Pre-AP Geometry 1 Unit 2: Deductive Reasoning. Pre-AP Geometry 1 Unit 2 2.1 If-then statements,...

Pre-AP Geometry 1

Unit 2: Deductive Reasoning

Pre-AP Geometry 1 Unit 2

2.1 If-then statements, converse, and biconditional statements

Conditional Statements

• Conditional Statement-– A statement with two parts (hypothesis and conclusion) – Also known as Conditionals

• If-then form– A way of writing a conditional statement that clearly showcases

the hypothesis and conclusion p→q• Hypothesis-

– The “if” part of a conditional statement– Represented by the letter “p”

• Conclusion– The “then” part of a conditional statement– Represented by the letter “q”

Conditional Statements

• Examples of Conditional Statements– If today is Saturday, then tomorrow is Sunday.– If it’s a triangle, then it has a right angle.– If x2 = 4, then x = 2.– If you clean your room, then you can go to the

mall.– If p, then q.

Conditional Statements

• Example 1• Circle the hypothesis and underline the conclusion in

each conditional statement

– If you are in Geometry 1, then you will learn about the building blocks of geometry

– If two points lie on the same line, then they are collinear

– If a figure is a plane, then it is defined by 3 distinct points

Conditional Statements

• Example 2• Rewrite each statement in if…then form

– A line contains at least two points

– When two planes intersect their intersection is a line

– Two angles that add to 90° are complementary

If a figure is a line, then it contains at least two points

If two planes intersect, then their intersection is a line.

If two angles add to equal 90°, then they are complementary.

Conditional Statements

• Counterexample– An example that proves that a given

statement is false

• Write a counterexample– If x2 = 9, then x = 3

Conditional Statements

• Example 3– Determine if the following statements are true

or false. – If false, give a counterexample.

• If x + 1 = 0, then x = -1• If a polygon has six sides, then it is a decagon.• If the angles are a linear pair, then the sum of the

measure of the angles is 90º.

Conditional Statements

• Converse– Formed by switching the if and the then part.

• Original– If you like green, then you will love my new shirt.

• Converse– If you love my new shirt, then you like green.

Biconditional Statements

• Can be rewritten with “If and only if”• Only occurs when the statement and the

converse of the statement are both true.• A biconditional can be split into a conditional and

its converse.• p if and only if q• All definitions can be written as biconditional

statements

Example

• Give the converse of the statement. – If the converse and the statement are both

true, then rewrite as a biconditional statement

1. If it is Thanksgiving, then there is no school.

2. If an angle measures 90º, then it is a right angle.

Quiz- Get out a piece of paper and answer the following questions:

Underline the hypothesis and circle the conclusion. Then, write the converse of the statement. If the converse and the statement are true, rewrite as a biconditional statement. If not, give a counterexample.1. If a number is divisible by 10, then it is divisible by 5.

2. If today is Friday, then tomorrow is Saturday.3. If segment DE is congruent to segment EF, then

E is the midpoint of segment DF.

Assignment

• Lesson 2.1

• P. 35 #2-30 even

Pre-AP Geometry 1 Unit 2

2.2: Properties from Algebra

p. 37

Properties of equality

• Addition property– If a = b, then a + c = b + c

• Subtraction property– If a = b, then a – c = b – c

• Multiplication property– If a = b, then ac = bc

• Division property– If a = b, then

cbca

Reasoning with Properties from Algebra

• Reflexive property– For any real number a, a = a–

• Symmetric property– If a=b, then b = a– If

• Transitive Property– If a = b and b = c, then a = c– If ∠D ∠E and ∠E ∠F, then ∠D ∠F

• Substitution property– If a = b, then a can be substituted for b in any equation or expression

• Distributive property– 2(x + y) = 2x + 2y

EFEF

DEFGthenFGDE ,

Two-column proof

• A way of organizing a proof in which the statements are made in the left column and the reasons (justification) is in the right column

• Given: Information that is given as fact in the problem.

Reasoning with Properties from Algebra

• Example 1– Solve 6x – 5 = 2x + 3 and write a reason for each step

Statement Reason

6x – 5 = 2x + 3 Given

4x – 5 = 3

4x = 8

x = 2

Reasoning with Properties from Algebra

Example 2• 2(x – 3) = 6x + 6• • • •

1. Given

2.

3.

4.

5.

Reasoning with Properties from Algebra

• Determine if the equations are valid or invalid, and state which algebraic property is applied

– (x + 2)(x + 2) = x2 + 4

– x3x3 = x6

– -(x + y) = x – y

Warmup

• With a partner, Complete proof # 11 and 12 on p. 40

Proving TheoremsLesson 2.3

Pre-AP Geometry

Proofs

Geometric proof is deductive reasoning at work.

Throughout a deductive proof, the “statements” that are made are specific examples of more general situations, as is explained in the "reasons" column.

Recall, a theorem is a statement that can be proved.

Vocabulary

Definition of a Midpoint

The point that divides, or bisects, a segment into two congruent segments.

If M is the midpoint of AB, then AM is congruent to MB

Bisect

To divide into two congruent parts.

Segment Bisector

A segment, line, or plane that intersects a segment at its midpoint.

Midpoint Theorem

If M is the midpoint of AB, then AM = ½AB and MB = ½AB

Proof: Midpoint Formula

Given: M is the midpoint of Segment AB

Prove: AM = ½AB; MB = ½AB Statement

1. M is the midpoints of segment AB2. Segment AM= Segment MB, or AM = MB 3. AM + MB = AB4. AM + AM = AB, or 2AM = AB 5. AM = ½AB 6. MB = ½AB

Reason

1. Given2. Definition of midpoint 3. Segment Addition Postulate4. Substitution Property (Steps 2 and 3) 5. Division Prop. of  Equality6. Substitution Property. (Steps 2 and 5)

The Midpoint FormulaThe Midpoint Formula

If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of segment AB has coordinates:

2

,2

2121 yyxxM

221 xx

M x

221 yy

M y

The Midpoint Formula

Application:

Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2).

Midpoint Formula

Application:

Find the coordinates of the other endpoint B(x, y) of a segment with endpoint C(3, 0) and midpoint M(3, 4).

Vocabulary

Definition of an Angle Bisector

A ray that divides an angle into two adjacent angles that are congruent.

If Ray BD bisects angle ABC, then ABD is congruent to DBC

Angle Bisector Theorem

If BX is the bisector of ∠ABC, then the measure of ∠ABX is one half the measure of ∠ABC and the measure of ∠XBC one half of the ∠ABC.

A

X

CB

Proof: Angle Bisector TheoremGiven: BX is the bisector of ∠ABC.

Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m ∠ABC

Statement Reason

1. BX is the bisector of ∠ABC 1. Given

2. m∠ABX + m∠XBC = m∠ABC

2. Angle addition postulate

3. m∠ ABX = m∠ XBC 3. Definition of bisector of an angle

4. m∠ ABX + m∠ ABX = m∠ ABC; 2 m∠ ABX = m∠ ABC

4. Substitution property

5. m∠ ABX = ½ m∠ ABC; m∠ XBC = ½ m∠ ABC

5. Division property

Reasons used in proofs

1. Given

2. Definitions

3. Postulates

4. Theorems

Page 50

Pre-AP Geometry 1

• Complementary Angles–Two angles that have a sum of 90º–Each angle is a complement of the other.

Non-adjacent complementary Adjacent angles complementary angles

• Supplementary Angles–Two angles that have a sum of 180º–Each angle is a supplement of the other.

• Example 1–Given that G is a supplement of H and

mG is 82°, find mH.

–Given that U is a complement of V, and mU is 73°, find mV.

• Example 2 T and S are supplementary.

The measure of T is half the measure of S. Find mS.

• Example 3 D and E are complements and D and F

are supplements. If mE is four times mD, find the measure of each of the three angles.

• Vertical angles are congruent–Given: angle 1 and angle 2 are vertical angles–Prove 1 2∠ ≅ ∠

Statement Reasons

1. 1.

2. 2.

3. 3.

4. 4.

31 2

32°

Find x and the measure of each angle.

2x + 10

∠A

Page 56

Pre-AP Geometry 1

• Two lines that intersect to form right angles

• We use the symbol to show that lines ⊥are perpendicular. Line AB Line CD⊥

A

D

C

B

• Theorem 2-4: If two lines are perpendicular, then they form congruent adjacent angles

• Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular

• Theorem 2-6: If the exterior sides of two adjacent angles are perpendicular, then the angles are complementary.

Unit 2.6: Planning a proof

p. 60

Pre-AP Geometry 1

September 11, 2008

Parts of a proof

1. Statement of the theorem you are trying to prove

2. A diagram to illustrate given information

3. A list of the given information

4. A list of what you are trying to prove

5. A series of Statements and Reasons that lead from the given information to what you are trying to prove.

Example proof of theorem 2-7If 2 angles are supplements of congruent angles, then the two angles

are congruent.Given: ∠2 ≅ ∠4∠1 and ∠2 are supplementary∠3 and ∠4 are supplementaryProve: ∠1 ≅ ∠3

1 2

3 4

Statement Reason

1. ∠1 and ∠2 are supplementary∠3 and ∠4 are supplementary

1. Given

2. m ∠1 +m ∠2 =180; m ∠3 + m∠4 =180

2. Definition of supp. ∠’s

3. m ∠1 +m ∠2 = m ∠3 + m∠4 3. Substitution property

4. ∠2 ≅ ∠4 4.given

5. ∠1 ≅ ∠3 5. Subtraction property of equality

Theorem 2-8:

• If two angles are complements of congruent angles, then the two angles are congruent.

• Prove theorem 2-8. Use the proof from theorem 2-7 (p. 61) to help. You may do this with a partner. Due at end of hour. Make sure you include all 5 parts (p. 60).