Download Unit 2 Packet (Green ch3)

Name:
Unit 2 ~ Congruent Triangles
Assignments
Advanced Geometry
Ch 3 Green Book
*All problems refer to the WRITTEN EXERCISES* (Green Textbook)
#
Class Date
Class Activity
Homework
1
Day 1
Section 3.4 Angle
Relationships
• Complete classwork
• Read section 3.4
• Become familiar with the following terms
about angles: acute, obtuse, right, adjacent,
complementary, supplementary, and vertical.
• Do p. 106 #5, 7, 13; p. 110 #12, 15, 20;
• Do p. 118 #17, 21, 25 (Two column proof not
required. Just explain it in your own words.)
2
Day 2
Introduction to Proofs
Fill in the Missing Statements
& Fill-in Proofs
• Complete classwork
• Study first group of reasons (from “given” to
“vertical angles theorem”).
3
Day 3
Section 3.5 Congruence
• Complete classwork
• Read section 3.5
• Do p. 122 #1, 3, 7, 12, 15, 17, 21–23
4
Day 4
Section 3.6 More Ways of
Proving Congruence
• Complete classwork
• Read section 3.6
• Do p. 127 #1, 3, 6, 7, 15, 19, 21, 22, 23, 25, 28
5
Day 5
Section 3.7 CPCTC
• Complete classwork
• Read section 3.7
• Do p. 130 #3, 5, 7, 12, 15, 19, 25, 27, 31
6
Day 6
Section 3.8 Isosceles
Triangles
• Complete classwork
• Read section 3.8
• Do p. 135 #1, 9, 15, 19, 23, 25, 27, 31, 33
7
Day 7
Section 3.9 Geometric
Inequalities
• Complete classwork (up to “D-Stix” Lab)
• Read section 3.9
• Do p. 140 #1, 3, 11, 13, 15, 17–19, 21, 26–28
8
Day 8
Section 3.9 continued
• Complete classwork
• Do p. 140 #5, 9, 12, 16, 23-25
9
Day 9
Review for Opportunity
• Unit 2 Review Sheet
• Study for Opportunity!!
10
Day 10
Unit 2 (Chapter 3)
Opportunity
• Algebra Review worksheet
Due Date
Extra Help Times: Math Center 7:30-8:15am M-F, Wednesdays 2:50-3:30pm, X Block, and by appointment
Use Canvas or email to ask questions as well!
Definitions, Theorems, Postulates, Corollaries, and
Properties used in Geometry Proofs
Add to this list as needed – Must be a reason in the text or proven/given in class.
• Given
Properties
• Reflexive property
• Transitive property
• Addition property of equality (APE)
• Subtraction property of equality (SPE)
• Substitution property (Subst.)
• Property of Inequalities
Segments & Angles:
• Definition of perpendicular bisector
• Definition of segment bisector
• Definition of angle bisector
• Definition of midpoint
• Definition of right triangle
• Definition of perpendicular lines
• Angle Addition Postulate ( ∠ + Post.)
• Segment Addition Postulate (Seg. + Post.)
• Definition of Supplementary
• Definition of Complementary
• Supplements to Congruent Angles are Congruent (SCAC)
• Complements to Congruent Angles are Congruent (CCAC)
• Vertical Angles Theorem (VAT)
Triangles:
• SSS
• SAS
• ASA
• AAS
• HL
• All right angles are ≅
• All radii of a circle are ≅
• Definition of Congruent Triangles/Corresponding Parts of ≅ Δ ’s are ≅ (CPCTC)
• Definition of an Isosceles Triangle
• Isosceles Triangle Theorem (ITT)
• Converse of Isosceles Triangle Theorem (CITT)
• Remote Interior Angles Theorem (RIAT)
• Largest Side Opposite Largest Angle (LSOLA)
• Exterior Angle Inequality Theorem
• Triangle Inequality Theorem ( Δ ≠ Thm)
• Triangle Sum Theorem
• 3rd Angles Theorem
• Definition of Median
1
Unit 2 Intro (Day 1)
Green Book: Euclidean Geometry… read together p. 99, paragraph 2-3
Geometry literally means
Postulate:
Example: Through any two points, there exists exactly one line.
Theorem:
Example: If two lines intersect, then they intersect in exactly one point.
A
• Notation
B
A
B
A
A
•
5
40o
C
B
B
Bisectors

CD bisects AB at P means:
C
1.
2.
P
A
3.
B
D
NOTE:

BD bisects ∠ABC means:
A
D
1.
2.
B
C
NOTE:
2
Day 1 Practice:
1. Two angles are complementary if the sum of their measures is 90°.
a.) If ∠A and ∠B are complementary and congruent, find the measure of each angle.
b.) ∠1 and ∠2 are complementary. If m∠1 = 3x – 23 and m∠2 = 4x + 1, find x.
2. Two angles are supplementary if the sum of their measures is 180°. If ∠A and ∠B are
supplementary and the measure of ∠A is 12 less than twice the measure of ∠B, find the
measure of each angle.
3. If ∠A and ∠B are each complementary to ∠C, what can you conclude? Draw a diagram
of the situation and explain.
4. If ∠A and ∠B are each supplementary to ∠C, what can you conclude? Draw a diagram of
the situation and explain.
3
5. When two lines intersect, as shown below, they form four separate angles, which
are labeled here as ∠1, ∠2, ∠3, and ∠4.
a.) Suppose m∠1 = 20°. Find the other three angles.
Pairs of
“opposite” angles such as ∠1 and ∠3 are called vertical angles.
b.) Explain why vertical angles are always congruent.

6. In the diagram below, OX bistects ∠AOB. Also m∠XOY = 90°.
a.) If m∠1 = 20°, find m∠2, m∠3, and m∠4.
b.) If m∠1 = 65°, find m∠2, m∠3, and m∠4.

b.) Prove that OY bisects ∠BOC. (You do not need to do a 2-column proof, but give an
argument for why this is true no matter what angle measure is given.)
4
End
Intro to Proofs (Day 2)
Part I: Read each statement and draw a picture that fits the situation being described. Then write the
statement and reason that follows directly from the one given. The first problem below is done as an
example for you.
Statements
Reasons
Picture
ex 1) ∠ 1 and ∠ 2 are supplementary 1) Given
2) m ∠ 1 + m ∠ 2 = 180º
(a) 1) F is the midpoint of AB
2) ____________________
2) Def. of supplementary
1
2
1) Given
2) ____________________
(b) 1) ∠ 1 and ∠ 2 are complementary 1) Given
2) ____________________
(c) 1) OC bisects ∠DOB
2) ____________________
(d) 1) EF bisects AB at point D.
2) ____________________
2) ____________________
1) Given
2) ____________________
1) Given
2) ____________________
(e) 1) Point G is on EF (not necessarily the midpoint)
2) ____________________
(f) 1) m ∠ 1 + m ∠ 2 = m ∠ ABC;
m∠1 = m ∠3
2) ____________________
(g) 1) ∠ 4 and ∠ 5 are vertical angles
1) Given
2) ____________________
1) Given
NO PICTURE
NEEDED
2) ____________________
1) Given
2) ____________________
2) ____________________
(h) 1) AB ⊥ BD
2) ____________________
1) Given
2) ____________________
5
Part II: Now let’s reverse the process. The last statement of a proof is given. Again, draw a picture that
fits each situation. Write a statement that directly precedes the last statement and write the reason for the
last statement.
Statements
Reasons
Picture
(a) 4) ____________________
5) ∠2 and ∠3 are supplementary
(b) 8) ____________________
9) OC bisects < JOK
5) ____________________
9) ____________________
(c) 6) ____________________
7) AB ⊥ BD
7) ____________________
(d) 5) ____________________
6) T is the midpoint of JR
6) ____________________
(e) 7) ____________________
8) AB = 12 AC
8) ____________________
A
B
6
C
Fill-in-the-Blanks Proofs
1) Given: DE = AF
EF bisects AB
Prove: DE =
1
AB
2
*Hint: Mark your diagram!
Statements
Reasons
1) EF bisects AB
1) ____________________
2) F is the midpoint of AB
1
3) AF = AB
2
2) ____________________
4) DE = AF
4) ____________________
5) ____________________
5) ____________________
3) ____________________
2) Given: ∠1 = ∠3
Prove: ∠PXR = ∠QXS
Statements
Reasons
1) ∠1 = ∠3
1)
2) ∠ 2 = ∠ 2
2)
3) ∠1 + ∠2 = ∠3 + ∠2
3) ____________________
4) ∠PXR = ∠1 + ∠2
∠QXS = ∠3 + ∠2
4) ____________________
5) ____________________
5) ____________________
7
3) Given: ∠2 and ∠3 are complementary
A
Prove: AB ⊥ CD
3
C
E
1
D
2
B
Statements
Reasons
1) ∠2 and ∠3 are complementary
1) ____________________
2) m∠2 + m∠3 = 90 o
2) ____________________
3) m∠2 = m∠1
3) ____________________
4) m∠1 + m∠3 = 90 o
4) ____________________
5) m∠1 + m∠3 = m∠AED
5) ____________________
6) ____________________
6) Substitution property (
7) ____________________
7) ____________________
)
4) Given: m∠BOA = m∠EOD
OC bisects ∠DOB
Prove: OC bisects ∠AOE
Statements
Reasons
1) m∠BOA = m∠EOD
OC bisects ∠DOB
1) ____________________
2) m∠BOC = m∠COD
2) ____________________
3) m∠BOC + m∠BOA = m∠COD + m∠EOD
3) ____________________
4) m∠BOC + m∠BOA = m∠COA
m∠COD + m∠EOD = m∠COE
4) ____________________
5) ____________________
5) Substitution property(
6) ____________________
6) ____________________
)
8
End
3-5 Congruent Triangles (Day 3)
E
Congruent Triangles:
B
D
A
F
C
What is the least amount of information we can know about a pair of
triangles to determine that they are congruent?
1. Using graph paper and a ruler, draw two triangles using line segments of 2”, 3”,
and 4” in each triangle. What conclusion can you make about these two
triangles?
2. Choose any 3 lengths and draw 2 more triangles with the same 3 lengths for the
sides. Does your conclusion above remain true for these two triangles?
3. On a blank page, use your ruler to draw a 2” segment that lines up with one side
of the angle drawn below.
4. Use your ruler to draw a 4” segment that lines up with the other side of the same
angle.
5. Connect the two lines that you drew to form a triangle.
6. Repeat this process with the second angle below.
What can you conclude about the third connector side of the two triangles?
What can you conclude about the two triangles?
Are there other combinations that create congruent triangles? Investigate with a partner until
you find 2 more combinations that will give you congruent triangles.
9
Congruence Theorems:
WCombinations that do NOT work
(do not prove Δs ≅ ):
X
Y
Z
Ex:
A
Q
B
Given: PQ ⊥ AB , ∠A ≅ ∠B ,
Q is the midpoint of AB .
Prove: ∆APQ ≅ ∆BPQ (2-column proof)
1. Given: XW ≅ XZ
WY ≅ ZY
Prove: ∆XWY ≅ ∆XZY
10
W
21
V
X 
Y it necessarily follow
2. (a) Consider the diagram below. If XY bisects ∠ WXZ, does
that ∠ 1 ≅ ∠ 2? Explain.
Z
End
(b) Given: ∠ WYV ≅ ∠ ZYV

XY bisects ∠ WXZ
Prove: ∆XWY ≅ ∆XZY
3. Draw and label a diagram corresponding to each situation below. Identify what is given
and what is to be proved. Then state the theorem you would use to prove it.
(a) If a line perpendicular to AB passes through the midpoint of AB, and segments are
drawn from any other point on that line to A and B, then two congruent triangles are
formed.
Diagram?
Given?
Prove?
Theorem to prove it?
(b) If pentagon ABCDE has all equal sides and has right angles at B and E, then diagonals
AC and AD form congruent triangles.
Diagram?
Given?
Prove?
Theorem to prove it?
11
HL postulate:
3-6 More Triangle Congruence (Day 4)
We saw before that SSA is not a viable postulate. Why not?
Does SSA ever work? If so, when?
Ex. 1 Given: ∆XWZ and ∆XYZ are right ∆s; WX ≅ YX .
Prove: ∆XWZ ≅ ∆XYZ
Ex. 2 Given: XY ⊥ AB; XA ≅ XB .
Prove: ∆AYX ≅ ∆BYX
12
Proofs Practice:
1. Given: m ∠ B = m ∠ C = 90  ,
∠ BAD ≅ ∠ CDA.
Prove: ∆ABD ≅ ∆DCA
A
B 1
1
A
1
2C
D
2
D
2
hint: redraw to separate ∆’s
2. Given: m ∠ B = m ∠ C = 90  ,
BA ≅ CD .
Prove: ∆ABD ≅ ∆DCA
3. Given: ∠ 1 ≅ ∠ 2,
∠ BAD ≅ ∠ CDA.
Prove: ∆ABD ≅ ∆DCA
13
4. (a) p. 127 #20
(b) p. 128 #27
Given: ∠1 ≅ ∠4; ∠2 ≅ ∠3.
Prove: ΔABD ≅ ΔBAC
Given: JK ⊥ KM ; LM ⊥ KM ; ∠4 ≅ ∠1
Prove: ΔJKM ≅ ΔLMK
Note: You need to use both the addition
property and the angle addition postulate.
Note: You need to use the angle addition
postulate and the subtraction property.
End
5. Draw and label a diagram corresponding to each situation below. Identify what is
given and what is to be proved. Then state the theorem you would use to prove it.
(a) In an isosceles triangle (i.e., two of its sides are congruent), if the angle between
the congruent sides is bisected, then two congruent triangles are formed.
Diagram?
Given?
Prove?
Theorem to prove it?
(b) In an isosceles triangle, if a segment is drawn from the angle between the congruent
sides to the midpoint of the opposite side, then congruent triangles are formed.
Diagram?
Given?
Prove?
Theorem to prove it?
14
Def. of ≅ Δ ’s (CPCTC)
Vertical
A Theorem:
A Angles
B B
C
D
D
3-7 CPCTC (Day 5)
C
E
E
Proof:
If you knew that ΔABC ≅ ΔEDC, what could you conclude
about AB and DE?
Ex. 1.
Given: AE & BD bisect each other.
Prove: AB ≅ DE
15
GR
R
2
P
1
3
X
X
4
1. Given: GH ⊥ JK
JG ≅ KG
Prove:J ∠J ≅ ∠K
2. Given: ∆PRX ≅ ∆PSX
Prove: ∆PRQ ≅ ∆PSQ
H SS
Q
Q
K
3. Given: ∆QRX ≅ ∆QSX
Prove: ∆PRX ≅ ∆PSX
Hint: ∠3 and ∠RXQ are supplementary.
∠4 and ∠SXQ are supplementary.
16
1 2
M
4. Given: ∆DEF ≅ ∆DCF; ∠2 ≅ ∠3 .
E
M
5. Given: MR ≅ PR ; MX ≅ PX
R
P
RX
X
A
B
Prove: ∆EFA ≅ ∆CFB
S1 2 3 4
3 4 5D6
3 4
End
C
F
P
S
Prove: ∠3 ≅ ∠4
E
6. Given: RS ⊥ MP ; MR ≅ PR .
Prove: MX ≅ PX
17
B
If AE = DE,
B
which angles
are
congruent?
E
3-8 Isosceles Triangles
(Day 6)
1
2
C
A
1. Draw BD that bisects ∠ ABC.
Is this always possible?
A
A
B
C
C
D
2. Draw BD ’ that bisects AC . (This is the median of ∆ABC) Is this always possible?
3. Can you make a triangle such that one segment bisects both an angle and the opposite
side at the same time?
Prove the Isosceles Triangle Theorem:
4. Given: AB ≅ CB
Prove:
5. Given: AE ≅ DE ; ∠1 ≅ ∠2 .
(Hint: need auxiliary line BD)
Prove: ∆AEB ≅ ∆DEC.
18
L
Q
6. Given: ML ≅ NL ; ∠PNM ≅ ∠QMN P Prove:
Try
separatingInstead of a
median,
the
draw . . .
≅
∆MPN
∆NQM
triangles.
N
M
7. The converse of the Isosceles ∆ Theorem is:
If
, then
.
Consider the following attempt at its proof.
Given ∆ABC with ∠ A ≅ ∠ C. Draw median BD from vertex B
to side AC. Thus, AD ≅ CD; and since BD ≅ BD, we have
∆ABD ≅ ∆CBD. So, AB ≅ BC by CPCTC.
(a) Explain what is wrong with this proof.
(b) Now write a correct proof.
19
8. (a) Do p. 136 #17
(b) Do p. 137 #26
Hint: Use what you just proved.
Given:
∠1 ≅ ∠2; ∠3 ≅ ∠4;
∠5 ≅ ∠6.
Prove: LK bisects MN
(c) Do p. 137 #32
Hint: You need to prove 2 pairs of
Given:
CX ≅ DX ;
AD ≅ BC.
Prove: AC ≅ BD
≅ Δ.
RS ⊥ PQ;
End.
∠PRS ≅ ∠PSR
Prove: RQ ≅ SQ
Given:
9. Draw any triangle, neatly, using a ruler. Carefully measure your side lengths in mm. Draw
the three medians of your triangle. Try this with a couple of different triangles. Explain what
you observe.
A
What if you draw the three angle bisectors instead?
20
B
Corollary to the Triangle Sum Theorem
90o
Triangle Sum Theorem
The 3-9
The sum of the measures of the
A
40o
triangle isTriangle Sum
°. Theorem
angles of a right triangle
are 7)
Geometric Inequalities
(Day
angles of a
Xo
C
.
D
Place triangle
here.
Corollary to the Triangle Sum Theorem
Definition of Corollary:
Remote Interior Angles Theorem
Definition of Exterior Angle:
Find the value of x. _______ What can you conclude?
21
The measure of an
angle of a triangle is equal to
the Inequality Theorem
of the
measures of the two
terior Angles
(EAIT)
angles.
Let’s show this with paper…
Place triangle
on this line.
Now let’s prove it!
Consider ∆ABC shown below. Prove that m∠4 = m∠1 + m∠2 .
A corollary…
22
arger Side Opposite Larger Angle (LSOLA):
P
S
1
What is your
2nd to last step?
Q
2
End
3
R
3. What’s wrong with the following argument?
D-Stix” Lab
Here BC > AB. Also ∠ 1 is opposite BC and ∠ 2 is opposite AB. Since the longer side is
opposite a larger angle, it follows that m ∠ 1 > m ∠ 2.
You have 4 colored sticks. There are 4 possible ways to choose exactly 3 of them. List these
choices by color.
a.)
b.)
c.)
d.)
The lengths of the sticks are: red 12 in.
green 7 in.
blue 5 in.
yellow 4 in.
Put these numbers in the lists above.
4. Given m ∠ Q > m ∠ 2
Prove: PR > RQ
continued…
3-9 Geometric Inequalities continued (Day 8)
23
each
of the Inequality
4 choices attempt
to put
sticks end to end to form a triangle. In some cases this
Triangle
Theorem
( Δthe
≠ Thm):
S
Q
ot be done. State yes or no for each choice.
P
1
2
b.)
c.)
d.)
Write a sentence or two describing when it can and cannot be done.
3
R
o write a theorem regarding a requirement on three numbers that form the sides of a triangle.
Then…)
nsion: State a set of four numbers that could not be the sides of a quadrilateral.
1. If ∆ABC has AB = 3cm and BC = 7cm, what are the possible lengths of AC?
Explain (remember, side lengths do not have to be integers).
Proofs Practice:
2. Given: PR > RQ; RS > PS.
Prove: m ∠ Q > m ∠ 3
24
Z
Q
E
3. Given: XZ = XY; ZW = ZY.
Prove: ZY > WY
2
X
K
P
Y
W
3 1
A
End
C
R
4. Given: QR > QP
PR ≅ PQ
Prove: m∠P > m∠Q
5. Given: AC ≅ AE
AE ≅ KE
Prove: m∠1 > m∠2
25