Beginning
Segment
Proofs
Name: _________________
Period: ______
Teacher: _______________
Table of Contents
Day 1 : SWBAT: draw conclusions by applying geometric definitions.
Pages 1-4
HW: page 5
Day 2: SWBAT: draw conclusions by applying geometric definitions.
Pages 6-7
HW: pages 8-9
Day 3: SWBAT: apply definitions and theorems to write geometric proofs.
Pages 10-11
HW: pages 12-13
Day 4: SWBAT: apply the properties of equality and the Addition Postulate to
write geometric proofs.
Pages 14-19
HW: pages 20-22
Day 5: SWBAT: apply the Subtraction postulate to write geometric proofs.
Pages 23-26
HW: pages 27-29
Day 6: SWBAT: apply the multiplication and Division postulate to write geometric
proofs.
Pages 30-33
HW: pages 34-37
Day 7: SWBAT: apply postulates, definitions and theorems to write geometric
proofs.
Pages 38-43
Day 8: SWBAT: apply postulates, definitions and theorems to write geometric
proofs.
Pages 44-45
Day 9: Exam
The Building Blocks of Proof
All from www.wikipedia.com
Part of the problem you may have with this unit is that many of the things I will
ask you to formally prove is either (a) intuitively obvious, or (b) has been proven
by someone else already. But that is the burden of proof, to prove even the
painfully obvious, using only the smallest accepted truths, until you have shown
without a shadow of a doubt that you’ve attained an irrefutable truth.
Definitions:
A definition is a statement of the meaning of a word. There is a big difference
between the word “definition” and “property.” For example, there may be one
acceptable definition for the word “parallelogram,” but it has many properties.
Postulate (or Axiom):
“a statement or assumption that is agreed by everyone to be so obvious or selfevident that no proof is necessary, and which can be used to prove other
statements or theorems.”
Example: Partition postulate – A whole is equal to the sum of its parts.
Theorem:
“a proposition that has been… proved on the basis of explicit assumptions.” A
key property of theorems is that they possess proofs, not merely that they are
true.”
Example: Pythagorean theorem – the sum of the squares of the legs of a right
triangle are equal to the square of its hypotenuse.
Corollary:
“a proposition that follows with little or no proof from one other theorem or
definition. “
Examples
Theorem: If 2 sides of a triangle are congruent, then the angles opposite
those sides are also congruent.
Corollary: If a triangle is an equilateral triangle, then it is equiangular.
1
Review of Definitions
1:
The midpoint of a line segment divides the segment into two congruent segments.
Ex: If M is the midpoint of AB , then _____________________________________________.
2:
The bisector of a line segment intersects a line segment at its midpoint.
Ex: If DE bisects AB , then _____________________________________________.
3: An angle bisector divides an angle into two congruent angles.
Ex: If ⃗⃗⃗⃗⃗
𝐵𝐸 bisects ABC, then ___________________________________________.
2
4. Perpendicular Lines (segments) intersect to form right angles.
Ex: If AB  CD then _____________________________________________.
** Note **
5. The perpendicular bisector of a line segment is perpendicular to the line segment and bisects the line
segment.
Ex:
Then __________________ and ______________________.
6. An altitude is a line segment drawn from any vertex of the triangle that is perpendicular to and ending
in the line containing the opposite side.
Ex:
, then ________________________
3
7. A median is a line segment drawn from any vertex of a triangle to the midpoint of the opposite side.
Ex:
Then ______________________.
8.
Ex: If ∆ABC is an equilateral triangle, then _________________ and ____________________.
9.
Ex: If
Ex: If
Then _________________________________.
Then _________________________________.
4
∡3 are supplementary
5
Day 2
Homework – Day 1
6
7
Day 2 - HW
1.
Write a conclusion using  .
a)
If B is the midpoint of AD , then _____________.
b)
If BO bisects AD , then _____________.
c)
If BE bisects ∡DBO, then _____________.
d)
If BE bisects AD , then _____________.
2.
a)
Name two congruent segments _____________
b)
Name two congruent angles _____________
c)
CE + EF = _____________.
d)
FG + EF = _____________.
e)
CF - EF = _____________.
f)
m∡CHF - m∡EHF = _____________.
g)
∡CHE + ∡EHG  _____________.
3.
H
Given: HE is the angle bisector of CHF
F is the midpoint of CG
C
E F
G
State whether each of the following statements is True or False.
Given: ∡APQ is a right angle.
P is the midpoint of AB
a)
RP bisects AB
b)
∡1 is a right angle.
c)
AB + AP = PB
d)
AP  PB
e)
QP bisects AB
f)
A line has exactly one bisector.
g)
If RP = PB then P is the midpoint of RB
h)
The endpoint of AB is A.
i)
∡APR and ∡RPB is equal to 180.
j)
∡1 is adjacent to ∡3.
k)
∡1 and ∡2 form a linear pair.
8
4.
Write a valid conclusion(s) based on the given information.
a)
DC bisects ∡ACB _____________________
b)
BE is an altitude to ∆ACB _____________________
c)
DC and BE bisect each other__________________
5.
Write a valid conclusion based on the given information.
a)
F is the midpoint of DC _________________________
b)
AE is the perpendicular bisector of FD _________________________________
c)
CF + FE = _____________
d)
AB – AF = _____________
3. Write a valid conclusion based on the given information.
a)
SF is a median to ∆SAD ____________________
b)
DL bisects SF ____________________
c)
DL is an angle bisector____________________
4. Write a valid conclusion based on the given information.
a)
TM is the perpendicular bisector of GP ____________________
b)
∡TMP - ∡TMR  ____________________
c)
∡TMQ + ∡QMG  ____________________
9
Day 3 - Proofs with Definitions
1. Given:
Prove:
RT bisects QS at M.
QM  MS
____ is the midpoint of ____
____  ____
2. Given:
Prove:
JT  CP
∡CJT  ∡PJT
____ and ____ are right ∡𝒔
____  ____
3. Given:
Prove:
⃗⃗⃗⃗⃗⃗⃗
𝐷𝐵 bisects ∡ABC
∡ABD  ∡DBC
______  _______
10
O
4. Given:
Prove:
OR is a median to ∆OPS
PR  RS
S
P
R
____ is the midpoint of ____
____  ____
A
5. Given:
Prove:
____
AM is an altitude to ∆ABC
∡BMA  ∡AMC
B
M
C
____
____ and ____ are right ∡𝒔
____  ____
11
Day 3 - Homework
1.
2.
(3)
3.
(3)
4.
∡1 & ∡2
are right ∡s
(3)
(3)
5.
6.
(3)
(3)
(4)
(4)
(3)
(3)
12
7.
8.
9.
10.
13
Day 4 - Proving Segment Relationships
A proof is an argument that uses logic, definitions, properties, and previously proven
statements to show that a conclusion is true.
You learned in Chapter 1 that segments with equal lengths are congruent and
that angles with equal measures are congruent. So the Reflexive, Symmetric,
and Transitive Properties of Equality have corresponding properties of
congruence. An important part of writing a proof is giving justifications to
show that every step is valid.
Example 1:
14
Example 2:
Example 3:
15
If equal quantities are added to equal quantities, the sums are
equal.
OR
If congruent quantities are added to congruent quantities, the sums
are equal.
Addition of Segments
Addition of Angles
Writing Proofs
16
Example 3:
Example 4:
17
You Try!
Given: ̅̅̅̅
𝐾𝑅 ≅ ̅̅̅̅
𝑆𝑂
̅̅̅̅̅
𝑅𝑀 ≅ ̅̅̅̅
𝑃𝑆
Prove: ̅̅̅̅̅
𝐾𝑀 ≅ ̅̅̅̅
𝑂𝑃
Example 5:
You Try!
18
Summary
Write a valid conclusion for each.
1. Fdfdf
Conclusion: _______________
R
2. Given: ∡SRT  ∡VRY
X
Conclusion: _______________
S
T
Y
V
19
Day 4 – Homework
2.
20
3.
21
5. Fddfd
6.
22
Day 5 – The Subtraction Postulate
Subtraction of Segments
Example 1:
Example 2:
Subtraction of Angles
Example 3:
23
Writing Proofs
Example 4:
Example 5: Given: ∡STE  ∡WET
∡STW  ∡WES
Prove: ∡WTE  ∡SET
S
W
T
E
24
Example 6:
Example 7:
O
Given: ∡TOB  ∡WOM
Prove: ∡TOM  ∡WOB
X
T
M
W
B
25
SUMMARY
1.
REASON:
2.
REASON:
26
Day 5 – HW
1.
27
3.
4.
28
5.
6.
29
Day 6 - Applying the Multiplication/Division Postulate
Writing Proofs
 Look for key words like: midpoint, bisect, and trisect
 The Multiplication property is used when the segments or
angles in the given are less than those in the conclusion!
 * Small to Big*
30
Ex 1:
31
Writing Proofs
 Look for key words like: midpoint, bisect, and trisect
 The Division property is used when the segments or angles in
the given are greater than those in the conclusion!
* Big to Small*
Example 2:
32
Summary
 Look for key words like: midpoint, bisect, and trisect
 The Multiplication property is used when the segments or
angles in the given are less than those in the conclusion!
* Small to Big*
 The Division property is used when the segments or angles in
the given are greater than those in the conclusion!
* Big to Small*
1.
REASON:
2. 2.
REASON:
33
Day 6 – HW
1.
2.
34
3.
4.
35
5.
36
6. *Challenge*
37
Day 7 - REVIEW: Proofs with Postulates
1.
B
2. Given:
Prove:
AC  BC
BC  BA
AC  BA
C
A
38
3. Given:
Prove:
TS  YX , SR  XR
T
R
TR  YR
Y
4. Given:
Prove:
S
X
∡1  ∡2, ∡3  ∡4
∡CDA  ∡CBA
39
A
5. Given:
Prove:
AD  AE , DB  EC
D
AB  AC , ∆ABC is isosceles
E
X
B
C
C
6. Given: AE  BD
Prove: AD  BE
X
A
D
E
B
40
7.
41
8.
42
9.
43
Day 8 - REVIEW: Proofs with Postulates
44
45