Download 1 Math 1: Semester 2 3-3 (Angle Addition) 0

Math 1: Semester 2
3-3 (Angle Addition)
0-5:00
Attendance
QD. A pair shoes costs $50.95 and they are on sale for 25% off. What is the final cost?
$38.21
5:00-25:00
Lesson
Objective- To measure angles using a protractor and determining the bisector of an angle
NotesAngle Addition Postulate- For any angle, PQR, if a point, B, is the interior then,
two angles can be formed, which equal the original angle, so PQB + BQR = PQR
P
B
Q
Angle Bisector-
R
separates the angle into two, equal angles
Q
W
P
R
is the angle bisector of
P&m
QPW = m
WPR
Examples
#1) Angles addition
Find the following anglesa) m KNM
M
L
110°
25°
K
N
1
b) m
2
D
C
75°
2
A
c) m
B
JKL
L
(2x-10)°
J
4x°
K
M
#2 Angles bisector
If
bisects
F
CFE and m
C
1
2
CFE = 70; find m
CFD and m
DFE
D
E
35°
ActivityHave students draw an angle (encourage large angles to make the measuring easier), then
have them find the angle bisector by using a protractor.
25:00-50:00
Homework: (108-109) & angle bisector construction (using compass)
Nearing Proficient:
Proficient:
Adv. Proficient:
11-22 (odds)
11-22 (all)
11-22 (all); 23-25
2
Math 1: Semester 2
3-5 (Complementary & Supplementary Angle)
0-5:00
Attendance
5:00-25:00
Lesson
Objective- To identify the properties of complimentary and supplementary angles
NotesComplementary Angles-
Two angles sums equal 90 degrees
Supplementary angles-
Two angles sums equal 180 degrees
Supplementary Postulate- If two angles form a linear pair, then they are
supplementary angles
Examples
#1) Determine if the following are complementary
AGF & BGC
AGB &
BGC
FGE &
CGD
FGE & FGA
Yes
No
Yes
No
A
B
F
55°
80°
90°
G
C
10°
90°
E
D
#2 Name a pair of supplementary angles, which are not adjacent angles?
AGB & DGE
#3 Find the measure of an angle that is angle that is supplementary to
145
BGC?
25:00-50:00
Activity: Group work
Have students work with the two angles learned today by creating a table- one rewording
the mathematical definition and example through illustration
3
Homework: Worksheet 3-4 & 3-5
Math 1: Semester 2
3-4 (Adjacent & Linear Pair Angles)
0-5:00
Attendance
5:00-25:00
Lesson
Objective- To understand the relationship of angles, which share a common side- forming
adjacent angles and linear pairs
NotesAdjacent Angles-
Angles that share a common side & have the same vertex
Common side
1
2
Same vertex
Linear Pair- Adjacent angles, which their angles added up equal 180 degrees
1
2
Examples
#1) Adjacent angles
Determine whether
a) No
1
1&
2
2 are adjacent angles
b) Yes
1 2
c) No
4
1
2
#2 Linear Pair
Using the figure below construct as many linear pairs as possible
a) MCA
ACE
b) HCE
MCH
c) TCM
TCE
d) ACH
not possible
T
H
E
C
A
M
25:00-50:00
Activity
Have students choose the one of the following:
#1 VerbalAsk the student to help a friend who confuses adjacent and linear angles by writing a
paragraph explaining the difference between these two types of angles. They may draw
sketches to aid in their writing.
#2 KineticAsk students to look for angles at home. They should notice that many angles are right
angles. Have them note which angles are not right angles and where they appear.
Encourage students to use a protractor to verify their observations. Also, have them note
where they see adjacent angles and linear pairs.
Homework: (113-114)
Nearing Proficient:
Proficient:
Adv. Proficient:
8-19 (odds)
8-19 (all)
8-19 (all); 22-26
5
Math 1: Semester 2
3-6 (Congruent Angles)
0-5:00
QD
Attendance
If a dinner will cost you $65 but you have a coupon for 25% off. How much is
the final bill?
5:00-25:00
Lesson
Objective- Students will identify congruent and vertical angles and learn their uses.
ReviewDo the 5-minute check
NotesCongruent Angles- Two angles that have the same measure
Vertical anglesFormed by two intersecting lines; of the four angles formed, they
are only the nonadjacent angles
A
D
B
C
Thm Vertical angle
Vertical angles are congruent
Thm 3-2 & 3-3
If two angles are congruent, then there compliments or supplements are congruent
Thm 3-4 & 3-5
If two angles are complementary or supplementary to the same angle, then they
are congruent
Thm. 3-6
If two angles are congruent and supplementary, then each angle is right
Thm. 3-7
All right angles are congruent
6
Examples
1) Find the values of x
a)
b)
100°
75°
x°
(x-18)°
x = 100
2) Suppose
A
B. IF
B = 47, find the supplementary angle to
A
1st A = 47
2nd – The supplement would be 180 – 47 = 133°
3) Given:
1 is supplementary to
3 is supplementary to
2 = 105 degrees
Find m 1 & m 3
2
2
Since both angles (1 & 3) are supplementary to
130-105 = 75 degrees
2, they both have the same measure-
25:00-50:00
Activity: Logic
Have students write a short paragraph explaining why vertical angles are congruent.
They cannot use examples but must use the facts (theorems) learned in this chapter.
After students are finished, pair them and have them explain their method to each other.
Homework: (126-127)
Nearing Proficient:
Proficient:
Adv. Proficient:
9-14 (odds)
9-14 (odds); 15-20 (odds)
9-14 (all); 15-20 (all)
Math 1: Semester 2
3-7 (Perpendicular Lines)
0-5:00
Attendance
7
QD
If a big screen TV costs $ 1200 but it’s discounted by 15% off. How much is
the TV?
5:00-25:00
Lesson
Objective- Students will identify, construct, and use properties of perpendicular lines.
NotesPerpendicular lines- Lines that intersect at a 90 degree angle, use symbol
What statements can be said of the figure below?
1
2
4
3
a)
1 is a right
or 90 degrees, so l m
b) 1 & 3 are vertical angles, so 1
3 and m 3 = 90
c) 1 & 4 form a linear pair, so m 1 + m 4 = 180; 1 &
supplementary; and m 4 = 90
d) 2 & 4 are vertical angles, so 2
4 and m 2 = 90
4 are
Thm. If two lines are perpendicular, then all four angles are right angles
Examples
M
7
P
Q
1
2
R
5
6
8
S
O
3
4
N
1) Answer the questions by referring to the figure abovea) PRN is an acute angle
False
b) 4
8
True
c) QS OP
False
8
d)
2)
7 is an obtuse angle
True
Answer the questions by referring to the figure below
Find m 1 and m 2.
Given: AC BD; m 1 = 8x-2; m 2 = 16x –4
A
B
2
F
E
1
D
C
Since, AC BD, we know that Find m
So,
8x – 2 + 16x – 4 = 90
24x – 6 = 90
24x = 96
x=4
m
m
1+m
2 = 90
Substitute expressions
Combine like terms
Substitute x into the expressions
1 = 8(4) – 2 = 30 degrees
2 = 16(4) – 4 = 60 degrees
25:00-50:00
Activity: Visual
Have students work in pairs and construct a classroom with out any perpendicular lines
where the floors, ceiling, doors, windows, and walls meet.
Homework: (132) 8-24; 25-28
Nearing Proficient:
Proficient:
Adv. Proficient:
8-24 (odds)
8-24 (all)
8-24 (all); 25-28
Math 1: Semester 2
4-1 (Parallel Lines & Planes)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Understand how parallel lines form at the intersection of planes.
9
Review:
Point, ray, and segment.
ActivityHave students place letters to mark the eight corners of the room.
List on the board the points.
Construct/identify the twelve segments (edges), which make up the outlines of the room.
From the points, construct/identify the six planes (faces) that make up the ceiling, floor,
and walls.
NotesParallel Lines-
Lines in the same plane, which do not intersect
Skew Lines-
Lines that are not in the same plane and they do not intersect
B
C
D
A
G
E
F
25:00-50:00
Homework: (145-46) 12-31; 40-45 and worksheet
Nearing Proficient:
Proficient:
Adv. Proficient:
12-21; 40-45 and worksheet
12-21; 24-27; 28-31 (odd); 40-45 and worksheet
12-31; 40-45 and worksheet
Math 1: Semester 2
4-2 (Parallel Lines & Transversals)
0-5:00
5:00-25:00
Lesson-
Attendance
10
Objective:
lines.
Understand the relationships of angles formed by transversal and parallel
Review:
Vertical, supplementary, and complementary angles
NotesDay 1:
Transversal- a line that intersects two or more lines at two different points
Ask: How many angles are formed? 8
Exterior
1
3
2
4
Interior
5
6
7
Interior
Exterior
: lie between the two lines
Alternate Interior
:
Consecutive Interior
Exterior
8
that lie on opposite side of the transversal & in the
interior sides of the two lines
:
that lie on the same side of the transversal & in the interior
of the two lines
: that lie outside the two lines
Alternate Exterior
:
on the opposite sides of the transversal & in the exterior of
the two lines
Activity:
Ask: Use the diagram to fill in which angles fit which definitions
1
3
2
4
11
5
6
7
Vertical
:
Interior
:
Alternate Interior
:
Consecutive Interior
Exterior
8
:
:
Alternate Exterior
:
Day 2:
Thm. Alternate Interior
If two parallel lines are cut by a transversal, then each pair Alt. Int.
Ex.
’s are congruent
and
12
Thm. Consecutive Interior
If two parallel lines are cut by a transversal, then each pair Consecutive Int.
supplementary
’s are
Ex.
Thm. Alternate Exterior
If two parallel lines are cut by a transversal, then each pair Alt. Ext.
’s are congruent
Ex.
1
3
2
4
5
6
7
8
Corresponding Angles:
. All of the pairs
are congruent to each other.
Thm: If a transversal is perpendicular to on of the two parallel lines, then it is
perpendicular to the other line.
Examples
#1)
1
104°
2
3
1-76°
3-98°
5-76° 6-98°
8-104° 9-82°
11-104°
13-98°
2-82°
4- 104°
7-76°
10-98°
12-76°
14-82°
13
4
5
7
6
8
11
9
12
13
82°
10
14
#2)
26°
124° 56°
56°
124°
25:00-50:00
26°
98° 82°
124°
98° 82°
Homework: (152-53) Day 1: 13-26 Day 2- worksheet 4.2
Day 3- worksheet 4.3
Math 1: Semester 2
4-4 (Proving Lines Parallel)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Showing how to lines can be parallel using a transversal.
QD. Fill in the numbered angles
14
1
4
104°
2
3
5
6
82°
7
11
8
9
12
13
1-76°
3-98°
5-76° 6-98°
8-104° 9-82°
11-104°
13-98°
2-82°
4- 104°
7-76°
10-98°
12-76°
14-82°
10
14
NotesThm. Corresponding Angles
1
a
2
b
Thm. Alternate Interior Angles
1
a
2
b
Thm. Alternate Exterior Angles
1
a
2
b
Thm. Consecutive Interior Angles
1
a
15
2
b
Examples
Find x so that
#1)
a
84°
3x
b
x = 28
a
b
#2)
10x°
5x°
10x + 5x = 180
15x = 180
x = 12
25:00-50:00
Homework: (166)
Nearing Proficient:
Proficient:
Adv. Proficient:
9-20(odds)
9-20 (all)
9-20 (all); 21-23
Math 1: Semester 2
5-1 (Classifying Triangles)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
triangles.
Students will understand all parts of a triangle and how to classify varying
16
NotesTriangle-
parts
A
angle
side
vertex
B
ΔABC•
•
•
C
The sides are
The vertices are A, B, & C
The angles are
Types of Triangles (by angles)A) Acute
B) Obtuse
C) Right
All acute angles
One obtuse angle
One Rt. angle
Types of Triangles (by sides)A) Scalene
B) Isosceles
C) Equilateral
17
No sides congruent
at least two sides
congruent
All sides
congruent
ExamplesClassify each triangle by its angles and sides
#1
Right scalene
30°
#2
120°
Obtuse, isosceles
30°
#3
Find all side lengths of the triangle
A
2n + 2
B
10
2n – 2
C
25:00-50:00
Activity: Group
Have students work in pairs. Student #1 labels an index card for each of the seven
possible angle/side classifications for triangles: acute/scalene, acute/isosceles,
acute/equilateral, right/scalene, right/isosceles, obtuse/scalene, and obtuse/isosceles.
Student #2 prepares seven more cards, each picturing one of the seven triangles
possibilities. Shuffle the cards, arrange them face down, and have the students take turns
matching one set of cards to the other. Have them keep track of their scores.
18
Homework: (191) 8-16; 18-22; 26,27
Nearing Proficient:
Proficient:
Adv. Proficient:
8-16; 18-22;
8-16; 18-22; 26,
8-16; 18-22; 26,27
Math 1: Semester 2
5-2 (Angles of a Triangle)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Students will understand the relationship of all angles of a triangle.
NotesThm. Angle Sum
The sum of the measures of the angles of a triangle is 180°
x
x° + y° + z° = 180°
* This thm. is used to find missing angles measures
z
y
Ex. 1
Find m
So,
in ΔMNP if m
80 + 45 + m
= 180
125 + m
= 180
-125
-125
_________________
m
= 65°
Ex.2
Find the value of each variable in ΔABC
19
75°
B
x°
y°
58°
A
x is a vertical angle with 75° and so x = 75
last,
x° + y° + 58° = 180°
75 + y + 58 = 180
133 + y = 180
-133
-133
y = 47
C
Thm. The acute angles of a right triangle are complementary (=90°)
x° + y° = 90°
x°
y°
Equiangular Triangle-
A triangle, which all three angles are congruent or 60°
Ex.
Since it is a rt. Triangle both angles J & K equal
180.
So,
(x + 15)° + (x + 9)° = 90°
2x + 24 = 90
-24 -24
2x = 66
2
2
J
(x + 15)°
(x + 9)°
L
x = 33
Now, plug-in 78 into each angle expression
K
25:00-50:00
Activity:
Homework: (196-97) 8-19, 22, 23; 26-29
Nearing Proficient:
Proficient:
Adv. Proficient:
8-16, 17, 26-29
8-19, 26-29
8-19, 22, 23; 26-29
20
Math 1: Semester 2
5-3 (Translation, reflection, & rotation)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Students will understand certain transformations move figures in a plane
Activity: Auditory
Play a simple melody, then play it one octave higher. Ask how they are similar and
different. Use this discussion to introduce translations.
NotesTranslation- slide a figure; the initial figure is the pre-image & the slided figure is the
image
Reflection- flipping a figure (the figure has been flipped over the horizontal line)
21
Rotation- turning a figure (the figure has been rotated 90°)
Ex. Go over the six examples on pg. 198 (answers: rotation, reflection, translation,
reflection, rotation, and translation).
Mapping- Each point of a pre-image is paired with exactly one point of an image. When
all points have been mapped, this is called a Transformation, this symbol, →, is used to
indicate a mapping.
Ex. ∆ABC → ∆EFG
A
B
Preimage
A
B
C
C
→
→
→
Image
E
F
G
E
F
G
25:00-50:00
Homework: (201)
Nearing Proficient:
Proficient:
Adv. Proficient:
9-24
9-24
9-24
Math 1: Semester 2
5-4 (Congruent Triangles)
0-5:00
5:00-25:00
Lesson-
Attendance
22
Objective:
Students will understand the relationship between segments and angles of
congruent triangles.
ActivityMaterials: two pieces of graph paper, scissors, & straight-edge
Step 1: Have students go to page 203 and copy the two triangles in “hands-on”
section at the top of the page.
Step 2: After the triangles are constructed, the students will cut them out. Match
the same parts up together and compare.
Step 3: Have the students answer the following questions. Students should attach
the cut-out triangles and questions to turn in.
1) Identify all of the pairs of angles and sides that match or correspond.
2) ∆ABC is congruent to ∆FDE. What is true about their corresponding
sides and angles?
NotesCorresponding partsparts of congruent triangles that “match” up; slashes (-, =)
are used to identify congruent segments and arcs ( )are used for congruent angles.
Thm. Congruent Triangles
If two triangles are congruent, then all the corresponding parts are congruent.
If the corresponding parts of two triangles are congruent, then the two triangles are
congruent.
Examples
#1 If ∆ABC ∆FDE, name the congruent angles and sides. Then draw the triangles,
using arcs and slashes marks to show congruent angles and sides.
B
D
23
A
C
F
E
Ex. 2
∆UVW is congruent to ∆GHI.
Since, both triangles are congruent, angle V and H are congruent because they are
corresponding parts. So, we can set the measures of both angles equal to each other.
90 = 3x + 15
-15
-15
75 = 3x
Subtract 15 from both sides
Divide both sides by 3
75 = 3x
3 3
25 = x
25:00-50:00
Homework:
Day 1: Homework: (206) 11-25
Day 2: Class activity and worksheet
Math 1: Semester 2
5-5 (SSS & SAS)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Students will understand the relationship among sides and angles in
showing congruent between triangles
ActivityMaterials: one piece of paper, scissors, & ruler
Step 1: Have students cut out three segments form the piece of paper: 4” long, 5”,
and 6”.
24
Step 2: Ask students to form as many triangles as they can from the three
segments and keep track of their constructions on paper.
Step 3: Have students compare their triangles with those of other students.
Step 4: Have each student make a guess (conjecture) based on their results and
observations.
NotesPostulate: SSS (side-side-side)
If three sides of one triangle are congruent to three corresponding sides of another
triangle, then the triangles are congruent.
B
S
A
R
C
T
Included Angle- the angle formed by two given sides of a triangle
C
A
B
Post: SAS (Side-Angle-Side)
If two sides & the included angle of one triangle are congruent to the corresponding sides
and included angle of another triangle, then the two triangles are congruent.
25
B
S
A
R
C
T
Examples
Write a congruence statement for each pair of triangles
#1)
Hint: Sketch the two triangles by
constructing the segments and angles, which are congruent to each other
K
N
J
L
M
O
So, the congruence statement:
Tell whether each pair of triangles are congruent; why (which postulate); and write a
congruence statement.
#2)
B
A
Y
C
X
Z
Yes, SAS,
#3)
D
E
M
F
N
26
O
Yes, SSS,
25:00-50:00
Homework: (213)
Nearing Proficient:
Proficient:
Adv. Proficient:
8-15; 19; 22
8-16; 19; 22
8-18; 20-22
Math 1: Semester 2
5-6 (ASA & AAS)
0-5:00
5:00-25:00
Lesson-
Attendance
Objective:
Students will understand the relationship among sides and angles in
showing congruent between triangles
NotesIncluded Side- the side that falls between two angles
C
A
B
Postulate: ASA (angle-side-angle)
If two angles & the included side of one triangle are congruent to the corresponding
angles & the included side of another triangle, then the triangles are congruent.
B
S
27
A
R
C
T
Thm. AAS (Angle-Angle- Side)
If two angles & a non-included side of one triangle are congruent to the corresponding
angles and non-included side of another triangle, then the two triangles are congruent.
(note: the side is not between or included between the angles)
B
S
A
R
C
T
Examples
Name the needed information to make the triangles congruent
#1) Using ASAK
J
N
L
M
O
Needed congruent part:
28
#2) Using AASB
A
Y
C
X
Z
Needed congruent part:
#3) Determine whether each pair of triangles are congruent by SSS, SAS, ASA, or AAS
D
E
M
F
N
O
Yes, by SSS
25:00-50:00
Activity: Group & visual
Have students work in pairs to construct four examples showing two triangles congruent
by SSS, SAS, ASA, & AAS. Have them use one sheet of paper for each example.
Homework: (218-19)
Nearing Proficient:
Proficient:
Adv. Proficient:
15-22; plus worksheet (evens)
15-22; plus worksheet
15-22; plus worksheet
29