Algebraic and Geometric Proofs
Notes & Homework Assignment
Name_______________________________ Class Period_______
When you solve an equation, you use properties of real numbers. Segment lengths
and angle measures are real numbers, so you can also use these properties to write
logical arguments about geometric figures.
Algebraic Properties of Equality
Let a, b, and c be real numbers.
Addition Property
Multiplication Property
Substitution Property
Distributive Property
If a = b, then a + c = b + c.
If a = b, then ac = bc.
If a = b, then a can be substituted for b in any
equation or expression.
a(b + c) = ab + ac
Write reasons for each step: Solve 2x + 5 = 20 – 3x
Equation
Reason
2x + 5 = 20 – 3x
Given
Write reasons for each step: Solve -4(11x + 2) = 80.
Statement
Reason
-4(11x + 2) = 80
Given
Solve the equation and state a reason for each step.
I.
II.
III.
IV.
Statement
Reason
5x – 18 = 3x + 2
Given
Statement
Reason
55z – 3(9z + 12) = -64
Given
Statement
Reason
3(4v – 1) – 8v = 17
Given
Before exercising, you should find your target heart rate. This is the rate at which you
achieve an effective workout while not placing too much strain on your heart. Your target
heart rate (in beats per minute) can be determined from your age a (in years) using the
equation a = 220 – 10/7(r),
a) Solve the formula for r and write a reason for each step.
b) Use the result to find the target heart rate for a 16 year old and a 51 year old.
Statement
Reason
PROPERTIES The following properties of equality are true for all real numbers. Segment lengths
and angle measures are real numbers, so these properties of equality are true for segment lengths
and angle measures.
Reflexive Property of Equality
Real Numbers
a = a (remember segment lengths and angle
measures are real numbers!)
Reflexive Property of Congruency
Segments
AB  AB
Angles
A  A
Symmetric Property of Equality
Real Numbers
If a = b, then b = a (again, lengths & measures)
Symmetric Property of Congruency
Segments
If AB  CD, then CD  AB
Angles
If A  B, then B  A
Transitive Property of Equality
Real Numbers
If a = b and b = c, then a = c
Transitive Property of Congruency
Segments
If AB  CD and CD  EF , then AB  EF
Angles
If A  B and B  C, then A  C
Mini-Proofs:
Given: AB  CD and CD  EF
Statement
1. AB  CD
2. CD  EF
3. AB  EF
Given: M is between A & B and MB = PQ
Statement
1. M is between A & B
2. AM + MB = AB
3. MB = PQ
4. AM + PQ = AB
Prove: AB  EF
Reason
1. Given
2. Given
3. ____________________
Prove: AM + PQ = AB
Reason
1. Given
2. ____________________
3. Given
4. ____________________
1. Solve the equation and write a reason for each step: 14x + 3(7 – x) = -1
1
2. Solve the formula A  bh for b.
2
In exercises 3–7, use the property to complete the statement.
3. Substitution Property of Equality: If AB = 20, then AB + CD = __________.
4. Symmetric Property of Equality: If m1  m2, then ___________.
5. Addition Property of Equality: If AB = CD, then _____ + EF = ______ + EF.
6. Distributive Property: If 5(x + 8) = 2, then ___x + ____ = 2.
7. Transitive Property of Equality: If m1  m2 and m2  m3, then ____________.
8. Show that the perimeter of
triangle ABC is equal to the
perimeter of triangle ADC.
Statement
AD = AB
DC = BC
AC = AC
AD + DC + AC = AB + BC + AC
Given: AD = AB
DC = BC
Reason
9. Given: m1  m4,
mEHF  90,
mGHF  90
Prove: m2  m3
Statements
1) m1  m4
Reasons
1)
2) mEHF  90, mGHF  90
2)
3) mEHF  mGHF
3)
4) mEHF  m1  m2 and
mGHF  m3  m4
5) m1  m2  m3  m4
4)
6) m1  m2  m3  m1
6)
7) m2  m3
7)
5)
10.
Use the property to complete the statement.
11. Reflexive Property of Congruence: _____  SE.
12. Symmetric Property of Congruence: If _____  _____, then RST  JKL.
13. Transitive Property of Congruence: If F  J and _____  _____, then F  L.
Use the choices A-K to name the property demonstrated by the exercises. Choices may be used more than
once and each exercise may have more than one correct choice.
A.
B.
C.
D.
E.
F.
Associative Property
Commutative Property
Distributive Property
Reflexive Property
Symmetric Property
Transitive Property
G.
H.
I.
J.
K.
Substitution Property
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
1.
6x2 + x = x(6x + 1)
________
2.
(m1 + m2) + m3 = m1 + (m2 + m3)
________
3.
(m1 + m2) + m3 = (m2 + m1) + m3
________
4.
If AB + BC = AC, then BC + AB = AC
________
5.
2(AB)(MN) = (AB)(2)(MN)
________
6.
If mA = mB and mB = 35, then mA = 35
________
7.
If AB + BC = AC, then AC = AB + BC
________
8.
If mP – mT = 75 and mP = 115, then 115 – mT = 75
________
9.
BD = BD
________
10.
If PQ + QR = MN and MN = ST + UV, then PQ + QR = ST + UV ________
11.
If AB + BC = AC and BC = 15 cm, then AB + 15 = AC
________
12.
mABC = mABC
________
13.
AB + BC = PQ, therefore AB = PQ – BC
________
14.
mA = mB, therefore mA + 90 = mB + 90
________
15.
2(PQ) = 16 m, therefore PQ = 8 m
________
16.
mP =
17.
If m1 + m2 + m3 + m4 = 360 and m2 + m3 = 180,
18.
1
(mQ), therefore 2(mP) = mQ
2
________
then m1 + m4 = 180
________
If mP – 86 = 150, then mP = 236
________