Download Beginning Proofs

Geometry A
HW #
Row:
Beginning Proofs 2
Name:
KNOW YOUR ALGEBRAIC PROPERTIES:
Properties of equalities and congruence:
1.
Addition/Subtraction P of E : If a = b, and c = d, then a + c = b + d and a – c = b – d.
2.
Multiplication P of E : If a = b, and c = d, then ac = bd
3.
Division P of E : If a = b, c = d, and c and d  0, then a / c = b / d.
4.
Reflexive P of E: For all real number a, a = a.
5.
Symmetric P of E : If a = b, then b = a.
6.
Transitive P of E : If a = b and b = c, then a = c.
7.
Substitution Property : a and b are real numbers. If a = b, then a can be replaced by b and b can be
replaced by a.
8.
Reflexive P of Congruence : a  a.
Reflexive Prop. a = a
9.
Symmetric P of Congruence : If a  b, then b  a. Symmetric Prop. If a = b, then b = a
10.
Transitive P of Congruence : If a  b and b  c, then a  c. Transitive Prop. If a = b and b = c, then a = c
*What is the difference between addition and the Addition Property of Equality?
*What is the difference between the commutative property of additon/multiplication and the Symmetric Property?
*What is the difference between the Transitive Property and the Substitution Property?
Example 1.
Given: AB @ CD
A
Prove: AC @ BD
B
C
D
Proof:
Ref
Statements
Reasons
1. AB @ CD ; AB = CD
G
2. BC @ BC ; BC = BC
Fig.
2. Reflexive Property; Def. of 
3. AB + BC = BC + CD
1, 2
3. Addition P of E
4. AB + BC = AC
Fig.
4. Segment Addition Postulate (SAP)
5. BC + CD = BD
Fig.
5. SAP
6. AC = BD; AC @ BD
4, 5
6. Substitution; Def. of 
1. Given; Definition of Congruence
Example 2.
P
Given: RPT  UPS
Prove: RPS  UPT
1
2
3
R
U
S
Statements
1. RPT  UPS; mRPT = mUPS
T
Proof:
Ref
G
Reasons
1. Given; Def. of 
2. m2 = m2
Fig.
2. Reflexive Property
3. mRPT  m2 = mUPS  m2
1, 2
3. Subtraction P of E
4. m1 + m 2 = mRPT
Fig.
4. Angle Addition Postulate (AAP)
5. m2 + m3 = mUPS
Fig.
5. AAP
6. m1 = mRPT  m 2
4
6. Subtraction P of E
7. m3 = mUPS  m2
5
7. – P of E
Page 1
Geometry A
HW #
Row:
Beginning Proofs 2
6,7
8. m1 = m3
9. RPS  UPT
8
Exercise 1:
Given: m1 = m3; m2= m4
Prove : mABC = mDEF.
Name:
8. Substitution
9. Def. of congruence
G
C
1
F
H
2
4
B
A
D
Proof:
Ref
Statements
Reasons
1. m1 = m3
1. Given
2. m2= m4
2.
3. m1 + m2 = m3 + m4
3.
4. m1 + m2 =
4.
5. m3 + m4 =
5.
______________
6.
6. mABC = mDEF
Exercise 2:
Given: ST = RN; IT = RU.
Prove: SI = UN.
S
R
T
I
U
Statements
T
N
Proof:
Ref
Reasons
1. ST = RN
1. Given
2. IT = UN
2.
3. SI + IT =
3.
4. RU + UN =
____________
4.
5. SI + IT = RU + UN
5.
6.
6. Substitution Property
=
7.
7.
Page 2
3
E
Geometry A
HW #
Row:
Beginning Proofs 2
Name:
Exercise 3: Prove Theorem 2-1 (The Midpoint Theorem)
If M is the midpoint of AB , then AM = ½ AB and MB = ½ AB.
Given: M is the midpoint of AB
Prove: AM = ½ AB and MB = ½ AB
Statements
A
M
B
Proof:
Ref
Reasons
1. M is the midpoint of AB
1. Given
2. AM  MB (AM = MB)
2.
3. AM + MB = AB
3.
4. AM + AM = AB
4.
5. 2AM = AB
5.
6. AM = ½ AB
6.
7. MB = ½ AB
7.
Exercise 4: Prove Theorem 2-2 (The Angle Bisector Theorem)
If BX is the angle bisector of ABC, then mABX = ½ mABC and mXBC = ½ mABC.
Given:
Prove:
A
X
B
C
Statements
Proof:
Ref
Reasons
1.
1. Given
2.
2. Definition of angle bisector
3. mABX + mXBC = mABC
3.
4.
4. Substitution
5.
5. Addition
6.
6.
7.
7.
Page 3
Geometry A
HW #
Row:
Beginning Proofs 2
Name:
Exercise 5: Prove Theorem 2-7 (The Supplement Theorem)
If two angles are supplements of  angles (or the same angle), then the two angles are .
(Supplements of  angles are .)
Restatement:
Given: A  B; C is supplementary to A,
and D is supplementary to B.
Prove: C  D
Statements
A
C
B
Proof:
Ref
D
Reasons
1. C is supplementary to A
1. Given
2. D is supplementary to B
2.
3.
3. Def. of supplementary
4.
4.
5. mA + mC = mB + mD
5.
6.
6. Given
7. mC = mD; (C  D)
7.
Exercise 6: Prove Theorem 2-8 (The Complement Theorem)
If two angles are complements of  angles (or the same angle), then the two angles are .
( Complements of  angles are .)
Restatement:
Given:
Diagram:
Prove:
Proof (Add more lines if needed):
Statements
Ref
1.
1. Given
2.
2.
3.
3.
4.
4.
5.
5.
Page 4
Reasons