Lesson 2-6

Algebraic Proof

5-Minute Check on Lesson 2-55-Minute Check on Lesson 2-55-Minute Check on Lesson 2-55-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 eachstatement 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. Which statement is not supported by a postulate?Standardized Test Practice:

A

C

B

D

R and S are collinear. M lies on LM.

P, X and Y must be collinear. J, K and L are coplanar.

5-Minute Check on Lesson 2-55-Minute Check on Lesson 2-55-Minute Check on Lesson 2-55-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 eachstatement 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. Which statement is not supported by a

postulate?Standardized Test Practice:

A

C

B

D

R and S are collinear. M lies on LM.

P, X and Y must be collinear. 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 NumbersReflexive 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 & Subtraction

For all numbers a, b, and c, if a = b, 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

2(5 – 3a) – 4(a + 7) = 92 Original equation

Algebraic Steps Properties

10 – 6a – 4a – 28 = 92 Distributive Property–18 – 10a = 92 Substitution

Property –18 + 18 – 10a = 92 + 18 Addition Property

Solve 2(5 – 3a) – 4(a + 7) = 92

–10a = 110 Substitution Property

Division Property

a = – 11 Substitution Property

Answer: a = – 11

IfWrite a two-column proof. then

Statements ReasonsProof:

1. Given1.

2. 2. Multiplication Property

3. 3. Substitution

4. 4. Subtraction Property

5. 5. Substitution

6. 6. Division Property

7. 7. Substitution

1. Given

2. Multiplication Property

3. Substitution4. Subtraction Property

5. Substitution

6. Division Property

7. Substitution

Proof:Statements Reasons

1.

2.

3. 4.

5.

6.

7.

Write a two-column proof.

a.

Proof:Statements Reasons

1. Given

2. Multiplication Property

3. Distributive Property

4. Subtraction Property

5. Substitution

6. Subtraction Property

7. Substitution

1.

2.

3.

4.

5.

6.

7.

Write a two-column proof.

b. Given: Prove: a = –5

Read the Test Item

Determine whether the statements are true based on the given information.

A I only B I and II C I and III D I, II, and III

MULTIPLE- CHOICE TEST ITEM

then which of the following is a valid conclusion?

I

II

III

If and

Answer: B

If and then which of the following is a valid conclusion?

I.

II.

III.

MULTIPLE- CHOICE TEST ITEM

A I only B I and II C I and III D II and III

Answer: C

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.

Given:

m leg 1 22 cm

Prove: m leg 3 22 cm

Proof:

Statements Reasons

1. Given1.

2. Transitive Property2.

3. Definition of congruencem leg 1 m leg 33.

4. Givenm leg 1 22 cm4.

5. Transitive Propertym leg 3 22 cm5.

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