Download Postulate: A statement of fact.

Name:
Geometry
Unit 1: Essentials of Geometry
Extra Practice Worksheets (with answer keys) are available at popkoskimath.pbworks.com
These are optional and are not counted as homework. You are encouraged to complete these on your own
and/or after school to practice your skills and prepare for tests.
1-2: Points, Lines, and Planes
1-3: Measuring Segments
TERM
Point
DIAGRAM
DEFINITION
Indicates a location. Named w/ capital letter.
Line
A set of continuous points that extends endlessly in both
directions. Named with a lower case letter or by two points.
Plane
A flat surface with no thickness. Extends indefinitely in all
directions. Formed by three noncollinear points. Could also
be named with a capital letter.
Line Segment
Part of a line consisting of two points. Named with its two
endpoints.
Ray
Part of a line consisting of an endpoint and the set of points
on other side. Named by endpoint then another point on line.
Opposite Rays
Rays that have the same endpoint and lie on the same line.
Collinear Points
Points that are on the same line.
Coplanar Points
Points that are on the same plane.
Angle
Union of two rays that have the same endpoint (vertex).
Named with three points, the vertex, or label interior with a
number.
Acute Angle
An angle whose measure is greater than 0 and less than 90
Right Angle
An angle whose measure equal to 90
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Obtuse Angle
An angle whose measure is greater than 90 and less than
180
Reflex Angle
An angle whose measure is greater than 180 and less than
360
Straight Angle
An angle whose measure is equal to 180
Parallel Lines
Lines that do not intersect and have equal slopes
Skew Lines
Lines that are in different planes that do not intersect, but are
NOT parallel.
Perpendicular
Lines
Lines that intersect forming right angles and have negative
reciprocals for slopes
Linear Pair
When the exterior sides of adjacent angles lie on opposite
rays.
Midpoint
Divides a segment into two congruent segments.
Bisector
(Angles/Segments)
Divides an angle or segment into two congruent angles or
segments.
Segment/Angle
Addition Postulate
If point X is between points A and B, then AX+XB = AB
Congruent
-Shapes: same size and shape
-Segments: same length
-Angle: same measure
-A figure with no dimension cannot be measured (no length)
-A one-dimensional figure has a length.
-A two-dimensional figure has a length and a width
-A three-dimensional figure has a length, width, and height
Dimension
If OX is between the sides of <AOB, then
mAOX+XOB = mAOB
2
Intersection of two
lines
1 point
Intersection of two
planes
1 line
Intersection of line
and plane
1 point
Practice: Using the figure at right, answer the following questions.
For #1 – 8, write TRUE or FALSE
1.
____________ Points A, D, and E are coplanar.
2.
____________ Points E, H, and B are coplanar.
3.
____________ Points B, C, F, and E are coplanar.
4.
____________ The intersection of plane ADB and plane CHG is CG .
5.
____________ AE&BF are coplanar.
6.
____________ EF&DC are coplanar.
7.
____________ AB& GF are coplanar.
8.
____________ GH & point B are coplanar.
9.
Shade the plane that contains points B, C, and H.
10.
Darken the intersection of planes ABF and ADC.
Practice:
Practice:
If EG = 59, what are EF and FG?
Q is the midpoint of PR . What are PQ, QR and PR?
3
Postulate: A statement of fact.
Important Postulates and Theorems
To Remember!
Use the diagram below to answer the following.
Give two other names for line PQ:
Give two other names plane R:
Name three points that are collinear:
Name four points that are coplanar:
Give two other names for line ST:
Name a point that is not coplanar with points Q, S, and T:
Draw a sketch of the given question then solve.
1. Line RS bisects PQ at point R. Find RQ if PQ = 14 cm.
2. Point T is a midpoint of UV. Find UV if UT = 4 ½ yards.
In the diagram, M is the midpoint of the segment. Find the indicated length.
3. Find LN
4. Find MR
5. Name two planes that intersect in the given line: AB
HW 1-2: p. 16 #15, 18, 25, 32 – 36, 40 – 45
HW 1-3: p. 24 #20, 40, 43
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1-4: Measuring Angles
1-5: Exploring Angle Pairs
Angles:
Name the angle at the right.
______________ ______________ ______________
Note: You may also see this referred to as A, but we will NOT be using this label in proofs in this class!
Angle Measures:
1.
Classify each of the following angles according to their measures
2.
Write an equation/inequality to express the possible measures
________________________
________________________
________________________
________________________
How To Mark Congruent Angles  Given: 1  3 and 2  4
Practice:
If mRQT = 155, what are mRQS and mTQS?
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Angle Pairs
Definition
Example
Adjacent Angles:
Vertical Angles:
Complimentary Angles:
Supplementary Angles:
Practice:
Practice:
KPL and JPL are a linear pair.
What conclusions can be made from the diagram?
What are the measures of KPL and JPL?
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Important Postulates and Theorems To Remember!
VERTICAL ANGLES ARE CONGRUENT
1. DEF is a straight angle. What is the mDEC
and mCEF?
2. Are BFD and CFD adjacent angles? Explain.
3. Are AFB and EFD are vertical angles? Explain.
Additional Examples
1) Name the three angles in the diagram. Why can’t you just call this angle B???????
_______ , or _______
_______ , or _______
_______ , or _______
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2) Name the three angles in the diagram. Why can’t you just call this angle G???????
_______ , or _______
_______ , or _______
_______ , or _______
3) What type of angles do the x and y axis form in the coordinate plane? _________________
4) Given that m  GFJ = 155o, find
a) m  GFH __________
b) m  HFJ __________
5) Given that  VRS is a right angle, find
a) m  VRT __________
b) m  TRS __________
6) Identify all pairs of congruent angles in the diagram. If m  P = 120o, what is m  N?
 _______   _______
 _______   _______
m  N = ______o
7) WY bisects  XWZ, and m  XWY = 29o.
Find m  XWZ _________o
8
8) Identify all pairs of congruent angles in the diagram. If m  B = 135o, what is m  A?
 _______   _______
 _______   _______
m  A = ______o
9) KM bisects  LKN and m  LKM = 78o.
Find m  LKN _________o
10) Two angles form a linear pair. The measure of one angle is 4 times the measure of the
other. Find the measure of each angle.
11) The exterior sides of two adjacent angles are opposite rays. The measure of one angle
is 3 times the measure of the other. Find the measure of each angle.
12) In the diagram:
a) All points shown are _________________.
b) Points A, B, and C are ________________.
c)  DBE and  EBC are ______________ angles.
d)  ABC is a ____________ angle.
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13) Find m<EGT and m<TGC if EG is perpendicular to CG, and m<EGT = 7x + 2 and
m<TGC = 4x.
14) Find the value of each angle.
HW 1-4: p. 31 – 32 #18 – 23, 30
HW 1-5: p. 38 – 39 #25 – 28, 32 – 36
Extra Practice: Angle Pair Relationship Worksheet
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1-6 (Day 1): Basic Constructions
Refer to my website popkoskimath.pbworks.com for videos and printable instructions!
Constructions:
_________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________
1. Congruent Segments
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2. Congruent Angles
Proof: This construction works by creating two congruent triangles. The angle to be copied has the
same measure in both triangles (Side-Side-Side Triangle Congruence Theorem).
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3. Perpendicular Bisector
Proof: This construction works by effectively building 4 congruent triangles that result in right angles
being formed at the midpoint of the line segment.
13
4. Angle Bisector
Proof: This construction works by effectively building two congruent triangles. The angle has the
same measure in both triangles (Side-Side-Side Triangle Congruence Theorem).
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1-6 (Day 2): Applied Constructions
1. a) Construct an equilateral triangle whose sides are all the same length as XY .
b) Using this equilateral triangle, construct a 30 angle.
2. Construct AB so that AB = MN + OP
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3. Construct KL so that KL = OP  MN.
4. Construct A so that mA = m1 + m2
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5. Construct B so that mB = m1 - m2
6. Construct C so that mC = 2m2.
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7. Construct a segment whose length is 14 AB.
HW 1-6: Finish 1-6 Class Notes
Extra Practice: Constructions WS (Angle Bisector)
Constructions WS (Perpendicular Bisector)
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1-7 (Day 1): Midpoint and Distance in the Coordinate Plane
Midpoint of a Line Segment:
Midpoint Formula:
Find the coordinates of the midpoint of AB .
Practice:
Practice:
EF has endpoints E(7, 5) and F(2, -4).
What are the coordinates of the midpoint of EF ?
The midpoint of CD is (-2, 1). One endpoint is
C(-5, 7). What are the coordinates of D?
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Length of a Line Segment (Distance):
Distance Formula:
Find the length of AB .
Practice:
What is the distance between U(-5, 4) and V(3, 2)?
Express your answer in simplest radical form.
Practice:
In circle O, a diameter has an endpoints at (-5, 4)
and (3, -6). What is the length of the diameter?
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Find the coordinates of the midpoint of the segment with the given endpoints.
1. L(4, 2) and P(0, 2)
2. G(-2, -8) and H( -3, -12)
Using the given point R and the midpoint M, find the other endpoint.
3. R(6, 0), M(0, 2)
4. R( 3, 4), M(3, -2)
Horizontal & Vertical Lines
Find the length of line segment GF: _________
Find the length of line segment HJ: _________
So, GF _____ HJ
Find the length of the line segment. Round to the nearest tenth of a unit. If there are two segments
compare the lengths.
5.
6. RS: R(5, 4), S(0, 4)
TU: T(-4, -3), U(-1, 1)
HW 1-7 (Day 1): p. 54 – 55 #10, 13, 16, 19, 22, 25, 28, 51
Extra Practice: Midpoint WS 1, Midpoint WS 2, Midpoint WS 3
Distance WS 1, Distance WS 2
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1-7 (Days 2&3): Partitioning a Line Segment
1. AB is a directed line segment from A(1,3) to B(8,3).
What are coordinates of point P that partitions
the segment in the ratio 1:2?
2. AB is a directed line segment from A(7,9) to B(1,3).
What are coordinates of point P that partitions the
segment in the ratio 1:3?
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3. AB is a directed line segment from A(0, 1) to B(8,3).
What are coordinates of point P that partitions the
segment in the ratio 1:3?
4. The endpoints of XY are X (2, -6) and Y (-6, 2).
What are the coordinates of point P on XY such that
XY is ¾ of the distance from X to Y?
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*5. AB is a directed line segment from A(2, 2) to B(4,7).
What are coordinates of point P that partitions the
segment in the ratio 1:2?
Partition Point Formula: The section formula is just a fancier version of the midpoint formula. If a line
segment has endpoints (x1, y1) and (x2, y2), and a partition point P will separate the line segment into a ratio of
m:n, then students should plug the numbers into the section formula to find the coordinates of P.
Watch for starting and ending points! x1 and y1 are where you begin…x2 and y2 are where you end!
m=
n=
x1 =
y1 =
x2 =
y2 =
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Partition Point Formula Practice:
1. Find L on JK such that JL: LK = 4:1, when J = (-3, 5) and K = (1, -10).
m=
n=
x1 =
y1 =
x2 =
y2 =
2. Find R on MP such that MR:RP = 5:2, when M = (-10, 8) and P = (-2, 11).
m=
n=
x1 =
y1 =
x2 =
y2 =
3. Point C lies on directed line segment from A(5, 16) to B (-1, 2) and partitions the segment into a
ratio of 1 to 2. What are the coordinates of C?
m=
n=
x1 =
y1 =
x2 =
y2 =
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More Practice:
1. Points A(-2, -3) and B(8, 2) are the endpoints of AB. What are the coordinates of point C on AB such
that AC is 2/5 the length of AB?
2. LM is the directed line segment from L (-4, 1) to M (5, -5). What are the coordinates of the point that
partitions the segment in the ratio of 2 to 3?
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3. Points X (6, 4) and Y(-4, -16) are the endpoints of XY. What are the coordinates of point Z on XY
such that XZ is 4/5 the length of XY?
m=
n=
x1 =
y1 =
x2 =
y2 =
*4. Points X (6, 4) and Y(-4, -16) are the endpoints of YX. What are the coordinates of point Z on YX
such that YZ is 4/5 the length of YX?
m=
n=
x1 =
y1 =
x2 =
y2 =
HW 1-7 (Days 2&3): Midpoint Practice WS/Finish 1-7 Class Notes
Extra Practice: Directed Line Segment WS
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1-8: Perimeter, Circumference and Area (With Review of Polygons)
Classifying Polygons
Polygon
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Convex Polygon
____________________________________________________________________________________
Concave Polygon
____________________________________________________________________________________
n-gon _____________________________________________________________________________________
Equilateral Polygon _________________________________________________________________________
Equiangular Polygon ________________________________________________________________________
Regular Polygon____________________________________________________________________________
Name
Sides
Names of Polygons
Name
Sides
3
11
4
12
5
n
6
7
8
9
10
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1. Tell whether the figure is a polygon and whether it is convex or concave.
a)
b)
c)
2. The head of a bolt is shaped like a regular hexagon. The expressions shown represent side lengths of
the hexagonal bolt. Find the length of a side.
3. The expressions (4x + 8)o and (5x – 5)o represent the measures of two congruent angles . Find the
measure of an angle.
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1. Find the dimensions of the garden,
path.
2. What is the circumference and area of the including the
circle in terms of ? What is the
circumference and area of the circle to the
nearest tenth?
3. What is the area and perimeter of EFG?
4. Graph quadrilateral JKLM with vertices
J(-3, -3), K(1, -3), L(1, 4) and M(-3,1).
What is the perimeter of JKLM?
5. You are designing a poster that will be 3 yds wide and 8 ft high. How much paper do you need to make the
poster? Leave your answer in terms of feet.
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6. What is the area of the figure at the right? All angles are right angles.
7. Triangle JKL has vertices J(1, 6), K(6, 6) and L(3, 2). Find the approximate perimeter and
area of triangle JKL. Round all answers to the nearest tenth.
8. The base of a triangle is 24 feet. Its area is 216 square feet. Find the height of the triangle.
HW 1-8: p. 64-65 #14, 22, 32-33, 41-42
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Unit 1 Common Core Test Questions
1.
2.
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3.
4.
5.
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Unit 1 Common Core Test Questions Answer Key
1.
2.
3.
4.
5.
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