Download 1 - Collingswood High School

1. 1
Patterns and Inductive Reasoning
What comes next?
a.
b.
c. 1, 4, 16, 64, …
Steps for inductive reasoning…
1. Look for a pattern-
2. Make a conjecture-
3. Verify the conjecture-
d. Conjecture: The sum of the first n odd positive integers is
.
e. Show the conjecture is false by finding a counterexample.
Conjecture: For all real numbers x, the expression x2 is greater than or equal to x.
Describe the pattern, then predict the next number.
f. 2, 6, 18, 54, …
i. 0, 1, 4, 9, …
g. 3, 0, -3, 0, 3, 0, …
j. 256, 64, 16, 4, …
h. 13, 7, -5, …
k. 1, -2, 4, -8, …
l. Use examples to complete the conjecture.
The sum of any two odd numbers is…
Show the conjecture is false by finding a counterexample.
m. Everyone who takes geometry is a junior.
n. Only girls can be smart.
o. All prime numbers are odd.
p. If the product of two numbers is even, then the two numbers must be even.
q. The sum of two numbers is always greater than the larger number.
Use a calculator to explore the pattern. Write a conjecture based on what you
observe.
r.
101 * 34 = ?
s.
11 * 11 = ?
101 * 25 = ?
111 * 111 = ?
101 * 97 = ?
1111 * 1111 = ?
101 * 49 = ?
11,111 * 11,111 = ?
1.2
Points, Lines, and Planes
definition-
undefined terms-
points-
line-
plane-
collinear points-
coplanar points-
a. Name three points that are collinear.
b. Name four points that are coplanar.
c. Name three points that are not collinear.
Term
Line
Line segment
Definition
Symbol
Picture
Ray
Opposite rays
endpointsinitial point-
GEOMETRIC DRAWING…
-Draw 3 noncollinear points J, K, L
-Then draw
-Then draw
-Last draw
-Draw the intersection of a plane and a line.
-Draw the intersection of two planes.
-Draw two points, A and B. Then sketch AB. Add point C on the segment between A
and B.
-Draw points A, B, C, D, E, with no three collinear. Then sketch AB, BC, CD, DE, and
AE
1.3
Segments and Their Measures
postulate-
length-
between-
Segment Addition Postulate-
Distance Formula-
Find the distance between the two points.
a. (1, 2) and (4, 6)
b. (-2, -3) and (0, 0)
c. A and B
congruent segments-
d. B and C
Sketch the 3 collinear points. Then write the Segment Addition Postulate for the
points.
e. A is between P and L
f. R is between T and G
g. I is between A and E
g. Q is between W and Y
Use the distance formula to determine if JK
KL
h. J (3, -5)
i. J (0, -8)
K (-1, 2)
K (4, 3)
L (-5, -5
L (-2, -7)
In the diagram, AE = 20, BD = 6, and AB = BC = CD. Find each length.
j. BC
n. CD
k. AB
o. DE
l. CA
p. BE
m. DA
q. CE
1.4
Angles and Their Measures
anglesidevertex-
NAMING ANGLES
MEASURING ANGLES
Angles that have the same measure are called
.
*Lengths are equal.
Segments are congruent.*
*Measures are equal.
Angles are congruent.*
Angle Addition Postulate
CLASSIFYING ANGLES
acute
right
obtuse
straight
adjacent anglesClassify the angle.
a. m A = 180º
c. m
C = 90º
b. m
d. m
D = 34º
B = 106º
Use the diagram to answer the questions.
e. Is
ABD
DBE?
f. Is
DBA
EBC?
g. Are
DBE and
EBC adjacent?
h. Are
ABD and
CBE adjacent?
Name the vertex and sides of the angle. Then estimate its measure.
1.5
Segment and Angle Bisectors
midpoint-
bisects-
segment bisector-
Midpoint Formula
Find the coordinates of the midpoint.
a. (1, 2) and (3, 6)
b. (-3, -5) and (-7, 10)
The midpoint of AB is C. Use the coordinates of A and C to find the coordinates of
B.
c. A (3, 4) and C (7,8)
d. A(-4,0) and C (0,-2)
Find the coordinates of the midpoints.
g. B(7,6) M(-5, 4)
h. A(0,0) B(-8,6)
Find the coordinates of the other endpoints of a segment with the given endpoint
and midpoint M.
i. D(-2, -4) M(5, -4)
j. Q(4,2) M(7,3)
Use the marks on the diagram to name the congruent segments and congruent
angles.
k.
l.
m.
1.6
Angle Pair Relationships
vertical angles-
linear pair-
Ex. 1
a) Identify all vertical angles
b) Identify all linear pairs
c) Identify adjacent angles
1
2
3
4
Vertical angle measures…
Linear pair measures…
Ex. 2
Angle 5 has a measure of 56º, what are the measures of the other 3 angles?
5
Ex. 3
6
8
7
Solve for x and y. Then give the measures of the four angles.
(y + 20) º
(3x +5) º
(4y – 15) º (x + 15) º
Complementary Angles-
Supplementary Angles-
Ex. 4 Complementary or Supplementary? Adjacent or nonadjacent?
a.
b.
c.
d.
Ex. 5
Ex. 6
Ex. 7
A is complementary to B, which is supplementary to
what are the measures of A and B?
D is complementary to E and supplementary to
are the measures of the other two angles?
C. If m A is 76º,
F. If m E is 34º, what
Find x and y, and then find the measures of all four angles.
(3x + 20) º
yº
(5x – 50) º
1.7
Intro to Perimeter, Circumference, and Area
s
l
w
a
h
c
b
r
Find the area and perimeter or circumference of the figure.
a.
5 in.
b.
4 in.
c.
12 ft.
d.
2 m.
8m
6m 8m
13 m
e.
7.5 mm
f.
14 yd
10 yd
13 yd
15 yd
g. A rectangle has an area of 36 in2 and a length of 3 in. What is its perimeter?
h. A triangle has an area of 48 ft2 and a base of 4 ft. What is its height?