Unit 2: Reasoning and Proof Geometry 1 Unit 2: Reasoning and Proof

2.1 Conditional Statements Geometry 1 Unit 2 2.1 Conditional Statements

Conditional Statements A statement with two parts If-then form A way of writing a conditional statement that clearly showcases the hypothesis and conclusion Hypothesis- The “if” part of a conditional Statement Conclusion The “then” part of a conditional Statement

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. The first statement is true. The second statement is false, triangles do not have to have a right angle. This was put together with a subject (a triangle) and predicate (it has a right angle) The third statement is false, x could equal -2.

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 Statements do not have to be true. The last one is clearly false.

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 Negation In most cases you can form the negation of a statement by either adding or deleting the word “not”. In most cases you can form the negation of a statement by either adding or deleting the word “not”.

Conditional Statements Examples of Negations Statement: Negation : Statement: Mr. Ross is not more than 6 feet tall. Negation: Mr. Ross is more than 6 feet tall I am doing my homework. Negation:

Conditional Statements Example 4 Write the negation of each statement. Determine whether your new statement is true or false. Stanfield is the largest city in Arizona. All triangles have three sides. Dairy cows are not purple. Some VGHS students have brown hair.

Conditional Statements Converse Formed by switching the if and the then part. Original If you like green, then you will love my new shirt. If you love my new shirt, then you like green.

Conditional Statements Inverse Formed by negating both the if and the then part. Original If you like green, then you will love my new shirt. If you do not like green, then you will not love my new shirt.

Conditional Statements Contrapositive Formed by switching and negating both the if and then part. Original If you like green, then you will love my new shirt. If you do not love my new shirt, then you do not like green.

Conditional Statements Write in if…then form. Write the converse, inverse and contrapositive of each statement. I will wash the dishes, if you dry them. A square with side length 2 cm has an area of 4 cm2.

Conditional Statements Point-line postulate: Through any two points, there exists exactly one line Point-line converse: A line contains at least two points Intersecting lines postulate: If two lines intersect, then their intersection is exactly one point

Conditional Statements Point-plane postulate: Through any three noncollinear points there exists one plane Point-plane converse: A plane contains at least three noncollinear points Line-plane postulate: If two points lie in a plane, then the line containing them lies in the plane Intersecting planes postulate: If two planes intersect, then their intersection is a line

2.2: Definitions and Biconditional Statements Geometry 1 Unit 2 2.2: Definitions and Biconditional Statements

Definitions and 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.

Definitions and Biconditional Statements Example 1 An angle is right if and only if its measure is 90º A number is even if and only if it is divisible by two. A point on a segment is the midpoint of the segment if and only if it bisects the segment. You attend school if and only if it is a weekday. Conditional: If an angle is a right angle, then its measure is 90º. Converse: If an angle’s measure is 90º, then it’s a right angle. Conditional: If a number is even, then it is divisible by two. Converse: If a number is divisible by two, then it is even. Conditional: If a point on a segment is the midpoint of the segment, then it bisects the segment. Converse: If a point on a segment bisects the segment, then it is the midpoint of the segment.

Definitions and Biconditional Statements Perpendicular lines Two lines are perpendicular if they intersect to form a right angle A line perpendicular to a plane A line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it The symbol is read, “is perpendicular to.

Definitions and Biconditional Statements Example 2 Write the definition of perpendicular as biconditional statement.

Definitions and Biconditional Statements Example 3 Give a counterexample that demonstrates that the converse is false. If two lines are perpendicular, then they intersect.

Definitions and Biconditional Statements Example 4 The following statement is true. Write the converse and decide if it is true or false. If the converse is true, combine it with its original to form a biconditional. If x2 = 4, then x = 2 or x = -2

Definitions and Biconditional Statements Example 5 Consider the statement x2 < 49 if and only if x < 7. Is this a biconditional? Is the statement true? Yes No

Geometry 1 Unit 2 2.3 Deductive Reasoning

Deductive Reasoning Symbolic Logic Statements are replaced with variables, such as p, q, r. Symbols are used to connect the statements.

Deductive Reasoning Symbol Meaning ~ not Λ and V or → if…then ↔ if and only if

Deductive Reasoning Example 1 Let p be “the sum of the measure of two angles is 180º” and Let q be “two angles are supplementary”. What does p → q mean? What does q → p mean?

Deductive Reasoning Example 2 p: Jen cleaned her room. q: Jen is going to the mall. What does p → q mean? What does q → p mean? What does ~q mean? What does p Λ q mean?

Deductive Reasoning Example 3 Given t and s, determine the meaning of the statements below. t: Jeff has a math test today s: Jeff studied t V s s → t ~s → t

Deductive Reasoning Deductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument.

Deductive Reasoning Law of Detachment Given: p → q Given: p When you have a true conditional statement and you know the hypothesis is true, you can conclude the conclusion is true. Given: p → q Given: p Conclusion: q

Deductive Reasoning Example 4 Determine if the argument is valid. If Jasmyn studies then she will ace her test. Jasmyn studied. Jasmyn aced her test. Valid

Deductive Reasoning Example 5 Determine if the argument is valid. If Mike goes to work, then he will get home late. Mike got home late. Mike went to work Invalid

Deductive Reasoning Law of Syllogism Given: p → q Given: q → r Given two linked conditional statements you can form one conditional statement. Given: p → q Given: q → r Conclusion: p → r

Deductive Reasoning Example 6 Determine if the argument is valid. If today is your birthday, then Joe will bake a cake. If Joe bakes a cake, then everyone will celebrate. If today is your birthday, then everyone will celebrate. valid

Deductive Reasoning Example 7 Determine if the argument is valid. If it is a square, then it has four sides. If it has four sides, then it is a quadrilateral. If it is a square, then it is a quadrilateral.

2.4 Reasoning with Properties from Algebra Geometry 1 Unit 2 2.4 Reasoning with Properties from Algebra 39

Reasoning with Properties from Algebra Objectives Review of algebraic properties Reasoning Applications of properties in real life

Reasoning with Properties from Algebra 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

Reasoning with Properties from Algebra Reflexive property For any real number a, a = a Symmetric property If a=b, then b = a Transitive Property If a = b and b = c, then a = c Substitution property If a = b, then a can be substituted for b in any equation or expression Distributive property 2(x + y) = 2x + 2y

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 Subtraction property of equality Addition property of equality Division property of equality

Reasoning with Properties from Algebra Example 2 2(x – 3) = 6x + 6 Given

Reasoning with Properties from Algebra Determine if the equations are valid or invalid. (x + 2)(x + 2) = x2 + 4 x3x3 = x6 -(x + y) = x – y Invalid. Valid Invalid

Reasoning with Properties from Algebra Geometric Properties of Equality Reflexive property of equality For any segment AB, AB = AB Symmetric property of equality If then Transitive property of equality If AB = CD and CD = EF, then, AB = EF

Reasoning with Properties from Algebra Example 3 In the diagram, AB = CD. Show that AC = BD A B C D Statement Reason AB = CD AB + BC = BC + CD AC = AB + BC BD = BC + CD AC = BD 1. Given 2. Addition property of equality 3. Segment addition postulate 4. Segment addition postulate 5. Substitution property of Equality

In the diagram, ABC = DBF. Show that  ABD = CBF A C D B F In the diagram, ABC = DBF. Show that  ABD = CBF Statement Reason

2.5: Proving Statements about Segments Geometry 1 Unit 2 2.5: Proving Statements about Segments

Proving Statements about Segments Key Terms: 2-column proof A way of proving a statement using a numbered column of statements and a numbered column of reasons for the statements Theorem A true statement that is proven by other true statements

Proving Statements about Segments Properties of Segment Congruence Reflexive For any segment AB, Symmetric If , then Transitive If and ,then

Proving Statements about Segments J K L Example 1 In triangle JKL, Given: LK = 5, JK = 5, JK = JL Prove: LK = JL Statement Reason 1. 1. Given 2. 2. Given 3. 3. Transitive property of equality 4. 5. 5. Given 6. 6. Transitive property of congruence

Proving Statements about Segments Duplicating a Segment Tools Straight edge: Ruler or piece of wood or metal used for creating straight lines Compass: Tool used to create arcs and circles A B C D Steps Use a straight edge to draw a segment longer than segment AB Label point C on new segment Set compass at length of segment AB Place compass point at C and strike an arc on line segment Label intersection of arc and segment point D Segment CD is now congruent to segment AB

2.6: Proving Statements about Angles Geometry 1 Unit 2 2.6: Proving Statements about Angles

Proving Statements about Angles Properties of Angle Congruence Reflexive For any angle A, Symmetric Transitive

Proving Statements about Angles Right Angle Congruence Theorem All right angles are congruent.

Proving Statements about Angles Congruent Supplements Theorem If two angles are supplementary to the same angle, then they are congruent. 1 2 3

Proving Statements about Angles Congruent Complements Theorem If two angles are complementary to the same angle, then the two angles are congruent. 4 5 6

Proving Statements about Angles Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2

Proving Statements about Angles Vertical Angles Theorem Vertical angles are congruent 1 2 3 4

Proving Statements about Angles Example 1 Given: Prove: A 1 2 4 3 C B Teacher edition page 110 example 1 Statement Reason 1. 2. 3. 4.

Proving Statements about Angles Example 2 Given: Prove: 1 2 4 3 Teacher edition page 110 example 2 Statement Reason 1. 2. 3. 4.

Proving Statements about Angles Example 3 Given: are right angles Prove: A D C B Teacher edition page 110 example 3 Statement Reason 1. 2. 3. 4.

Proving Statements about Angles Example 4 Given: m1 = 24º, m3 = 24º 1 and 2 are complementary 3 and 4 are complementary Prove: 2  4 Statement Reason 1. 2. 3. 4. Teacher edition page 111 example 4 1 2 3 4

Proving Statements about Angles Example 5 In the diagram m1 = 60º and BFD is right. Explain how to show m4 = 30º. Teacher edition page 111 example 5 1 2 3 4 A F E D C B

Proving Statements about Angles Example 6 Given: 1 and 2 are a linear pair, 2 and 3 are a linear pair Prove: 1  3 Statement Reason 1. 2. 3. Teacher edition page 111 example 6 1 2 3