel lines

and a transversal have measures of 2 108x and 4 60x . If the larger angle

is more than three times the smaller angle, find the measures of the angles.

8. Two alternate interior angles formed by parallel lines and a transversal

are represented by 2 10 41x x and 8 40x . Find the measures of the

angles.

9. Base angles of an isosceles triangle are 2 2 18x x and 6 30x . Find the

measure of the VERTEX angle of the triangle.

10. The angles in a triangle are represented by 24 20x , 26 5 60x x , and 24 99x . If x is a positive number, find the measures of the angles of the

triangle.

19

11. Two acute angles in a right triangle are 210 40x x and 22 4 48x x . If

x is negative, find the measures of the angles to the nearest degree.

12. Two alternate exterior angles formed by parallel lines and a transversal

are represented by 2 3 20x x and 12 30x . If the angles are not right

angles, find their measures.

13. The measures of the angles of a pentagon are 2 100x , 2 100x , 2 2 150x x , 2 3 50x x , and 2 110x . If x is an integer, find the measures

of the angles of the pentagon.

14. Two acute angles in a right triangle are 2 5 30x x and 2 30x . If x is

a positive number, find the measures of the angles in the triangle.

20

Parallel Line Proofs WS Date _____________

1. Given: 1 2 2. Given: 1 2

Prove: ||l n Prove: ||l n

Statement Reason Statement Reason

1) 1 2 1) 1 2

__________________________________ ____________________________________

2) ||l n 2) ||l n

__________________________________ ____________________________________

3. Given: ; ||AB n l n 4. Given: || ; ||k l l n

Prove: AB l Prove: ||k n

Statement Reason Statement Reason

1) ; ||AB n l n 1) || ; ||k l l n

_________________________________ ___________________________________

2) 2 is right angle 2) 1 2

_________________________________ ___________________________________

3) 1 2 3) 2 3

_________________________________ ____________________________________

4) 1 is right angle 4) 1 3

_________________________________ ____________________________________

5) AB l 5) ||k n

_________________________________ ____________________________________

l

n

3

2

1 l

n

1

2

3

1 l

n 2

A

B

k

l

n

1

2

3

m

21

5. Given: 1 & 2 are supplementary

Prove: ||l m

Statement Reason

1) 1 & 2 are supplementary

________________________________________________________________________

2) 01 2 180m m

________________________________________________________________________

3) 01 3 180m m

________________________________________________________________________

4) 1 2 1 3m m m m

________________________________________________________________________

5) 2 3m m

________________________________________________________________________

6) 2 3

________________________________________________________________________

7) ||l m

________________________________________________________________________

6. Given: || ; ||l n m n

Prove: ||l m

Statement Reason

1) || ; ||l n m n

________________________________________________________________________

2) 1 3

________________________________________________________________________

3) 2 3

________________________________________________________________________

4) 1 2

________________________________________________________________________

5) ||l m

l

m

1

2

3

l

m

n

1

2

3

p

22

7. Given: ;m l p l

Prove: ||p m

Statement Reason

1) ;m l p l

________________________________________________________________________

2) & are right anglesADC BED

________________________________________________________________________

3) ADC BED

________________________________________________________________________

4) ||p m

________________________________________________________________________

A

C

B

D

F

E

G H

l

m p

23

More Proofs with Parallel lines DATE: ___________

Complete the proofs below on separate paper.

1. Given: 1 4 2. Given: QT bisects PQS ; 1 4

Prove: l m Prove: QT RS

1

3. Given: 2 3 ; QR QS 4. Given: 2 3 ; QT bisects PQS

Prove: QT RS Prove: QT RS

5. Given: 5 7 ; 6 8

Prove: a c

24

k1

32

ED A

C

B

k1

32

ED A

C

B

k1

32

ED A

C

B

BE DBA

3 1

CD BE

BE DA

CD DA

1 2

3C

BE DA

CD DA

25

54

3

2

1

B C

AM N

87

65

4 3

21AB

CD

E

F

MN BC

1 2 3 180m m m

MN BC

1 4; 3 5

4 BAN

4 180m m BAN

1 180m m BAN

2 5m BAN m m

2 3m BAN m m

1 2 3 180m m m

AB CD EF

1 6 180m m

AB CD

AB CD EF

1 6 180m m

5 6

5 6 180m m

1 6 5 6m m m m

1 5m m

1 5

AB CD

26

4

3

2

1

PR

Q

L M

3x 30 x

5 4x

QL QM

QL LP

PQR 's

3 4 's

2 4 's

1 3

1 2

QP QR 's

LM PR 's

QLM

QL QM

LM PR

PQR

27

8

7

65

4

3

21

S

T

V

Q

U

R

8 7 4m x 2 2 17m x

4 3 11m x 5 7

23 2x x 22 25 100x x

x

AC BD02 52m

6, 7m m 8m

6 5 4m x 7 30m x 8 20m x