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Lesson 2-6
Algebraic Proof
5-Minute Check on Lesson 2-5
Transparency 2-6
In the figure shown, A, C, and DH lie in plane R, and B is on AC.
State the postulate that can be used to show each
statement is true.
1. A, B, and C are collinear.
2. AC lies in plane R .
3. A, H, and D are coplanar.
4. E and F are collinear.
5. DH intersects EF at point B.
6.
Standardized Test Practice:
Which statement is not supported by a
postulate?
A
C
R and S are collinear.
P, X and Y must be collinear.
B
D
M lies on LM.
J, K and L are coplanar.
5-Minute Check on Lesson 2-5
Transparency 2-6
In the figure shown, A, C, and DH lie in plane R , and B is on AC.
State the postulate that can be used to show each
statement is true.
1. A, B, and C are collinear. A line contains
at least two points.
2. AC lies in plane R . If two points lie in a plane, then the entire line
containing those points lies in that plane.
3. A, H, and D are coplanar. Through any 3 points not on the same line,
there is exactly one plane.
4. E and F are collinear. Through any 2 points, there is exactly one line.
5. DH intersects EF at point B. If two lines intersect, then their intersection
is exactly one point.
6. Standardized Test Practice:
Which statement is not supported by a
postulate?
A
C
R and S are collinear.
P, X and Y must be collinear.
B
D
M lies on LM.
J, K and L are coplanar.
Objectives
• Use algebra to write two-column proofs
• Use properties of equality in geometry proofs
Vocabulary
• Deductive argument – a group of logical steps
used to solve problems
• Two-column proof – also known as a formal
proof
Algebraic Properties
Properties of Equality for Real Numbers
Reflexive
For every a, a = a
Symmetric
For all numbers a and b, if a = b, then b = a
Transitive
For all numbers a, b, and c, if a = b and b = c,
then a = c
Addition
For all numbers a, b, and c, if a = b,
& Subtraction
then a + c = b + c and a – c = b - c
Multiplication
& Division
For all numbers a, b, and c, if a = b,
then ac = bc and if c ≠ 0, a/c = b/c
Substitution
For all numbers a and b, if a = b, then a may be
replaced by b in any equation or expression
Distributive
For all numbers a, b, and c, a(b + c) = ab + ac
Solve 2(5 – 3a) – 4(a + 7) = 92
Algebraic Steps
Properties
2(5 – 3a) – 4(a + 7) = 92
10 – 6a – 4a – 28 = 92
–18 – 10a = 92
–18 + 18 – 10a = 92 + 18
–10a = 110
Original equation
Distributive Property
Substitution Property
Addition Property
Substitution Property
Division Property
a = – 11
Answer: a = – 11
Substitution Property
Write a two-column proof. If
then
Proof:
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
3. Substitution
4.
4. Subtraction Property
5.
5. Substitution
6.
6. Division Property
7.
7. Substitution
Write a two-column proof.
a.
Proof:
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
4.
5.
3. Substitution
4. Subtraction Property
5. Substitution
6.
6. Division Property
7.
7. Substitution
Write a two-column proof.
b. Given:
Prove: a = –5
Proof:
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
4.
3. Distributive Property
4. Subtraction Property
5.
5. Substitution
6.
6. Subtraction Property
7.
7. Substitution
MULTIPLE- CHOICE TEST ITEM
If
and
then which of the
following is a valid conclusion?
I
II
III
A I only
B I and II
C I and III
D I, II, and III
Read the Test Item
Determine whether the statements are true based
on the given information.
Answer: B
MULTIPLE- CHOICE TEST ITEM
If
and
then
which of the following is a valid conclusion?
I.
II.
III.
A I only
Answer: C
B I and II
C I and III
D II and III
SEA LIFE A starfish has five legs. If the length of leg 1 is 22
centimeters, and leg 1 is congruent to leg 2, and leg 2 is
congruent to leg 3, prove that leg 3 has length 22 centimeters.
Prove: m leg 3
Given:
m leg 1
22 cm
22 cm
Proof:
Statements
Reasons
1.
1. Given
2.
2. Transitive Property
3. m leg 1
m leg 3
3. Definition of congruence
4. m leg 1
22 cm
4. Given
5. m leg 3
22 cm
5. Transitive Property
Summary & Homework
• Summary:
– Algebraic properties of equality can be
applied to the measures of segments and
angles to prove statements
• Homework:
– pg 97-8: 4-9, 15-18, 24, 25