Download Unit #4 Triangles

Problem of the Day—(+6pts Collect)
**Take/Return Test and Short Notes**
“If the midpoint of line CW is (7, 4) and point C is (9, -2),
what is the coordinate of point W?”
Section 4.0
“Triangles + Classifying Triangles”
What is a Triangle?
“A polygon with three angles (vertices) and three
side (edges) that are line segments.”
A
D
H
Example #1:
I. Name this Triangle?
II. Name the 3 Vertices?
III. Name the 3 Sides?
Classifying Triangles
**To Classify Triangles By (1) Angles and (2) Sides**
By Angles:
1. Right—One Right, 90, Angle .
2. Equiangular—All Angles are Equal.
3. Acute—All Angles are Less than 90.
4. Obtuse—One Angle more than 90.
By Sides:
5. Equilateral—All Sides are Equal, or Congruent
6. Isosceles—Two Sides are Equal, or Congruent.
7. Scalene—No Sides are Equal, or Congruent.
Ex. #2:
Classify each Triangle
I.
5
I.
Equiangular, Equilateral Triangle
II.
Obtuse, Isosceles Triangle
5
5
II.
12
12
104
III. Acute, Scalene Triangle
III. All Angles Less then 90 and All
Sides are not equal
Worksheet
“Classifying Triangles”
Problem of the Day
**Check Homework and Return Tests**
Review SOL Problem:
Write the Equation of the Circle whose center is (5, 7)
and an outside point is (8, 11)?
( x  h)  ( y  k )  r
2
2
2
Problem of the Day—6th Hour
**Check Homework and Return Tests**
2 Review SOL Problems:
1. Write the Equation of the Circle whose center is (5, 7)
and an outside point is (8, 11)?
( x  h)  ( y  k )  r
2
2.
2
2
QR bisects <PQS. If <PQR = 3x + 4 and <SQR = 5x – 10.
Find x and Find m<PQS?
“Triangle Angle Sum”
Investigation: Sum of a Triangle
Worth: + 5Points
1.
2.
3.
4.
5.
6.
Cut out the Triangle.
Number the Angles
Place the three angles adjacent,
or next to, each other to form
one angle.
Glue Together as so.
Answer Two Questions.
Collect with Name.
2
1
3
2
3
1
Q#1: What angle is made by Angles 1, 2, and 3
after being cut out and taped together?
Q#2: What is the sum, or what do the 3 angles, of
a triangle add up to be?
Triangle Angle-Sum Theorem
“The sum of the three angle measures of a triangle add
to be 180 Degrees.”
m<A + m<B + m<C = 180
B
A
C
How Many Degrees are Equiangular Triangles?
180 Degrees =
3 Angles
60 Degrees
Ex. #1:
Find m<B?
B
A
Angles = 180 Degrees
117
<A + <B + <C = 180
117 + <B + 33 = 180
<B + 150 = 180
<B = 30 Degrees
33
C
Example #2
Find <1, <2, and <3?
62
59
1
53
2
3
Ex. #3:
Find x and y?
y
x
41
x:
180 = 90 – 41- x
49 = x
y:
“Vertical Angles Equal”
If x = 49, then
y = 49
Ex. #4:
I. Find x?
II. Find m<A; m<B; m<C?
A x
2x + 11
B
2x + 4
C
What are Exterior Angles?
“Outside Angles formed by a side and an extension of
an adjacent side. For the exterior there is two inside
angles or remote interior angles.”
m<1 = m<2 + m<3
1
Exterior Angle
2
3
Remote Interior Angles
Ex. #5:
Find m<1?
40
1
m<1 = m<2 + m<3
<1 = 40 + 30
<1 = 70
30
Ex. #6:
Find m<2?
2
113
m<1 = m<2 + m<3
113 = <2 + 73
<2 = 40
73
Ex. #7:
Find x and y?
31
y
x
72
x = 77 and y = 103
Ex #8: Find x, y, and w?
x:
180 = 65 – 39 – x
76 = x
y:
180 – x = y
180 – 76 = y
104 = y
21
39
65
x
y
w
w:
180 – 21 – y = w
180 – 21- 104 = w
55 = w
Try Example #9 if needed
Find x, y, and w?
x = 70
y = 110
w = 30
40
30
80
x y
w
1.
Worksheets
“Triangle Angle Sums”
2. Quiz (Next Class)
--Name/Classifying Triangles
--Solving Missing 180-degree Angles of Triangles
Worth: +36pts
Due: Next Tuesday, November 27th
Problem of the Day
**Check HW, then Quiz, then Short Notes**
Try these SOL Triangle Problems:
1. I. B. 45 Degrees and II. C. 135 Degrees
2. C. 69 Degrees
3. B. 28 Degrees
Then Short Notes
“Congruent Polygons”
Congruent Polygons
“Two polygons that have matching corresponding
Angle and Side Parts.”
‘Name in correct Angle Order’
Turn
Turn
Ex. #I:
Tell the Congruent Triangles
G
GJH   ______
J
H
M
N
T
Ex. #II:
Find Missing Parts of these Congruent Polygons
Find x and y?
Find x, y, and w?
56
x
x
65
115
w
y
y
42 in
56
Ex. #III:
Name the Corresponding, Side and Angle, Parts
B
D
P
X
N
A
C
R
<X =<N
<B =<R
<A =<P
<C =<D
XB =RN
AC =DP
AX =PN
BC =DR
XBCA = NRDP
Try Examples 1, 2, & 3:
Name the Corresponding Parts
Example #1
I.
II.
III.
IV.
HEV
EH
EV
HV
Example #2
I.
A
II. B
III. V
IV. T
V. ABVT
Example #3
x = 90 degrees
y = 145 degrees
w = 72 degrees
1. Worksheet—Congruent Figures
2. SOL Homework #4
--1st, 5th, and 7th Blocks Due Monday, November 26th
--2nd and 6th Blocks Due Tuesday, November 27th
Problem of the Day (11/26 1,5,7th Hours)
**Check HW and Collect SOL HW #4 + Honors Activity (Tomorrow)**
Do and Complete Problems 1 thru 3
Problem #1
I.
Triangle RTH
II. TR
III. KQ
IV. MK
V. <H
VI. <Q
Problem #2
x = 90 degrees
y = 156 degrees
z = 114 degrees
Problem #3
x = 54 degrees
y = 54 degrees
w = 41cm
4 Types of Triangle Congruence
Example #1
Find x w/Congruent Triangles
Given:If QRS = TUV and
QS = 3x + 2 and TV = 7x – 6, Find x? Find QS and TV?
Given:If QRS = TUV and
QR = 5x + 2 and TU = 7x – 10, Find x? Find QR and TU?
1. SSS (Side-Side-Side)
“If all three sides are equal, then triangles are congruent.”
Shown:
Reflexive Property
“Triangles share middle line and equal to each other”
A
M
D
C
AC = AC by “Reflexive Property”
2. SAS (Side-Angle-Side)
“If two sides and the middle angle are equal, then
triangles are congruent.”
Shown:
3. ASA (Angle-Side-Angle)
“If two angles and the middle side are equal, then
triangles are congruent.”
Shown:
A
S
A
A
S
A
4. AAS (Angle-Angle-Side)
“If the two angles and the outside side are equal, then
triangles are congruent.”
Shown:
A
A
A
S
A
S
Vertical Angles
“Opposite Angles Equal”
Try Example #2:
Congruent by SSS, SAS, ASA, or AAS
I. SSS
II. ASA
III. SAS
IV. SAS
V. ASA
VI. SSS
VII. ASA
VIII.AAS
IX. SAS
X. AAS
XI. SAS
Example #3:
Tell if SSS, SAS, ASA, or AAS w/Givens
R
B
D
X
G
H
S
Given:
1. SG = SH
2. RG = RH
C
Given:
1. BX = HX
2. <CBX = <DHX
H
1. Worksheet—SSS, SAS, ASA, AAS
2. POP QUIZ
3. If not done, SOL HW #4

#1
Problem of the Day + POP QUIZ
**Check HW**
For #1-6 Pop Quiz, Tell if SSS, SAS, AAS, or ASA:
1.
2.
4.
5.
3.
6.
Proving the 4 Types of Triangle Congruence
Proofs: “Two Column”
Column 1:
Column 2:
Statements
1.
2.
3.
4.
Reasons
1.
2.
3.
4.
Hints:
Statement 1 Write Given Statement
Statement 2 Write Given Statement
Statement 3 You Provide
Statement 4 Re-write the Prove
Sentence from top
Hints:
Reason 1
Given
Reason 2
Given
Reason 3 You Provide
Vertical Angles (<PTS = <CTS) or
Reflexive Property ( ET = ET)
Reason 4 SSS, SAS, ASA, or AAS
Example #1
Reasons
1.
2.
3.
4.
Statements
1.
2.
3.
4.
Given
2. Given
1.
AB = CB
2. AD = CD
1.
3. BD = BD
3. Reflexive Property
4.
ABD =
CBD
4. SSS
Example #2
Statements
1.
2.
3.
4.
Reasons
1.
2.
3.
4.
1.
AC = EC
2. BC = DC
1.
3. <ACB = <ECD
3. Vertical Angles
4.
ACB =
Given
2. Given
ECD
4. SAS
Example #3
Statements
1. <Q = <S
2. <TRS = <RTQ
3. RT = RT
4.
QRT = STR
Reasons
1. Given
2. Given
3. Reflexive Property
4. AAS
Example #4
Statements
1. LK = LM
2. LP = LJ
3. <MLP = <KLJ
4.
JKL =
PML
Reasons
1. Given
2. Given
3. Vertical Angles
4. SAS
Example #5
Statements
1. JK = MK
2. <GKJ = <GKM
3. GK = GK
4.
GJK =
GMK
Reasons
1. Given
2. Given
3. Reflexive Property
4. SAS
Problem of the Day, then Activity
Given:
1. <A = <M
2. AX = MX
A
Y
X
Prove:
AXT = MXY
T
Reasons
Statements
M
1.
<A = <M
2. AX = MX
1.
3. <AXT = <MXY
3. Vertical Angles
4.
AXT =
Given
2. Given
MXY
4. ASA
Worth: +20 Points
“Fill in the Statement, Reasons, and Proofs”
Worksheet—Proving SSS, SAS, ASA, or AAS
2. Quiz
(Next Tuesday December 4th for 1st, 5th, and 7th Blocks)
(Next Wednesday December 5th for 2nd and 6th Blocks)
1.






Solving for Congruent Polygons
SSS, SAS, ASA, AAS Congruence
Proofs of SSS, SAS, ASA, and AAS
Proofs of CPCTC Problems
HL Theorem
Reflexive and Vertical Angle Properties
Problem of the Day
**Check Worksheet**
Do and Complete Proofs 1 and 2
Statements and Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Statements and Reasons
1.
1.
2.
2.
3.
3.
4.
4.
1. HL Theorem
2. Proving w/CPCTC
Right Triangle Parts
Hypotenuse (H)
Leg (L)
Leg (L)
Longest Side Opposite Right Angle = Hypotenuse
Two Shorter Sides touching Right Angle = Legs
Hypotenuse- Leg ‘HL’ Theorem
“If 1 Hypotenuse, 1 Leg, and 1 Right Angle are
Congruent, then the right triangles are congruent.”
**Two Right <‘s, Two Equal Legs, Two Equal Hypotenuses**
Ex #1:
Name the Two Right Triangles equal by ‘HL’
O
3
N
T
3
M
5
E
5
5
V
B
W
3
P
PMN = TVW
Ex #2:
Congruent by ‘HL’?
I. HL (yes/no)?
II. HL (yes/no)?
YES, by HL
NO, by HL
Ex #3a:
What extra information do you need to prove
these Triangles equal by ‘HL’
Q
R
B
T
D
C
<Q and <C are Right Angles
Ex #3b:
What extra information do you need to prove
these Triangles equal by ‘HL’
B
J
H
M
Legs BJ = BH
or
Legs JM = HM
Ex #3c:
What extra information do you need to prove
these Triangles equal by ‘HL’
B
J
H
M
Hypotenuse BM = BM
Problem of the Day
1. Fill in the 2 Missing Blanks
Prove:
Triangles BAH = MAC
B
H
th
(7
Hour)
2. What extra information
do you need to prove these
Right Triangles equal by
‘HL’ Theorem?
X
A
C
M
Proof:
1. BA = MA
1. Given
2. CA = HA
2. Given
3. __________ 3. Vertical Angles
4. BAH = MAC 4. ____________
E
A
J
CPCTC
“Corresponding Parts of Congruent Triangles are Congruent”
‘After proving Triangles Congruent by SSS SAS ASA or AAS,
you prove the Remaining Sides or Angles by CPCTC.’
Statement #5: Side = Side or Angle = Angle
Reason #5: “CPCTC”
Example #4
Statements
1.
2.
3.
4.
5.
Reasons
1.
2.
3.
4.
5.
1.
AC = EC
2. BC = DC
1.
3. <ACB = <ECD
3. Vertical Angles
ACB =
ECD
5. AB = ED
4. SAS
4.
Given
2. Given
5. CPCTC
Example #5
Statements
1.
2.
3.
4.
5.
Reasons
1.
2.
3.
4.
5.
1.
<DCH = <MCH
2. <DHC = <MHC
1.
3. CH= CH
3. Reflexive Property
DCH =
MCH
5. <D = <M
4. ASA
4.
Given
2. Given
5. CPCTC
+5 Points
Worksheet—HL Theorem and CPCTC
2. Quiz
(Tuesday December 4th for 1st, 5th, and 7th Blocks)
(Wednesday December 5th for 2nd and 6th Blocks)
1.






Solving for Congruent Polygons
SSS, SAS, ASA, AAS Congruence
Proofs of SSS, SAS, ASA, and AAS
Proofs of CPCTC Problems
HL Theorem
Reflexive and Vertical Angle Properties
Problem of the Day + Collect Honors Activity
**Check HW, Short Quiz, then Short Notes**
1. C. AC = EC
1. Given
2. Given
3. Vertical Angles
4. ASA
5. CPCTC
3. x = 5 and AC = 39
2.
Then Short Notes
Problem of the Day
th
(7
What is the Measurement of:
I. m<MTV?
II. m<TVB?
III. m<NVB?
T
63
M
65
V
B
N
Hour)
“Equilateral Triangles”
Equilateral Triangles
“If a triangle is Equilateral, then the triangle has (1) all
equal 60 angles and (2) all equal sides.”
XY = YW = WX = 12
<X = <Y = <W = 60
X
12
Y
W
Ex #1A.
Two sides of an Equilateral Triangle have lengths of 2x + 4
and x + 8.
I. Find x?
II. Find the Length of each Side?
Ex #1B.
Two sides of an Equilateral Triangle have lengths of 3x + 12
and 7x - 8.
I. Find x?
II. Find the Length of each Side?
Ex #2. Equilateral Triangle
Find x and w?
w
10x – 5
25
w
w
25
Ex #3. Equilateral Triangle
Find x?
8x + 4
Ex #4. Equilateral Triangle
Find x, y, z, and w?
11w
z
x
6w + 30
x
y
Worksheet—Equilateral Triangles
2. Late/Missing Work
1.
Problem of the Day
**Check Homework and Return Quiz**
In this Equilateral Triangle Problem,
Find x, y, w, and z?
x
2w + 10
8z – 12
5w - 2
x y
“Isosceles Triangles”
Investigation: Isosceles Triangles
Worth: + 12 Points
1.
2.
3.
4.
5.
6.
7.
8.
C
Make a 5in Line.
Open the compass up 4in.
A
B
Make two arcs in the middle to
create a connection point.
Draw two lines to the connection
point to create a triangle.
Label Angles A, B, and C.
DD
Draw a Straight Line from Angle
C down to make angle D.
Answer Questions.
Q#1: What are the Measures
of Angles A, B, C, and D?
Collect with Name.
Q#2: What do you notice about the
Measures of Angles A and B?
Isosceles Triangles
1. Two equal leg sides.
2. Two equal base angles below the two equal sides.
C
A
B
Base Angles <A = <B
AC = BC
Thus, Isosceles Triangles 2 Equal Sides and 2 Equal Base Angles.
Isosceles Triangle Word Problem #1
“If the legs of an Isosceles Triangle have lengths 2x + 4
and 1x + 8 and the base bottom side has a length 5x -2,
What is x? What is the length of the base?”
Isosceles Triangle Word Problem #2
“What is the measure of the top missing third vertex
angle of an isosceles triangle if both base angles
measure 42 Degrees and 42 Degrees?”
Isosceles Triangle Examples #1
Find x and y for Both?
I.
II.
32
23
y
54
y
x
x
112
x
x
Isosceles Triangle Example #2
I. Find Angle F? II. Find Angle E?
E
118
F
G
Isosceles Triangle Examples #3
Find x and w for Both?
I.
II.
w
w
x
x
47
x
130
Isosceles Triangle Examples #4
Find x, y, and w for Both?
III.
IV.
w
w
2x + 50
y
x
53
7x
y
+9 Points
1.
Worksheet on Isosceles Triangles
2. UNIT 4 TEST (Next Friday , Dec. 14th or Monday, Dec. 17th)








Triangles
Congruent Polygons
Congruent Triangles
SSS, SAS, ASA, AAS Problems
SSS, SAS, ASA, AAS Proofs
CPCTC and CPCTC Proofs
HL Theorem
Equilateral, Isosceles, and Right Triangle Problems and Solving for
Missing Variables
Problem of the Day
**Check HW and Test Friday/Monday**
1. Slope = 0
2. (x + 1)^2 + (y – 10)^2 = 25
3.
x = 32 and w = 45in
4. x = 10, y = 130, and w = 80
“Overlapping Congruent Triangle + O.C.T Proofs”
Overlapping Triangles
“Triangles that share a common side and/or common
angle on top of each other.”
Ex #1: Overlapping Triangles
What common sides and common angles are shared
between Triangles ACD and EDC?
A
E
C
D
Ex #2: Overlapping Triangles
What common sides and common angles are shared
between Triangles ACD and ECB?
A
E
B
D
C
Ex #3: Reasons in this Overlapping Triangle Proof
Statements
1. BA = DE
2. CA = CE
Reasons
C
3. <CAE = <CEA
B
D
4. AE = AE
X
5. Triangle BAE =
Triangle DEA
A
E
Given: 1. BA = DE 2. CA = CE
6. <ABE = <EDA
Prove: <ABE = <EDA
Overlapping Proof Scramble #1
Statements
1. <ZXW = <YWX
2. <ZWX = <YXW
3. WX = WX
4. Triangle ZWX = Triangle YXW
5. ZW = YX
Reasons
1. Given
2. Given
3. Reflexive Property
4. ASA
5. CPCTC
Overlapping Proof Scramble #2
Statements
1. CA = CE
2. BA = DE
3. <CAE = <CEA
4. AE = AE
5. Triangle BAE = Triangle DEA
Reasons
1. Given
2. Given
3. Base Angles of an
Isosceles Triangle
are Congruent.
4. Reflexive Property
5. SAS
+8 Points
1.
Worksheet on Overlapping Triangles
2. UNIT 4 TEST (Friday , Dec. 14th or Monday, Dec. 17th)








Triangles
Congruent Polygons
Congruent Triangles
SSS, SAS, ASA, AAS Problems
SSS, SAS, ASA, AAS Proofs
CPCTC and CPCTC Proofs
HL Theorem
Equilateral, Isosceles, and Right Triangle Problems and Solving for
Missing Variables
Problem of the Day
**Check HW then Short Lesson and Test Review**
1. Which is Parallel to show the
2.
y = 3/4x – 1
Ceiling Beam is Parallel to the
Floor Beam?
Which is Parallel to the Line above?
Ceiling Beam
A. y = 4/3x + 2
x
z
y
Floor Beam
y=w
B. w = x
C. w = z
D. x = z
A.
w
B. y = 3/4x – 5
C. y = -4/3x + 3
D. y = -3/4x + 6
Review Problem (+6pts)
<ACB = 56 degrees
x = 31
y=4
“Ratios and Proportions”
Ratio—compares two quantities in a fraction form
with one number over another number.
a
b
Proportion—two equal ratios.
a c

b d
To Solve Proportions
“Cross-Multiply”
a c

b d
ad = bc
Multiply across from upper left to
lower right and from upper right
to lower left.
x 5

y 8
What do you get?
8x = 5y
Ex #1:
Solve for x in these Proportions
x 2

35 5
“Cross-Multiply”
5x= 70
x = 14
x x 1

2
3
“Cross-Multiply”
3x = 2(x + 1)
3x = 2x + 2
1x = 2
x=2
Ex #2:
Solve for x in this Proportion
x  7 x 5

2
5
“Cross Multiply”
5(x – 7) = 2(x + 5)
5x – 35 = 2x + 10
3x – 35 = 10
3x = 45
x = 15
Problem of the Day (7th Hour)
Complete #1-2
1. Solve for x in this
Proportional Ratio
2. Find x and y in this
Isosceles Triangle Problem
2x 1 x  5

6
2
“Cross Multiply”
4x + 2 = 6x - 30
2 = 2x - 30
32 = 2x
x = 16
y = 52 and x = 5
STUDY FOR UNIT 4 TEST
(Friday , Dec. 14th or Monday, Dec. 17th)


Triangles
 Congruent Polygons
 Congruent Triangles
 SSS, SAS, ASA, AAS Problems
 SSS, SAS, ASA, AAS Proofs
 CPCTC and CPCTC Proofs
 HL Theorem
Equilateral, Isosceles, and Right Triangle Problems and
Solving for Missing Variables
Problem of the Day, then TEST
Name the Angle Included by the Sides DE and DG?
D
E
G
Honors #37: Fill in the Missing Statements/Reasons
Statements
1. HP = CK
2. GP = GK
G
3.
4.
Reasons
1. Given
2. Given
3.
4. Reflexive Property
5. Triangle HPK =
H
C
X
6.
Triangle CPK 5.
6.
P
K
Given: 1. HP = CK 2. GP = GK
Prove: HK = CP
Extra Credit
2. SOL Homework #5
3. Honors—Notes on Tessellations
1.