Geometry
Chapter 1 “Basics of Geometry”
Assign / Section / Homework08/25 / 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.
- 17, 15, 12, 8, …b. 48, 16, , , … 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:
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.
Ex 2: a. Draw 3 collinear points A, B, C.
b. Draw point D not collinear with ABC.
c. Draw .
d. Draw ray .
e. Draw segment .
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 5: Sketch a line that intersects a plane in one point.
Ex 6: Sketch two planes that intersect.Ex 7: Sketch two planes that do not intersect.
Describe their intersection.
Ex 8: Answer True or False for the following:
a)Points A, B, and C are collinear. _____
b)Points A, B, and C are coplanar. _____
c)Point F lies on . _____
d)lies on plane DEF. _____
e) and intersect. _____
f) is the intersection of plane ABC and plane DEF. ____
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 .
If , then B is between A and C.
Ex 1: RS = TU, ST = 9, RU = 33
a) Find RS b) Find SU.
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 =
Definition: Congruent Segments
If two segments are congruent, then.
If two segments have , then.
If AB = CD, then .
Ex 4: a) In example 3, is?
b) Ifis congruent to in example 3, then DE = .
1.4 Angles and Their Measures
Target Goals: Use angle postulates.
Classify as acute, right, obtuse, or straight.
Angle:
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 ______.
____
Adjacent Angles: Share a common ______and ______,
but have no ______in common.
Ex 1: Name the adjacent angles in the figure.
Interior and Exterior of an Angle:
ANGLE ADDITION POSTULATE:
If P is in the interior of , then.
Ex 2: Find the measure of the following angles:
a) b) If
then .
Ex 3: If the then, solve for x.
CLASSIFYING ANGLES:
An angle that measures greaterthanand less than is called an angle.
An angle that measuresis called a angle.
An angle that measures greater thanand less than is called an angle.
An angle that measures is called a angle.
1.5 Segment and Angle Bisectors
Target Goals: Bisect a segment.
Use the midpoint formula.
Bisect an angle.
Midpoint:
If a point is a midpoint of a segment, then it.
If a point , then it is the midpoint.
Bisect:
Segment Bisector:
THE MIDPOINT FORMULA
Midpoint Formula
If A ( x1, y1) and B ( x2, y2) then
M =
Ex 1: Find the coordinates of the midpoint of with endpoints A(-2, 3) and B(5, -2).
Ex 2: The midpoint of is M(1, 4). One endpoint is J(-3, 2). Find the coordinates of the other endpoint.
BISECTING AN ANGLE:
Angle Bisector:
If a ray is an angle bisector,then it.
If a ray,then it is the angle bisector.
Ex 3:RT bisectsQRS.Ex 4:KM bisects JKL.
Given that mQRS=, what areThe measures of the two congruent
the measures of QRT &TRS?angles are and . Find the measures of JKM and MKL.
Ex 5: Name the parts.Ex 6:Find the measure of ∠RST.
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:
**The sum of the measures on angles that form a linear pair is .
Vertical Angles:
Theorem: If two angles are vertical angles, then ______.
Ex 1: In the diagram shown, 1 has a measure of . Ex 2: Solve for x.
Find the m2 and m3.
Vertical Pairs:
Linear Pairs:
m∠1 = 60o
m∠2 =
m∠3 =
m∠4 =
Ex 3: Solve for x.
COMPLEMENTARY AND SUPPLEMENTARY ANGLES:
Complementary Angles: If two angles are ______, then their sum is ______.
If the sum of two angles is ______, then ______.
Complementary Complementary
AdjacentNonadjacent
Supplementary Angles: If two angles are ______, then their sum is ______.
If the sum of two angles is ______, then ______.
SupplementarySupplementary
AdjacentNonadjacent
Ex 4: Given that mA = 55, findit’sEx 5: X and Y are supplementary.
complement and it’s supplement. Find the measure of each angle if
mX = 6x – 1 and mY = 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.