Geometry Chapter 3 2013-2014 Parallel lines
Proving Parallel lines
Triangles
Angles of Polygons
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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!!!!
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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
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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.
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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.
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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:
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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.
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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
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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.
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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
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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?
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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
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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?
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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.
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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.
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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
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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
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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
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