Honors Geometry Lesson 2-1: Analyze Conditional Statements

Learning Target: At the end of today’s lesson we will be able to successfully analyze conditional statements.

Conditional StatementA logical statement that has two parts, a hypothesis and a conclusion, often written as an “___-_____” statement.

Hypothesis A hypothesis is the "_____" part of a conditional statement.

ConclusionA conclusion is the "______" part of a conditional statement.

Statements and definitions can be written as a conditional in if-then form.

Ex 1: Rewrite the conditional statement in if-then form.a. An angle whose measure is 90˚ is a right angles. ____________________________________________

b. All squares have 4 sides. ________________________________________________________

c. When x = 3›, x2 = 9. ______________________________________________________________

NegationThe negation (~) of a statement is the negative of the original statement. Statement 1: The ball is red. Statement 2: The cat is not black. Negation 1: _____________________________ Negation 2: ____________________________________

Given Initial Conditional Statements: If an angle measures 90˚, then it is a right angle.

Related Conditionals

Description Notation Example

ConverseThe converse of a conditional statement is formed by ________________ the hypothesis and conclusion.

Inverse

The inverse of a conditional statement is formed by _______________ both the hypothesis and conclusion.

Contrapositive The contrapositive of a conditional statement is formed by writing the _____________ and then _____________

both the hypothesis and conclusion.

Ex 2: Write the if-then form, the converse, the inverse, and the contrapositive of the conditional statement: "Olympians are athletes." Decide whether each statement is true or false.

● If-then form: _______________________________________________________________________

● Converse: ________________________________________________________________________

● Inverse: ___________________________________________________________________________

● Contrapositive: _____________________________________________________________________

PERPENDICULAR LINES:If two lines intersect to form a right angle, then they are perpendicular lines.

Write the converse:

Is this a true statement?

You can write "line l is perpendicular to line m" as____________.

Biconditional StatementsWhen a conditional statement and its converse are both true you can write them as a single statementthat contains the phrase “if and only if” this can also be abbreviated “iff” or can have the symbol

Example: Two lines are perpendicular if and only if they intersect to form a right angle.

Any valid definition can be written as a biconditional statement.

Ex 3: Determine if the following statements are valid definitions. If so, write it as a biconditional. If not, give a counterexample.

a) An angle whose measure is 90˚ is a right angles.

b) If ∠1 and ∠2 are a linear pair, then ∠1 and ∠2 are supplementary.

c) If ∠1 and ∠2 are a vertical angles, then ∠1 and ∠2 are congruent.

d) If two lines lie in the same plane and do not intersect, then they are parallel.

Honors Geometry Lesson 2-2: Use Inductive and Deductive Reasoning

Learning Target: At the end of today’s lesson we will be able to successfully use inductive reasoning.

Inductive ReasoningThe process of finding a pattern for specific cases and then writing a conjecture for the general case.

ConjectureA statement that is based on observation.

To show that a conjecture is true, you must show that it is true for all cases.

To show that a conjecture is false, you must find a counterexample.

CounterexampleA specific case for which the conjecture is false.

Ex 1: Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure.

Figure 4:

Description:

Ex 2: Describe the pattern in the numbers –1, –4, –16, –64 ,….Write the next three numbers in the pattern.

Description: ______,______,______

Ex 3: Given five noncollinear points, make a conjecture about the number of ways to connect different pairs of the points.

Conjecture:

Ex 4: Numbers such as 3, 4, and 5 are called consecutive numbers. Make and test a conjecture about the sum of any three consecutive numbers.

Conjecture:

Ex 5: A student makes the following conjecture about the difference of two numbers. Find a counterexample to disprove the student’s conjecture.

Conjecture: The difference of any two numbers is always smaller than the larger number.

Deductive ReasoningUsing facts, definitions, accepted properties, and the laws of logic to form a logical argument.

LAWS OF LOGIC:I. Law of Detachment: If the hypothesis of a true conditional statement is true, then the conclusion is also true.

Given: If you get a hit, then your baseball team will win. ……… You hit a home run. (Fact!) What can you conclude?________________________________________________________________

Ex 6: Use the Law of Detachment and the given true statements to make a valid conclusion if possible.a. If two angles have the same measure, then they are congruent.

You know that m∠A = m∠B. Conclusion?________________________________________________

b. Jesse goes to the gym every weekday.

Today is Monday. Conclusion?________________________________________________

c. If 90° < m∠A < 180°, then A is obtuse.

The measure of ∠A is 125°. Conclusion?________________________________________________

d. If two adjacent angles form a right angle, then they are complementary.

∠A and ∠B are complementary. Conclusion?________________________________________________

e. If a quadrilateral is a rhombus, then it has four congruent sides.

Quadrilateral ABCD has four congruent sides. Conclusion? ________________________________________

II. Law of Syllogism

Ex 7: If possible, use the Law of Syllogism to write the conditional statement that follows from the pair of true statements.

a) If Cedric plays in a big game, then he gets nervous. If Cedric gets nervous, then he performs well. Statement? _____________________________________________________________________________

b) If Ron eats lunch today, then he will eat a sandwich. If Ron eats a sandwich, then he will drink a glass of milk.

Statement?______________________________________________________________________________

c) If x2 > 36, then x2 > 30. If x > 6, then x2 > 36.

If hypothesis p, then conclusion q. If these statements are true,

If hypothesis q, then conclusion r.

If hypothesis p, then conclusion r. then this statement is true

Statement?______________________________________________________________________________

d) If a polygon is regular, then all sides are congruent. If a polygon is regular, then all angles are congruent.

Statement?______________________________________________________________________________

Honors Geometry Lesson 2-3: Use Postulates and Diagrams

Axiom/Postulates: Rules that are accepted without proof. Theorems: Rules that are proved. *****Unlike the converse of a definition, the converse of a postulate or theorem cannot be assumed as true.

Point, Line, and Plane Postulates:

Name/Description IllustrationTwo Point PostulateThrough any two points there exists exactly one line.

Line-Point PostulateA line contains at least two points.

Line Intersection PostulateIf two lines intersect, then their intersection is exactly one point.

Three Point PostulateThrough any three noncollinear points there exists exactly one plane.

Plane-Point PostulateA plane contains at least three noncollinear points.

Plane-Line PostulateIf two points lie in a plane, then the line containing them lies in the plane.

Plane Intersection PostulateIf two planes intersect, then their intersection is a line.

For more details see page 84 in your text book.

Ex 2:

a.) Sketch a diagram showing b.) Sketch a diagram showing intersecting

perpendicular to intersecting at point X. at point W, so that

Additional Vocabulary:

A line is perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point.

You can assume only certain things from an illustration. What statement can you make from the following illustrations? Use definitions, theorems, or postulates as REASONS to justify each statement.

Illustrations Statement Reason

Honors Geometry Lesson 2-4: Reason Using Properties from Algebra Proofs

Algebraic Properties of Equality Let a, b, and c be real numbers.

Property Name Given Statement Property ConclusionAddition Property If a = b then

Subtraction Property If a = b then

Multiplication Property If a = b then

Division Property If a = b and c ≠ 0 then

Substitution Property If a = b then

Distributive Property a(b+c) = then

REFLEXIVE PROPERTY: - Real Numbers: For any real number a, __________________

- Segment Length: For any segment AB, __________________

- Angle Measure: For any angle A, ______________________

SYMMETRIC PROPERTY: - Real Numbers: For any real numbers a and b, if a = b, then ___________________

- Segment Length: For any segments AB and CD, if AB = BC, then _____________________

- Angle Measure: For any angles A and B, if m∠A = m∠B, then ________________________

TRANSITIVE PROPERTY:- Real Numbers: For any real numbers a, b, and c, if a = b and b = c, then _______________

- Segment Length: For any segments AB ,CD, and EF if AB = CD , and CD = EF, then______________

- Angle Measure: For any angles A, B, and C, if m∠A = m∠B and m∠B = m∠C, then _______________

Name the property of equality or congruence theorem that the statement illustrates.

1. If GH = JK, then JK = GH. _______________________________

2. If r = s, and s = 44, then r = 44. _______________________________

3. m∠N = m∠N _______________________________

4. If ∠5 ≅ ∠3, then ∠3 ≅ ∠5 _______________________________

5. If ∠R ≅ ∠T and ∠T ≅ ∠P, then ∠R ≅ ∠P _______________________________

6. If then _______________________________

Complete the following proofs.

Ex: 1 Given 6(x+2) = -12Prove: x = -4

Statements Reasons

1) 6(x+2) = -12 1) Given

2) _____________________________________ 2) _____________________________________

3) _____________________________________ 3) _____________________________________

4) _____________________________________ 4) _____________________________________

Ex 2: Given: Prove: x = 34

Statements Reasons

1) 1) Given

2) m∠ AOB+m∠BOC=180 ° 2) Linear Pair ⟹Supp Angles

3) _____________________________________ 3) _____________________________________

4) _____________________________________ 4) _____________________________________

5) _____________________________________ 5) _____________________________________

6)_____________________________________ 6) _____________________________________

Ex 3: Given: Prove: x = 5

Statements Reasons

1) 1) Given

2) XY + YZ = XZ 2) _____________________________________

3) _____________________________________ 3) _____________________________________

4) _____________________________________ 4) _____________________________________

5) _____________________________________ 5) _____________________________________

6)_____________________________________ 6) _____________________________________

Honors Lesson 2-5 Prove Statements about Segments and Angles

VOCABULARYTheorem

A theorem is a statement that can be proven. Once a theorem has been proven it can be used as a reason in future proofs.

ProofA proof is a logical argument that shows a statement is true. Proofs can be done in two-columns, as a flow chart,

or in a paragraph. The most common form used in high school geometry is the two-column form.

Two-column proofA two-column proof has numbered statements and corresponding reasons that show an argument in two columns. The statements are always in the left column and the reasons are in the right column.

Note: The last statement in a proof is ALWAYS what you are trying to PROVE.

Proof #1 Given: AC = AB + AB

Prove: AB = BC

Statements Reasons1.) AC = AB + AB 1.) ____________________________________

2.) ____________________ 2.) ____________________________________

3.) ____________________ 3.) ____________________________________

4.) ____________________ 4.)_____________________________________

Proof #2 Given: bisects ∠ABC

Prove: m∠ABC = 2 ∙ m∠1

Statements Reasons

1.) bisects ∠ABC 1.) ____________________________________

2.) ________________________________ 2.) ____________________________________

3.) ________________________________ 3.) Definition of Congruent Angles

4.) ________________________________ 4.) _____________________________________

5.) ________________________________ 5.) Substitution Property of Equality

6.) ________________________________ 6.) ____________________________________

Proof #3 Given: m∠2 = m∠3, m∠AXD = m∠AXC

Prove: m∠1= m∠4

Statements Reasons

1.) m∠AXD = m∠AXC , m∠ 2 = m∠3 1.) _____________________________________

2.) m∠AXD = m∠ ______ + m∠ ______ 2.)_____________________________________

3.) m∠AXC = m∠ ______ + m∠ ______ 3.)_____________________________________

4.) m∠ __ + m∠ ___ = m∠ ___ + m∠ ___ 4.) _____________________________________

5.) m∠ 1 + m∠ ______ = m∠ 3 + m∠ 4 5.) ______________________________________

6.) _____________________________ 6.) ______________________________________

Honors Geometry Lesson 2-6: Prove Angle Pair Relationships

Ex 1: Prove the Right Angle Congruence Theorem: All right angles are _________________________.

Given: ∠1 and ∠2 are right anglesProve: ∠1 ≅ ∠2

Ex 2: Given: Prove: ∠JKL ≅ ∠MLK

Theorem 2.4: Congruent Supplements Theorem:If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

If ∠l and ∠2 are supplementary and ∠3 and ∠2 are supplementary, then ________________________.

Ex 3: Prove that two angles supplementary to the same angle are congruent.

Given: are supplementary.are supplementary.

Prove:

Theorem 2.5: Congruent Complements Theorem:

If two angles are complementary to the same angle (or to congruent angles), then they are __________________________.

If ∠4 and ∠5 are complementary and ∠6 and ∠5 are complementary, then _____________.

Ex 4:Given ∠l and ∠2 are supplements. ∠l and ∠4 are supplements. m∠2 = 45°Prove m∠4 = 45°

LINEAR PAIR POSTULATEIf two angles form a linear pair, then they are supplementary.

∠l and ∠2 form a linear pair, so ∠l and ∠2 are supplementary and m∠1 + m∠2 = _______.

THEOREM 2.6 VERTICAL ANGLES CONGRUENCE THEOREM

Vertical angles are congruent.

Ex 5:Use the Linear Pair Postulate and the Vertical Angle Congruence Theorem to solve each.

a) If m∠3 = 121°, find m∠1, m∠2, and m∠4. b) Write and solve an equation to find x.

Use x to find m∠AED.

c) In the diagram at the right, m∠1 = 38° and m∠4 = 98°. Find the indicated angle measures.

Find m∠3 = _____Find m∠DGE = _____

Find m∠CGE = _____

Find m∠2 = _____

Find m∠AGC = _____