Notes- Unit 1 - Points, Lines, and Planesx
Transcript of Notes- Unit 1 - Points, Lines, and Planesx
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Geometry Chapter 1 – Points, Lines, Planes, and Angles
***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** ____ Algebraic Equations Review
____ Keystone Vocabulary Writing Assignment
____ (1-1) Points, Lines, and Planes Page 9-10 # 13-20, 21, 22, 25, 26, 27, 30-46
____ (1-2) Linear Measure – Day 1 Page 17 # 12-15, 22 – 27
____ (1-2) Linear Measure – Day 2 Page 17-19 # 28-39, 58, 62
____ (1-3) Distance – Day 1 Page 25 # 13-30
____ (1-3) Midpoints – Day 2 Page 26 # 31-42
____ Take Home Test on Sections 1-1 through 1-3
____ (1-4) Angle Measure – Day 1 Page 33-34 # 4-6, 12-24
____ (1-4) Angle Measure – Day 2 Page 34 # 25-33
____ (1-4) Angle Measure – Day 3 Page 34 # 34 – 39, 52-60, 61, 63, 65
____ (1-5) Angle Relationships – Day 1 Page 42 # 11 – 25 (skip # 17)
____ (1-5) Angle Relationships – Day 2 1-5 Practice WS
____ (1-5) Angle Relationships – Day 3 Page 42 # 17, 27-30, 31-35, 37
____ (1-6) Polygons – Day 1 Page 49 – 50 # 12-25, 29-34
____ (1-6) Polygons – Day 2 Page 49 – 50 #26 – 28
____ Chapter 1 Review WS
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Date: ______________________
Section 1 – 1: Points, Lines, and Planes Notes
A Point: is simply a _______________. Example:
� Drawn as a ________.
� Named by a ______________ letter.
� Words/Symbols:
A Line: is made up of ____________ and has no thickness or __________.
� Drawn with an _________________ at each end.
� Named by the _____________ representing two points on the line or a lowercase
script letter.
� Points on the same _______ are said to be _____________.
� Words/Symbols: Example:
A Plane: is a _______ surface made up of ____________.
� Drawn as a ____________ 4-sided figure.
� Named by a _____________ script letter or by the letters naming three
___________________ points.
� Points that lie on the same plane are said to be _______________.
� Words/Symbols: Example:
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Example #1: Use the figure to name each of the following.
a.) Name a line that contains point P.
b.) Name the plane that contains lines n and m.
c.) Name the intersection of lines n and m.
d.) Name a point not on a line.
e.) What is another name for line n.
f.) Does line l intersect line n or line m? Explain.
Example #2: Draw and label a figure for the following relationship.
a.) Point T lies on WR. b.) AB intersects CD in plane Q at point P.
Example #3:
a.) How many planes appear in this figure?
b.) Name three points that are collinear.
c.) Are points A, B, C, and D coplanar? Explain.
d.) At what point do DBsuur
and CAsuur
intersect?
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CRITICAL THINKING
1.) Why do chairs sometimes wobble?
Include the following in your answer:
• an explanation of how the chair legs relate to points in a plane, and
• how many legs would create a chair that does not wobble.
2.) Complete the figure below to show the following relationship:
Lines a, b, and c are coplanar and lie in plane Q. Lines a and b intersect at
point P. Line c intersects line b at point R, but does not intersect line a.
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Date: ______________________
Section 1 – 2: Linear Measure Notes – Part 1
Measure Line Segments
� A line segment, or ______________, is a measurable part of a line that consists of
two points, called _________________, and all of the points between them.
� A segment with endpoints A and B can be named as _______ or _______.
� The length or _______________ of AB is written as ________.
Example #1: Use a metric ruler to draw each segment.
g.) Draw LM that is 42 millimeters long.
b.) Draw QR that is 5 centimeters long.
Example #2: Use a customary ruler to draw each segment.
a.) Draw DE that is 3 inches long.
b.) Draw FG that is 23
4 inches long.
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Calculate Measures
� Betweenness of Points: Point M is between points P and
Q if and only if P,Q, and M are ______________ and
__________________.
Example #4:
a.) Find LM. b.) Find XZ.
c.) Find DE.
d.) Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.
e.) Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21. Draw a picture!
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Date: ______________________
Section 1 – 2: Linear Measure Notes – Part 2
Example: Find the value of x and LM if L is between N and M, NL = 6x – 5,
LM = 2x + 3, and NM = 30. Draw a picture!
Measure Line Segments
Key Concept (Congruent Segments):
Two __________________ having the same Ex:
measure are __________________.
Symbol:
Example #1: Name all of the congruent segments found in the kite.
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Example #2: Find the measurement of RS.
Example #3: Use the figures to determine whether each pair of segments is congruent.
a.) ,AB CD b.) ,WZ XY
c.) ,HO HT d.) ,MH TH
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CRITICAL THINKING
1.) Explain the difference between a line and a line segment and why one of these can
be measured, while the other cannot.
2.) Refer to the figure to the right.
a.) Name three collinear points.
b.) Name two planes that contain points B and C.
c.) Name another point in plane DFA.
d.) How many planes are shown?
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Date: ______________________
Section 1 – 3: Distance Notes – Part 1
Distance Between Two Points
Key Concept (Distance Formulas):
� Number Line
� Coordinate Plane
The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by
d =
Example #1: Find the distance between E(-4, 1) and F(3, -1).
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Example #2: Use the number line to find QR.
Example #3: Use the number line to find CD.
Example #4: Use the number line to find AB and CD.
Example #5: Use the Distance Formula to find the distance between the following points.
a.) A(10, -2) and B(13, -7)
b.) X(-5, -7) and Y(-10, 7)
c.) G(-4, 1) and H(3, -1)
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Date: ______________________
Section 1 – 3: Midpoint Notes – Part 2
Midpoint of a Segment
Key Concept (Midpoint):
� The midpoint M of PQ is the point ___________________ P and Q such that
_____________________.
� Number Line: The coordinate of the midpoint
of a __________________ whose endpoints have
coordinates a and b is
Example #1: The coordinates on a number line of J and K are –12 and 16, respectively. Find the
coordinate of the midpoint of JK . Hint: Draw a number line!
Example #2: The coordinates on a number line of T and S are 5 and 8, respectively. Find the
coordinate of the midpoint of TS . Hint: Draw a number line!
� Coordinate Plane: The coordinates of the
_____________________ of a segment
whose endpoints have coordinates (x1, y1) and
(x2, y2) are
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Example #3: Find the coordinates of the midpoint of PQ for P(-1, 2) and Q(6, 1).
Example #4: Find the coordinates of the midpoint of GH for G(8, -6) and H(-14, 12).
Example #5: Find the coordinates of the midpoint of AB for A(4, 2) and B(8, -6).
Example #6: What is the measure of PR if Q is the midpoint of PR ? Segment Bisector: any segment, line, or plane that interests a
segment at its _______________
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CRITICAL THINKING
1.) Which equation represents the following problem?
Fifteen minus three times a number equals negative twenty-two. Find the number.
a.) 15 – 3n = -22
b.) 3n – 15 = -22
c.) 3(15 – n) = -22
d.) 3(n – 15) = -22
2.) Find the distance between points at (6, 11) and ( -2, -4).
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Date: ______________________
Section 1 – 4: Angle Measure Notes – Part 1
Measure Angles
� Degree: a unit of measure used in measuring
______________ and __________. An arc of a
circle with a measure of 1° is ___________ of the
entire circle.
� Ray: is a part of a ___________
� It has one ____________________ and extends
indefinitely in _________ direction.
� Symbols:
� Opposite Rays: two rays _________ and _________ such
that B is between A and C
Key Concept (Angle):
An angle is formed by two ______________________ rays that have a common
__________________.
� The rays are called ____________ of the angle.
� The common endpoint is the ______________.
� Symbols:
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� An angle divides a plane into three distinct parts.
� Points _____, _____, and _____ lie on the angle.
� Points _____ and _____ lie in the interior of the angle.
� Points _____ and _____ lie in the exterior of the angle.
Example #1:
a.) Name all angles that have B as a vertex.
b.) Name the sides of 5∠ .
c.) Write another name for 6∠ .
Example #2:
a.) Name all the angles that have W as a vertex.
b.) Name the sides of 1∠ .
c.) Write another name for WYZ∠ .
d.) Name the vertex of 4∠ .
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Date: ______________________
Section 1 – 4: Angle Measure Notes – Part 2
Measure Angles
Key Concept (Classify Angles):
RIGHT ANGLE: ACUTE ANGLE: OBTUSE ANGLE:
Model: Model: Model:
Measure: Measure: Measure:
Example #1: Measure each angle, then classify as right, acute, or obtuse.
a.) b.)
c.) d.)
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e.) f.)
Example #2: Measure each angle named and classify it as right, acute, or obtuse.
a.) TYV∠
b.) WYT∠
c.) TYU∠
d.) VYX∠
e.) SYV∠
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Date: ______________________
Section 1 – 4: Angle Measure Notes – Part 3
Congruent Angles
Key Concept (Congruent Angles):
Angles that have the same _____________________ are
congruent angles.
Arcs on the figure also indicate which angles are
___________________.
Example #1: State whether each pair of angles is congruent, and if so write a congruence statement.
a.) b.)
Example #2: Find the value of x and the measure of one angle.
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Angle Bisector: a _________ that divides an angle into _________ congruent angles.
Ex:
If PQuuur
is the angle bisector of ___________,
then _____________________.
Example #3: In the figure, QP and QR are opposite rays, and QT bisects RQS∠ .
a.) If 56 +=∠ xRQTm and 27 −=∠ xSQTm , find RQTm∠ .
b.) Find TQSm∠ if 1122 −=∠ aRQSm and 812 −=∠ aRQTm .
Example #4: In the figure, YU bisects ZYW∠ and YT bisects XYW∠ .
a.) If 1051 +=∠ xm and 2382 −=∠ xm , find 2∠m .
b.) If WYZm∠ =82 and 254 +=∠ rZYUm , find r.
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CRITICAL THINKING
1.) Mr. Lopez wants to cover the walls of his unfinished basement with pieces of
plasterboard that are 8 feet high, 4 feet wide, and ¼ inch thick. If the basement
measures 24 feet wide, 16 feet long, and 8 feet tall, how many pieces of plasterboard
will he need to cover all four walls?
2.) Each figure below shows noncollinear rays with a common endpoint.
a.) Count the number of angles in each figure.
b.) Describe the pattern between the number of rays and the number of angles.
c.) Make a conjecture of the number of angles that are formed by 7 noncollinear
rays and by 10 noncollinear rays.
d.) Write a formula for the number of angles formed by n noncollinear rays with a
common endpoint.
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Date: ______________________
Section 1 – 5: Angle Relationships Notes – Part 1
Pairs of Angles
Key Concept (Angle Pairs):
� Adjacent Angles: are two angles that lie in the same ____________, have a common
_____________, and a common ___________, but no common interior ____________
� Examples:
� Vertical Angles : are two non-adjacent angles formed by two __________________ lines
� Examples: Non-example:
� Linear Pair : a pair of ________________ angles whose non-common sides are opposite
__________.
� Example: Non-example:
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Example #1 : Name an angle pair that satisfies each condition.
a.) two angles that form a linear pair
b.) two acute vertical angles
c.) an angle supplementary to VZX∠
d.) two acute adjacent angles
Key Concept (Angle Relationships):
� Complementary Angles: two angles whose measures have a sum of ________
� Examples:
� Supplementary Angles: two angles whose measures have a sum of ________.
� Examples:
Example #2: Find the measures of two supplementary angles if the measure of one angle is 6 less
than 5 times the measure of the other angle.
Example #3: Find the measures of two complementary angles if the difference in the measures of the
two angles is 12.
Example #4: The measure of an angle’s supplement is 33 less than the measure of the angle. Find the
measure of the angle and its supplement.
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Date: ______________________
Section 1 – 5: Angle Relationships Notes – Part 2
Perpendicular Lines
� Lines that form right angles are _____________________.
Key Concept (Perpendicular Lines):
� Perpendicular lines intersect to form _________ right angles.
� Perpendicular lines intersect to form _________________
_______________ angles.
� ________________ and _________ can be perpendicular to lines
or to other line segments and rays.
� The right angle symbol in the figure indicates that the lines are ___________________.
� Symbol: _______ is read is perpendicular to.
Example #1: Find x so that KO HM⊥suur suuur
.
Example #2: Find x and y so that BE and AD are perpendicular.
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Assumptions:
Example #3: Determine whether or not each of the following statements can be assumed or not.
� All points shown are coplanar.
� P is between L and Q.
� PLPN ≅
� QPO∠ and OPL∠ are supplementary.
� PMPN ⊥
� L, P, and Q are collinear.
� LPMQPO ∠≅∠
� POPQ ≅
� PQLP ≅
� LMP∠ and MNP∠ are adjacent angles.
� LPN∠ and NPQ∠ are a linear pair.
� LPMOPN ∠≅∠
� ,,, POPNPM and LQ intersect at P.
Example #4: Determine whether each statement can be assumed from the figure below. Explain.
a.) 90m VYT∠ =
b.) TYW∠ and TYU∠ are supplementary
c.) VYW∠ and TYS∠ are complementary
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Date: ______________________
Section 1 – 5: Angle Relationships Extra Examples
Example #1: Two angles are complementary. One angle measures 24° more than the other. Find the measures of the angles. Example #2: Find the measures of two supplementary angles if the measure of one angle is 4 less than 3 times the measure of the other angle. Example #3: The measure of an angle’s supplement is 22 less than the measure of the angle. Find the measure of the angle and its supplement.
Example #4: Find the value of x so that ACsuur
and BDsuur
are perpendicular.
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CRITICAL THINKING
1.) A counterexample is used to show that a statement is not necessarily true. Find a
counterexample for the statement Supplementary angles for linear pairs.
2.) What kinds of angles are formed when streets intersect?
Include the following in your answer:
• the types of angles that might be formed by two intersecting lines, and
• a sketch of intersecting streets with angle measures and angle pairs identified.
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Date: ______________________
Section 1 – 6: Polygons Notes
Polygons
� A polygon is a ______________ figure whose sides are all segments.
� The sides of each angle in a polygon are called ___________ of the polygon, and the vertex of
each angle is a _____________ of the polygon.
� Examples:
� Polygons can be ________________ or ________________.
� Examples:
_____________________ ________________________
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Regular Polygon: a convex
polygon in which all the ________
are congruent and all the angles are
___________________.
Ex:
Example #1: Name each polygon by the number of sides. Then classify it as convex or concave, regular
or irregular.
a.) b.)
Perimeter
� The perimeter of a polygon is the sum of the _______________ of its sides, which are
_________________.
Example #2: Find the perimeter of each polygon.
a.) b.) c.)
Number of Sides
Polygon
3
quadrilateral
5
6
heptagon
octagon
9
decagon
12
n
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CRITICAL THINKING
1.) Refer to the figure to the right.
• Find the perimeter of pentagon LMNOP.
• Suppose the length of each side of pentagon LMNOP is doubled. What effect
does this have on the perimeter?
2.) Quadrilateral ABCD has a perimeter of 95 centimeters. Find the length of each side
if AB = 3a + 2, BC = 2(a – 1), CD = 6a + 4, and AD = 5a – 5.