Geometry
Chapter 1 “Basics of Geometry”
Assign
Section
Homework
08/25
Review
Worksheet #1
08/26
Intro to TI-Nspire
Worksheet #2
08/25
1.1 Patterns and Inductive
Reasoning
Worksheet #3
08/29
1.2 Points, Lines, and Planes
Worksheet #4
08/30
1.3 Segments and Their Measure
Worksheet #5
08/31
Review (1.1 – 1.3)
Worksheet #6
09/01
QUIZ (1.1 – 1.3)
Worksheet #7
09/02
TI-Nspire Activity
Worksheet #8
09/06
1.4 Angles and Their Measure
Worksheet #9
09/07
1.5 Segment and Angle Bisectors
Worksheet #10
09/08
Review (1.4 – 1.5)
Worksheet #11
09/09
1.6 Angle Pair Relationships
Worksheet #12
09/12
Review (1.4 – 1.6)
Worksheet #13
09/13
QUIZ (1.4 – 1.6)
Worksheet #14
In order to receive full credit, assignments must be neat, complete, on time, and all
work must be shown. Homework is your practice time, make it worthwhile. Assignments
are subject to change.
1.1 Patterns of Inductive Reasoning
Target goals: Find and describe patterns.
Use inductive reasoning to make real-life conjectures.
Find a counterexample.
Geometry developed when people began recognizing and describing patterns. In this section, you will be
describing visual and number patterns. You can use patterns to help make predictions.
VOCABULARY
Conjecture:
Inductive Reasoning:
Counter Example:
DESCRIBING A VISUAL PATTERN:
Ex 1: Sketch the next figure in the pattern.
Ex 2: Sketch the next figure in the pattern.
DESCRIBING A NUMBER PATTERN:
Sometimes, patterns allow you to make accurate predictions.
Ex 3: Describe a number pattern and then predict the next number.
a. 17, 15, 12, 8, …
b. 48, 16,
16 16
,
,…
3 9
c. 2, 4, 16, 256, . .
d. 4, -20, 100, -500,.
INDUCTIVE REASONING
1) Look for a pattern
2) Make a conjecture
3) Verify the conjecture (Are there counter examples?)
FINDING A COUNTEREXAMPLE
*To prove that a conjecture is true, you need to prove it is true in ALL cases.
*To prove that a conjecture is false, you need to provide a single
.
Ex 3: Find a counterexample to show the conjecture false.
Conjecture:
If the difference of two numbers is odd, then the greater of the numbers
must also be odd.
Counterexample: ___ - ___ = ___
The conjecture is
.
Ex 4: Find a counterexample to show the conjecture false.
Conjecture: The difference of two positive numbers is always positive.
Counterexample:
The conjecture is
.
1.2 Points, Lines, and Planes
Target goals: Understand and use the basic undefined terms and defined terms of geometry.
Sketch the intersections of lines and planes.
A definition uses known words to describe a new word. In geometry, some words, such as point, line, and
plane are undefined terms. Although these words are not formally defined, it is important to have a general
agreement about what each word means.
UNDEFINED TERMS:
Point:
Line:
Plane:
DEFINED TERMS:
Collinear Points:
Coplanar Points:
Line Segment:
Ray:
Opposite rays:
K
Ex 1: a. Name three points that are collinear.
b. Name four points that are not coplanar.
c. Name three points that are not collinear.
N
R
M
L
P
Ex 2: a. Draw 3 collinear points A, B, C.
b. Draw point D not collinear with ABC.
c. Draw AB .
d. Draw ray BD .
e. Draw segment CD .
f. Name opposite rays.
Ex 3: Draw a line. Label three points on the line and name a pair of opposite rays.
SKETCHING INTERSECTIONS OF LINES AND PLANES:
Two or more geometric figures
if they have one or more points in common. The
of the figures is the set of points the figures have in common.
Ex 4: Draw two intersecting lines.
Ex 6: Sketch two planes that intersect.
Describe their intersection.
Ex 5: Sketch a line that intersects a plane in one
point.
Ex 7: Sketch two planes that do not intersect.
E
F
Ex 8: Answer True or False for the following:
a)
b)
c)
d)
e)
f)
Points A, B, and C are collinear. _____
Points A, B, and C are coplanar. _____
Point F lies on DE . _____
DE lies on plane DEF. _____
BD and DE intersect. _____
BD is the intersection of plane ABC and plane DEF. ____
D
C
A
B
1.3 Segments and Their Measures
Target goals: Use segment postulate.
Use the Distance Formula to measure distances.
Postulates:
vs.
Theorems:
SEGMENT ADDITION POSTULATE:
If B is between A and C, then
.
A
If
B
C
, then B is between A and C.
Ex 1: RS = TU, ST = 9, RU = 33
R
S
T
A
B
a) Find RS
b) Find SU.
C
U
T
U
Ex 2: Y is between X and Z. Find the distance between points X and Z if the distance
between X and Y is 12 units and the distance between Y and Z is 25 units.
USING THE DISTANCE FORMULA
Distance Formula
If A (x1, y1) and B (x2, y2) are points in a coordinate plane,
then the distance between A & B is….
AB =
Ex 3: Find the length of the segments.
AC =
AD =
A (-1, 1)
C (3, 2)
D (3, -5)
Definition: Congruent Segments
If two segments are congruent, then
If two segments have
If AB = CD, then
.
, then
.
.
Ex 4: a) In example 3, is AC  AD ?
b) If DE is congruent to AC in example 3, then DE =
.
1.4 Angles and Their Measures
Target Goals: Use angle postulates.
Classify as acute, right, obtuse, or straight.
Angle:
B
A
C
Measure of an Angle: To indicate the measure of ∠A we write _________.
Angles are measured in _______________.
Congruent Angles: Angles that have the same measure are ____________________.
BAC ____ DEF
D
B
A
30
C
E
30
F
Adjacent Angles: Share a common _________ and ____________,
but have no __________ in common.
Ex 1: Name the adjacent angles in the figure.
X
Y
W
Z
Interior and Exterior of an Angle:
P
Q
R
ANGLE ADDITION POSTULATE:
R
If P is in the interior of RST , then
.
P
S
T
Ex 2: Find the measure of the following angles:
b) If mWXZ  48 and mYXZ  31
then mWXY  _______ .
a) mQRS  _______
Q
W
T
R
19o
23o
X
S
Ex 3: If the mABC  88 then, solve for x.
Y
Z
A
D
(2x)°
B
(x - 2)°
C
CLASSIFYING ANGLES:
An angle that measures greater than 0 and less than 90 is called an
An angle that measures 90 is called a
angle.
An angle that measures greater than 90 and less than 180 is called an
An angle that measures 180 is called a
angle.
angle.
angle.
1.5 Segment and Angle Bisectors
Target Goals: Bisect a segment.
Use the midpoint formula.
Bisect an angle.
A
B
M
Midpoint:
If a point is a midpoint of a segment, then it
.
If a point
, then it is the midpoint.
Bisect:
A
M
Segment Bisector:
THE MIDPOINT FORMULA
Midpoint Formula
If A ( x1, y1) and B ( x2, y2) then
B
M=
A
Ex 1: Find the coordinates of the midpoint of AB with endpoints A(-2, 3) and B(5, -2).
Ex 2: The midpoint of JK is M(1, 4). One endpoint is J(-3, 2). Find the coordinates of the other
endpoint.
B
BISECTING AN ANGLE:
B
D
A
C
Angle Bisector:
If a ray is an angle bisector, then it
If a ray
.
,then it is the angle bisector.
Ex 3: RT bisects  QRS.
Given that m  QRS= 60 , what are
the measures of  QRT &  TRS?
Ex 4: KM bisects  JKL.
The measures of the two congruent
angles are 2 x  7  and 4 x  41  .
Find the measures of  JKM and
 MKL.
Ex 5: Name the  parts.
Ex 6: Find the measure of ∠RST.
b g b
g
A
R
50o
S
B
C
D
T
1.6 Angle Pair Relations
Target Goals: Identify vertical angles and linear pairs.
Identify complementary and supplementary angles.
VERTICAL ANGLES AND LINEAR PAIRS
Linear Pair:
1
**The sum of the measures on angles that form a linear pair is
.
4
2
3
Vertical Angles:
Theorem: If two angles are vertical angles, then _____________________________.
Ex 1:
In the diagram shown,  1 has a measure of 60 .
Find the m  2 and m  3.
1
2
4
105
3
Vertical Pairs:
Linear Pairs:
m∠1 = 60o
m∠2 =
m∠3 =
m∠4 =
Ex 3: Solve for x.
3x°
Ex 2: Solve for x.
(2x + 5)°
b2 x  11g
COMPLEMENTARY AND SUPPLEMENTARY ANGLES:
Complementary Angles: If two angles are _________________________, then their sum is _______.
If the sum of two angles is ________, then __________________________.
Complementary angles may either be adjacent
or nonadjacent.
4
3
1
2
Each angle is the ________________ of the
other.
Complementary
Adjacent
Complementary
Nonadjacent
Supplementary Angles: If two angles are _________________________, then their sum is _______.
If the sum of two angles is ________, then __________________________.
7
5
8
Supplementary angles may either be
adjacent or nonadjacent.
6
Supplementary
Adjacent
Supplementary
Nonadjacent
Ex 4: Given that m  A = 55  , find it’s
complement and it’s supplement.
Each angle is the ________________ of the
other.
Ex 5:  X and  Y are supplementary.
Find the measure of each angle if
m  X = 6x – 1 and m  Y = 5x – 17.
Ex 6:  P and  Q are complementary. The measure of  Q is 4 times the measure of  P. Find the
measure of each angle.