Download GPS Geometry Summer Packet Name

GPS Geometry Summer Packet
Name ----------------
In order to complete this packet, we have included the geometry postulates, theorems, properties
and definitions that we expect you to know from middle school. Refer to them as you answer the
questions in this packet.
Look at this figure and fill in the blank for questions 1 to 6:
T
1. Name a ray that points up:
_
2. Name a ray that goes to the right:
_
3. Name 3 collinear points:
_
4. Name 2 lines:
----------------
5. Name a plane:
_
6. Name 3 line segments using point D:
_
Look at the following figure to name the parts requested in questions 7 to 10.
)(
A
L..
7. Name the red line segment:
_
8. Name the green ray:
_
I
I
9. Name 3 non-collinear points:
_
10. Name two rays that intersect at B:
_
Draw the figure described in each question. Label all the parts based on the question.
11. line m and line a that intersect at point C.
12. Points R, T, A and C where 3 points are collinear.
13. 3 lines AB, DE and LM, so that AB intersects DE, AB intersects LM,
-
-
but DE does not intersect LM.
14. Draw RS such that RS bisects AB at point M.
15. RS and RT are the same ray.
16. AB and Be are collinear.
17. Points X, Y, Q and Z are collinear and XY and QZ do not intersect.
Using this number line, answer questions 18 to 27:
B
•
-16
v
•
-14
18.
BA
19.
RN
20.
VX
21.
VP + PN
R
•
-9
A
•
-5
P
•
-4
T
o
w
J
•
•
2
6
N
•
12
c
x
15
23"
22. The midpoint of TN
23. AX
24. The coordinate of the midpoint of VN
25. If P is a midpoint of a segment that has
endpoint V, find the coordinate of the other
endpoint.
26. TJ -PR
27. BV + VP + PN
Segment Addition and Midpoint. In the following questions, all points are collinear.
28. IfR is between A and L, write an equation that can be used to solve problems.
29. IfRA= 10 and AL= 36, find RL.
30. If R is also a midpoint, write another equation that can be used
to solve problems.
31. If R is a midpoint and RA is 25, find RL.
)
32. X is between A and C; C is between T and L; L is not between
A and C, and T is not between X and C but is between A and X. Draw the picture.
AT
=
x + 5; TC
=
3x + 9; CL
=
33. If X is the midpoint of AL, and AT
find x and the measure of AL.
12; AL
=
=
62; find x
2x -6; TX
=
4x +8; XL
x=
_
=
8x - 24,
--:-
_
AL= -------34. IfQR = 5, RT = 8 and TV
(All points are collinear)
=
12, find QV.
Use this figure to answer questions 35 to 40:
35. Name an angle with vertex D.
_
36. Name the sides of the blue angle.
_
37. Name the vertex of the blue angle.
_
38. Name 3 angles in
t::.
DBE.
_
39. Name the red triangle. ---------40. Name all the angles with vertex B.
_
Draw and label the following based on the information in each question:
41. Two angles Ll and L2 where the vertex of Ll is in the interior of L2.
42. A triangle ABC, where LBCA is an obtuse angle.
Types of Angle Pairs:
Adjacent Angles, Complementary angles, Supplementary Angles, Linear Pair, Vertical Angles.
Use the following figure and the definitions of the types of angles to answer questions 43 to 52.
43. mL7 =47°, find mLIO.
44.
mL16
= 36°find
mL19.
_
_
45. mL20 =41°, find mL19 and mL16.
_
46. Name a linear pair.
47. Name 2 vertical angles.
_
48. Name a pair of supplementary angles.
_
49. Name a pair of adjacent angles.
50. Are
51.
L16
and
L17 adjacent
_
angles?
mL16 = 46° and mL17 = 67°,
mL12 and mL19.
mLll =
mL18 =
mL13=
_
mL19=
find
Explain:
mz n, mL18, mL14, mL13,
mL14 =
mL12 =
_
and
_
52. Are angles 8 and 2 supplementary angles?
_
Use the following figure for question 53 to 55.
53. Name a pair of vertical angles.
54. If
mL4 = 58°, mL3 = 49°
_
and
m.Z: = mLIO, find
all angle measures. Mark on the figure
below:
r
I
I
55. If
mL4=
4m + 1 and
mL7=
3m + 7, find
mLs.
56. The measures of one angle of a linear pair is 3 times the measure ofthe other angle. Find the
measures of the two angles.
57. Classify the angles in number 56.
_
Angle Bisector - a ray that cuts an angle into two congruent angles.
Given that
CT
58. mLACT
bisects
=
LACR,
63, find
use the following figure for questions 58 to 61.
mLACR
-----------
59. mLRCT=29,find mLACT
60.
mLACT = 4x
61.
mLACT=
+ 27 and
7x + 19 and
Find
mLACR =
_
134, find x
mLRCT = 9x +1,
---------
find x--------
mLACR =--------------
Using the following figure, identify the following angle pairs as alternate interior angles, corresponding
angles, alternate exterior angles, vertical angles, or consecutive interior angles, or none of these.
'"'
62. L 1 and L 16
63. L 6 and L 12
64. L 4 and L5
65. L 11 and L9
66. L 7 and L2
67. L 13 and L6
68. L 15 and L9
69. L 8 and L9
70.
L 9 and L2
71. L5 and L8
;
I
I
Using the following figure, identify the following angle pairs as alternate interior angles, corresponding
angles, alternate exterior angles, vertical angles, or consecutive interior angles, or none of these.
72. L. 1 and L.16
73. L. 6 and L.12
74. L. 4 and L. 5
75. L. 11 and L. 9
76. L. 7 and L.2
77. L. 13 and L. 6
78. L. 15 and L. 9
79. L. 8 and L.9
80.
L. 9 and L.2
81. L. 5 and L. 8
Using the following figure, identify the following angle pairs as alternate interior angles, corresponding
angles, alternate exterior angles, vertical angles, or consecutive interior angles, or none of these.
82. L 1 and L 18
83. L 6 and L 12
84. L 18 and L 13
85. L 11 and L6
86. L 7 and L 11
87. L 13 and L6
88. L 15 and L8
89.
L 8 and L9
90.
L 9 and L7
91. L 5 and L9
92. Using the following figure, determine the angle measures given that m L 8 = 78°
and m L 9 = 27°, find all the numbered angles. Mark on the drawing.
().,II b
b
Using the following figure, determine the angle measure and the rule that supports it. Give the letter of
the rule from the following choices:
A.
B.
C.
D.
E.
F.
If 2 parallel lines are cut by a transversal,
If 2 parallel lines are cut by a transversal,
If 2 parallel lines are cut by a transversal,
If2 parallel lines are cut by a transversal,
Vertical Angle Theorem
Linear Pair Postulate
then the alternate interior angles are congruent.
then the alternate exterior angles are congruent.
then the corresponding angles are congruent.
then the consecutive interior angles are supplementary.
CLI//;
q
,0
"
/:1..
b
Given: m L 3
=
89° and m L 9
=
101°
93. Find mL 14 -:-------94. Find mL 2
95. Find mL 12
96. Find mL 5
Rule: -------Rule:
Rule:
Rule:
97. Find m L 10
Rule:
98. FindmL16
Rule:
99. Find m L 8
Rule:
100. Using the following figure and the following given information, m z; 5 = 2x -16, m z; 1 = x + 12,
m L 14 = 15y + 8, m L 12 = 4y' + 1, find x and y and mark the measures of all angles.
a.
101. Using the following figure, find all missing angle measures. Mark the figure.
J IImIl /7
i: /IjJ
t
p
II b
How to tell when lines are parallel:
Alternate Interior Angles are Congruent
Alternate Exterior Angles are Congruent
Corresponding Angles are Congruent
Consecutive Interior Angles are Supplementary
Slope is another way to tell if lines are parallel or perpendicular on a coordinate plane. Parallel Lines
have the same slope.
Perpendicular Lines have slopes that when multiplied together they will equal -1.
Exercises:
Determine in each drawing which lines are parallel. If there are no lines parallel based on the
information shown, write not parallel.
tL
102.
6
103.
(104)
104.
e,
t
c.
105.
0
(10
e.
106.
107.
_
~----~--_+--~m
~---:....~_1_~
108.
_
~--~~~h
I
I
J...
109.
_
110.
_
~-_P4__I_----_?
111.
e
_
115
~
In the next set of problems, find x in each drawing that will allow the given lines to be parallel.
112.
_
113.
_
114.
_
115.
_
116.
_
117.
118.
_
_
119.
_
120.
_
121.
_
In this set of problems, determine if AB is parallel, perpendicular or neither to CD.
122. A (-1,1)
B(2,3)
C (2,2)
123. A (-2,1)
B(1,-2)
C (-1,-1) D (3,3)
124. A (-2,2)
B(3,2)
C(2,-I)
125. A (-1,2)
B(1,-2)
126. A (-2,1)
B(1,5)
C(1,-2)
C (5,2)
127. A (1,3)
B(6,3)
C(3,-I)
128. A (5,2)
B(8,3)
C (3,8)
D (5,4)
D(2,4)
D(2,-I)
D (2,-2)
D (-2,-1)
D (6,9)
8th Grade Geometry Terms, Postulates and Theorems
UNDEFINED TERMS:
Point - no measurable length, no measurable width, no measurable depth; named
with one capital letter.
Line - measurable length, no measurable width, no measurable depth; named
with 2 capital letters (representing the names of2 points on the line) and a symbol ofa
line above the letters, or with one cursive small letter. Example: AB
Plane - measurable length, measurable width, no measurable depth; named with
3 capital letters (representing the names of 3 points not on the same line) and the word
plane in front. Example: Plane ABC
DEFINITIONS:
Collinear - points, segments or rays on the same line.
Coplanar - points, lines, segments, or rays on the same plane.
Line segment or segment - that portion of a line with 2 distinct endpoints;
Named with 2 capital letters (the names of the endpoints) and a symbol ofa segment
above the letters. Example: AB
Ray - that portion of a line with 1 endpoint; named with 2 capital letters (the first
being the endpoint of the ray, and the second being a point on the ray) and a symbol of a
ray above the letters. Example: AB
Opposite Rays - Two rays that share the same endpoint and extend in opposite
directions to form a line.
Length of a Segment - a real number representing the length of a segment written with 2 capital letters (the names of the endpoints). Example: AB = 5 the length
of segment
AB is 5 units.
Midpoint - a point that bisects a segment. If A is the midpoint of CD, then
CA=AD.
Angle - the figure created when 2 rays share the same endpoint; Named with 3
letters (the first letter is the name of a point on one side, the second letter is the name of
the shared endpoint, and the third letter is the name of a point on the other side) and an
angle sign in front of the letters or a number with the angle sign in front of the number.
Example: L.ABC
Sides of an angle - the two rays that form the angle.
Vertex of an angle - the shared endpoint of the 2 rays that form the angle.
Interior of an angle - the points that are between the sides of the angle.
Exterior of an angle - the points that are not in the interior or on the angle.
Measure of an angle - a real number from 0 to 180, representing the measure of
the angle using a protractor. Named with a non-capital letter m in front of the name of
the angle. Example: m L.ABC = 80°, the measure of angle ABC is 80 degrees.
Adjacent Angles - Two angles that share a side, and share a vertex, but do not
share any interior points.
Acute Angle - an angle that measures less than 90°.
Right Angle - an angle that measures 90°.
Obtuse Angle - an angle that measures more than 90° but less than 180°.
Straight Angle - an angle that measures 180°.
Vertical Angles - two non-adjacent angles formed when 2 lines intersect.
Complementary Angles - two angles whose measures add up to 90°.
Supplementary Angles - two angles whose measures add up to 180°.
Bisector - to cut into 2 congruent parts.
Linear Pair - 2 adjacent angles whose non-shared sides are opposite rays.
Parallel Lines - coplanar lines that do not intersect. (Symbol!!)
Skew Lines - non-coplanar lines that do not intersect. .
Perpendicular Lines - two coplanar lines that meet to form a right angle.
(Symbol .1)
Congruent - to have the same measure.
Parallel Planes - two planes that do not intersect.
Transversal - a line that intersects 2 or more lines.
Corresponding Angles - the angles formed when 2 lines are cut by a transversal
and correspond to the same position at the points of intersection.
Alternate Interior Angles - the angles formed when 2 lines are cut by a
transversal and are between the lines, on opposite sides of the transversal and at different
points of intersection.
Alternate Exterior Angles - the angles formed when 2 lines are cut by a
transversal and are outside the lines, on opposite sides of the transversal and at different
points of intersection.
Consecutive Interior Angles - the angles formed when 2 lines are cut by a
transversal and are between the lines, on the same side of the transversal and at different
points of intersection.
Triangle - A closed figure with 3 angles and 3 sides; Named with 3 capital letters
representing the names of the vertices of the triangle and a symbol of a triangle in front of
the letters. Example: e.ABC
Scalene Triangle - a triangle with 3 non-congruent sides.
Isosceles Triangle - a triangle with 2 congruent sides.
Equilateral Triangle - a triangle with 3 congruent sides.
Acute Triangle - a triangle with 3 acute angles.
Right Triangle - a triangle with 1 right angle.
Obtuse Triangle - a triangle with 1 obtuse angle.
Equiangular Triangle - a triangle with 3 congruent angles.
PROPERTIES OF EQUALITY AND CONGRUENCE:
Reflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = a.
Transitive Property of Equality: If a = band b = c, then a = c.
Substitution Property of Equality: If a = b and a + c = d, then b + c = d.
Addition Property of Equality: If a = b, then a + c = b + c.
Subtraction Property of Equality: If a = b, then a - c = b - c.
Multiplication Property of Equality: If a = b, then a . c = b . c.
a
b
c
c
Division Property of Equality: If a = b, then - = - .
t
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:
-
AB
If AB
AB
~ CD, then CD ~. AB
--
-
If AB ~ CD and CD ~ XY, then AB ~ . XY
POSTULATES AND THEOREMS:
If point A is between points C and D, then AC + AD
= AD.
All right angles are congruent.
If 2 angles are supplementary
congruent to each other.
to the same or congruent angles, then they are
If 2 angles are complementary
congruent to each other.
to the same or congruent angles, then they are
If 2 angles form a linear pair then they are supplementary.
Postulate).
(Linear Pair
Vertical angles are congruent.
If 2 parallel lines are cut by a transversal, then the corresponding
congruent.
angles are
If 2 parallel lines are cut by a transversal, then the alternate interior angles
are congruent.
If 2 parallel lines are cut by a transversal, then the alternate exterior angles
are congruent.
If 2 parallel lines are cut by a transversal, then the consecutive interior
angles are supplementary.
If 2 lines intersect to form a linear pair of congruent angles, then the lines are
perpendicular.
If 2 sides of 2 adjacent acute angles are perpendicular,
complementary .
If 2 lines are perpendicular,
then the angles are
then they intersect to form 4 right angles.
In a coordinate plane, if 2 lines are parallel, then their slopes are equal. Any
2 vertical lines are parallel.
f
I
In a coordinate plane, if 2 lines are perpendicular, then the product of their
slopes is -1. Vertical and horizontal lines are perpendicular.
If 3 parallel lines intersect 2 transversals, then they divide the transversals
proportionally.
Pythagorean
Theorem
2
0
+ b2 = c2
The sum of the measures of the angles of a triangle is 180°.
I
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