Download Geometry: Mr. Miller`s Class – Lesson Ch 1.1 Identify Points, Lines

Geometry: Mr. Miller’s Class – Lesson 1
Ch 1.1 Identify Points, Lines, and Planes
Diagram #1:
Undefined terms: words that do not have a formal definition
Point – no dimension, represented by a dot
Line – one dimension, represented by a line with two arrowheads, extends without end
Through any two points, there is exactly one line. Any two points on a line will name it.
Plane – two dimensions, represented by the shape of a floor or wall, extends without end
Through any three points not on the same line, there is exactly one plane.
Three points that are not all on the same line can name a plane.
Collinear points – lie on the same line
Coplanar points – lie in the same plane
⃡ . Name a point that is not coplanar with points Q, S,
Guided Practice: Using diagram 1, Name two other names for 𝑆𝑇
and T.
Diagram #2:
Defined Terms: terms that can be described using known words, point or line.
Line segment – consists of endpoints A and B and all points on ⃡𝐴𝐵 that are between A and B.
⃡ that lie on the same side of A as B.
Ray – AB consists of the endpoint A and all points on 𝐴𝐵
⃡ between A and B, then 𝐶𝐴 and 𝐶𝐵 are opposite rays.
If C lies on 𝐴𝐵
Guided Practice: Give another name for ̅̅̅̅
𝐸𝐹 .
Are 𝐻𝐽 and 𝐽𝐻 the same ray? Are 𝐻𝐽 and 𝐻𝐺 the same ray? Explain
Diagram #3:
Intersection – set of points the figures have in common
Guided Practice: Sketch two different lines that intersect a plane at the same point. Use the diagram at the right. Name
the intersection of PQ and line k. Name the intersection of plane A and plane B. Name the intersection of line k and plane
A.
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Geometry: Mr. Miller’s Class – Lesson 2
Ch 1.2 Use Segments and Congruence
Postulate or axiom – a rule that is accepted without proof
Theorem – rule that can be proved
Postulate 1: Ruler Postulate - the points on a line can be matched one to one with the real numbers. The real number that
corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute
value of the difference of the coordinates of A and B (pg 9).
Diagram 1:
̅̅̅ to the nearest tenth of a centimeter.
Ex 1: Measure the length of ̅𝑆𝑇
Postulate 2: Segment Addition Postulate - If B is between A and C, then AB + BC = AC. If AB + BC + AC, then B is
between A and C (pg 10).
Ex 2: Apply the Segment Addition Postulate
Diagram #3:
Ex 3: Use the diagram to find GH.
Congruent segments (≅) – line segments that have the same length (direction does not matter)
Diagram #4:
̅ ≅ LM
̅̅̅̅
Ex 4. Determine whether JK
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Geometry: Mr. Miller’s Class – Lesson 3
Ch 1.3 Use Midpoint and Distance Formulas
Midpoint – point that divides (or bisects) the segment into two congruent segments.
Segment bisector – point, ray, line, line segment, or plane that intersects the segment at its midpoint.
Discovering the Midpoint Formula, What if the points were on a number line? P was located at 2 and Q was located at 8.
Where would M be located?
Diagram #1:
Ex 1: In the skateboard design, ̅̅̅̅̅
𝑉𝑊 bisects ̅̅̅̅
𝑋𝑌 at point T, and XT = 39.9
cm. Find XY.
Diagram #2:
Ex. 2: Point M is the midpoint of ̅̅̅̅̅
𝑉𝑊. Find the length of ̅̅̅̅̅
𝑉𝑀.
The Midpoint Formula - the coordinates of the midpoint of a segment are the averages of the x-coordinate and of the ycoordinates of the endpoints.
Midpoint Formula =
̅̅̅̅ are R(1, -3) and S(4,2). Find the
Ex. 3: a) The endpoints of 𝑅𝑆
coordinates of the midpoint M
Find and endpoint
b) The midpoint of ̅̅̅
𝐽𝐾 is M(2,1). One endpoint is J(1,4). Find the
coordinates of endpoint K
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Geometry: Mr. Miller’s Class – Lesson 4
Ch 1.3 Cont. Use Midpoint and Distance Formulas
The Distance formula is rally just a fancy Pythagorean Theorem
For Your Notebook:
Distance Formula – formula for computing the distance between two points in a coordinate plane. If A(x1, y1) and B(x2,
y2) are points in a coordinate plane, then the distance between A and B is …
̅̅̅̅ with endpoints R ( 2,3) and
Ex. 4: What is the approximate length of 𝑅𝑆
S(4, -1)?
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Geometry: Mr. Miller’s Class – Lesson 5
Ch 1.4 Measure and Classify Angles
Angle (symbol ∠ ) – consists of two different rays with the same endpoint
Sides – the rays are the sides of the angle
Vertex – the endpoint of the angle
Diagram: #1
Doodle an Example:
Ex. 1: Name the three angles in the diagram #1.
Protractor – measures an approximate angle – units: degrees ( º )


Postulate 3: Protractor Postulate – The measure of ∠𝐴𝑂𝐵 is equal to the absolute value of the difference between the
real numbers for 𝑂𝐴 and 𝑂𝐵 (pg 10).
Classifying Angles --- acute, right, obtuse, and straight
Diagram: #2
Ex. 2: Use the diagram to find the measure of the indicated angle.
(Classify)
a) ∠KHJ
b) ∠GHK
c) ∠GHJ
d) ∠GHL
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Postulate 4: Angle Addition Postulate – If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum of
the measures of ∠RSP and ∠PST (pg 24).
Diagram #3
Ex. 3: Given that m∠LKN = 145º, find m∠LKM and m∠MKN
Congruent angles – two angles that have the same measure
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Angle bisector – ray that divides an angle into two angles that are congruent. In the diagram below, 𝐵𝐷 bisects ∠ABC.
So, ∠ABC ≅ ∠DBC and m∠ABD = m∠DBC.
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Geometry: Mr. Miller’s Class – Lesson 6
Ch 1.5 Describe Angle Pair Relationships
Doodle Examples →
Complementary angles – the sum of the measures of two angles equals 90º.
Supplementary angles – the sum of the measures of two angles equals 180º.
Adjacent angles – two angles that share a common vertex and side, but have no common interior points.
Diagram 1:
Ex. 1: In the figure, name a pair of complementary angles, a pair of
supplementary angles, and a pair of adjacent angles.
Ex. 2: a) Given that ∠1 is a complement of ∠2 and m∠1 = 68º, find m∠2
Diagram 2:
b) Given that ∠3 is a supplement of ∠4 and m∠4 = 56º, find m∠3
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Angle Pairs
Linear pair – when noncommon sides are opposite rays from two adjacent angles
Vertical angles – when sides are from two pairs of opposite rays
Diagram 3:
Ex. 4: Identify all of the linear pairs and all of the vertical angles in
diagram 3.
Diagram 4:
Ex. 5: Two angles form a linear pair. The measure of one angle is 5
times the measure of the other. Find the measure of each angle.
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For your Notebook:
Things you can conclude from a diagram
 All points shown are coplanar
 Points A, B, and C are collinear, and B is between A and C.
⃡ , 𝐵𝐷, 𝐵𝐸 intersect at point B
 𝐴𝐶
 ∠DBE and ∠EBC are adjacent angles, and ∠ABC is a straight angle.
 Point E lies in the interior of ∠DBC
Things you CANNOT conclude
 ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐵𝐶
 ∠DBE ≅ ∠EBC
 ∠ABD is a right angle
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Geometry: Mr. Miller’s Class – Lesson 7
Ch 1.6 Classify Polygons
Doodle →
A figure that lies in a plane is called a plane figure. A polygon is a closed plane figure with the following properties.
1) It is formed by three or more lines segments called sides.
2) Each side intersects exactly two sides, one at each endpoint, so that no tow sides with a common endpoint are
collinear.
Each endpoint of a side is a vertex (vertices) of the polygon. A polygon can be named by listing the vertices in
consecutive order.
Doodle →
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Polygons
 Convex- no line that contains a side of the polygon contains a point in the interior of the polygon
 Concave – one that is not convex also called nonconvex
Diagram 1:
Ex. 1: Tell whether the figure is a polygon and whether it is convex or
concave.
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Classifying polygons – named by the number of its sides
n-gon --- n is the number of a polygon’s sides, may be used to name the polygon
equilateral polygon – all sides are congruent
equiangular polygon – all angles in the interior of the polygon are congruent
regular polygon – convex polygon that is both equilateral and equiangular
Diagram 2:
Ex. 2: Classify the polygon by the number of sides. Tell whether the
polygon is equilateral, equiangular, or regular. Explain your reasoning.
Diagram 3:
Ex. 3 A table is shaped like a regular hexagon. The expression show
represent side lengths of the hexagonal table. Find the length of a side.
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Geometry: Mr. Miller’s Class – Lesson 8
Ch 1.7 Find Perimeter, Circumference, and Area
Recall:
Perimeter – distance around a figure --- units (m) or (ft)
Circumference – distance around a circle --- units (m) or (ft)
Area – amount of surface covered by a figure --- units (m2)
For Your Notebook
Square, Rectangle, Triangle, Circle
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Diagram #1:
Ex. 1: Find the perimeter and area of the rectangular basketball court.
Diagram #2:
Ex. 2: You are ordering circular cloth patches for your soccer team’s
uniforms. Find the approximate circumference and area of the path
shown.
Diagram #3:
Ex. 3: Triangle QRS has vertices Q(1, 2), R(4, 6), and S(5, 2). What is
the approximate perimeter of triangle QRS?
Ex. 4: An ice-resurfacing machine is used to smooth the surface of the
ice at a skating rink. The machine can resurface about 279 square yards
of ice in one minute. About how many minutes does it take the machine
to resurface a rectangular skating rink that is 200 feet long and 90 feet
wide?
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