Geometry Honors Notes Name:________________________ Class Pd. _____ Date: ________ Teacher: _________
1-1 Understanding Points, Lines, and Planes
Objectives
1) Identify, name, and draw points, lines, segments, rays, and planes.
2) Apply basic facts about points, lines, and planes.
Vocabulary
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undefined term
line
collinear
segment
ray
postulate
point
plane
coplanar
endpoint
opposite rays
The most basic figures in geometry are undefined terms, which cannot be defined by using other figures.
The undefined terms ______, ______, and _______ are the building blocks of geometry.
TERM
Point
Line
Plane
Example 1: Naming Points, Lines, and Planes
A. Name four coplanar points
B. Name three lines.
Use the diagram to name two planes.
NAME
DIAGRAM
DEFINITION
NAME
DIAGRAM
Segment
Endpoint
Ray
Opposite rays
Example 2: Drawing Segments and Rays
A. a segment with endpoints M and N.
B. opposite rays with a common endpoint T.
Draw and label a ray with endpoint M that contains N.
Define a
POSTULATE:
Postulates Points, Lines, Planes
1-1-1
1-1-2
1-1-3
Example 3: Identifying Points and Lines in a Plane
Name a line that passes through two points.
Name a plane that contains three noncollinear points.
Intersection of Lines and Planes
1-1-4
1-1-5
Example 4: Representing Intersections
A. Sketch two lines intersecting in exactly one point.
B. Sketch a figure that shows a line that lies in a plane.
Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane.
When two lines intersect, their intersection is a _____________.
1-2 Measuring Segments
Objectives
1) Use length and midpoint of a segment.
2) Construct midpoints and congruent segments.
Vocabulary
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Coordinate
distance
length
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between
congruent segments
bisect
segment bisector
A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on
a ruler. The number is called a _____________________. The following postulate summarizes this concept.
1-2-1 Ruler Postulate
The distance between any two points is the _____________ _________ of the difference of the coordinates. If the
coordinates of points A and B are a and b, then the distance between A and B is _____________ or ____________.
The distance between A and B is also called the ____________ of AB, or AB.
Example 1: Finding the Length of a Segment
Find each length
Find
Find
Define
BC
XY
and
AC
and
XZ
Congruent segments
In the diagram, PQ = RS, so you can write PQ  RS. This is read as “segment PQ is congruent to segment RS.” Tick
marks are used in a figure to show congruent segments
1-2-2 Segment Addition Postulate:
Diagram:
Example 3A: Using the Segment Addition Postulate
.
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Y is between X and Z, XZ = 3, and XY =
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M is between N and O. Find NO
. Find YZ.
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E is between D and F. Find DF.
Define Midpoint Formula
Diagram/Example:
Example 4: Recreation Application
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The map shows the route for a race. You are at X, 6000 ft from the first checkpoint C. The second checkpoint D
is located at the midpoint between C and the end of the race Y. The total race is 3 miles. How far apart are the
2 checkpoints?
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You are 1182.5 m from the first-aid station. What is the distance to a drink station located at the midpoint
between your current location and the first-aid station?
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D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF.
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S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT.
1-3 Measuring Angles
Objectives
1) Name and classify angles.
2) Measure and construct angles and angle bisectors.
Vocabulary
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angle
vertex
interior of an angle
exterior of an angle
measure
degree
DEFINITIONS
acute angle
right angle
obtuse angle
straight angle
congruent angles
angle bisector
Diagram/Examples:
Angle
Vertex
Interior Angle
Exterior Angle
Example 1: Naming Angles
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A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name
three of the angles.
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Write the different ways you can name the angles in the diagram.
The_____________________ of an angle is usually given in degrees. Since there are 360° in a circle, one degree is
_______ of a circle. When you use a protractor to measure angles, you are applying the following postulate.
Postulate 1-3-1 Protractor Postulate
You can use the Protractor Postulate to help you classify angles by their
measure. The measure of an angle is the _______________ ____________ of
the difference of the real numbers that the rays correspond with on a
protractor.
If OC corresponds with c and OD corresponds with d,
mDOC = ________
or
________
Types of Angles
Sketch and Describe each type below.
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Example 2
Use the diagram to find the measure of each angle. Then classify each as acute,
right, or obtuse.
a. BOA
b. DOB
c. EOC
Define Congruent Angles
Diagram/Example:
1-3-2 Angle Addition Postulate:
Diagram:
Example 3: Using the Angle Addition Postulate
mDEG = 115°, and mDEF = 48°. Find mFEG
Define Angle Bisector
mXWZ = 121° and mXWY = 59°. Find mYWZ.
Diagram/Example:
Example 4: Finding the Measure of an Angle
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KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM.
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QS bisects PQR, mPQS = (5y – 1)°, andb mPQR = (8y + 12)°. Find mPQS.
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JK bisects LJM, mLJK = (-10x + 3)°, and mKJM = (–x + 21)°. Find mLJM.
1-4 Pairs of Angles
Objectives
1) Identify adjacent, vertical, complementary, and supplementary angles.
2) Find measures of pairs of angles.
Vocabulary
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adjacent angles
linear pair
vertical angles
complementary angles
supplementary angles
Define Adjacent Angles
Diagram/Example:
Define Linear Pair
Diagram/Example:
Example 1: Identifying Angle Pairs
Tell whether the angles are only adjacent, adjacent and form a linear pair, or not
adjacent.
AEB and BED
DEC and AEB
AEB and BEC
5 and 6
7 and 8
7 and SPU
Define Complementary Angles
Diagram/Example:
Define Supplementary Angles
Diagram/Example:
Example 2: Finding the Measures of Complements and Supplements
Find the measure of each of the following.
complement of E =
supplement of F =
Example 3: Using Complements and Supplements to Solve Problems
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An angle is 10° more than 3 times the measure of its complement. Find the measure of the complement.
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An angle’s measure is 12° more than ½ the measure of its supplement. Find the measure of the angle.
Example 4: Problem-Solving Application
Light passing through a fiber optic cable reflects off the walls of the cable in such a way that 1 ≅
2, 1 and 3 are complementary, and 2 and 4 are complementary.
If m1 = 47°, find m2, m3, and m4.
What if...? Suppose m3 = 27.6°. Find m1, m2, and m4.
Define Vertical Angles
Diagram/Example:
Example 5: Identifying Vertical Angles
Name the pairs of vertical angles:
1-5 Using Formulas in Geometry
Objective
1) Apply formulas for perimeter, area, and circumference.
Vocabulary
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perimeter
area
base
height
Define
Perimeter
Define
Area
Ex 1: Find the perimeter and area of each figure.
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diameter
radius
circumference
pi
Find the perimeter and area of a square with s = 3.5 in.
Ex 2: Find the amount of fabric used to make four rectangles. Each rectangle has a length of 6 ½ in. and a width of
2 ½ in.
Define Radius
Define Diameter
Define Circumference
Example 3: Finding the Circumference and Area of a Circle
Find the circumference and area of a circle with radius 14m.
1-6 Midpoint and Distance in the Coordinate Plane
Objectives
1) Develop and apply the formula for midpoint.
2) Use the Distance Formula and the Pythagorean Theorem to find the distance between two points.
Vocabulary
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Define
coordinate plane
leg
hypotenuse
Coordinate Plane
Diagram/Example:
Example 1:
Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).
Example 2: Finding the Coordinates of an Endpoint.
S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T.
Example 3.
Find EF and GH. Then determine if EF  GH.
Example 4: Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R
to S.
R(3, 2) and S(–3, –1)
Example 5: Sports Application
A player throws the ball from first base to a point located between third base and home plate and 10 feet from third
base.
What is the distance of the throw, to the nearest tenth?
1-7 Midpoint and Distance in the Coordinate Plane
Objectives
1) Identify reflections, rotations, and translations.
2) Graph transformations in the coordinate plane.
Vocabulary
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transformation
preimage
image
Define
transformation
Define
preimage
Define
image
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reflection
rotation
translation
Sketch
Example 1A: Identifying Transformation
Identify the transformation. Then use arrow notation to describe the transformation.
Example 2: Drawing and Identifying Transformations
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A figure has vertices at A(1, –1), B(2, 3), and C(4, –2). After a transformation, the image of the figure has
vertices at A'(–1, –1), B'(–2, 3), and C'(–4, –2). Draw the preimage and image. Then identify the
transformation.
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A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has
vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the
transformation.
Example 3: Translations in the Coordinate Plane
Find the coordinates for the image of ∆ABC after the translation (x, y)  (x + 2, y - 1). Draw the image.
Example 4: Art History Application
The figure shows part of a tile floor.
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Write a rule for the translation of hexagon 1 to hexagon 2.
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Use the diagram to write a rule for the translation of square 1 to square 3.
Add-