Unit 2: Deductive Reasoning Pre-AP Geometry 1 Unit 2: Deductive Reasoning 1

2.1 If-then statements, converse, and biconditional statements Pre-AP Geometry 1 Unit 2 2.1 If-then statements, converse, and biconditional statements p. 33 in text 2

Conditional Statements 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” 3

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. 4

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. 5

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. 6

Conditional Statements Counterexample An example that proves that a given statement is false Write a counterexample If x2 = 9, then x = 3 7

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. 8

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. 9

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 10

Example Give the converse of the statement. If the converse and the statement are both true, then rewrite as a biconditional statement If it is Thanksgiving, then there is no school. 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

2.2: Properties from Algebra p. 37 Pre-AP Geometry 1 Unit 2 2.2: Properties from Algebra p. 37 p. 33 in text 14

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 15

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

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 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 19

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 Invalid. Valid Invalid 20

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

Lesson 2.3 Pre-AP Geometry Proving Theorems Lesson 2.3 Pre-AP Geometry 22

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. 23

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. 24

Midpoint Theorem If M is the midpoint of AB, then AM = ½AB and MB = ½AB 25

Proof: Midpoint Formula Given: M is the midpoint of Segment AB Prove: AM = ½AB; MB = ½AB Statement 1. M is the midpoints of segment AB 2. Segment AM= Segment MB, or AM = MB 3. AM + MB = AB 4. AM + AM = AB, or 2AM = AB 5. AM = ½AB  6. MB = ½AB Reason 1. Given 2. Definition of midpoint 3. Segment Addition Postulate 4. Substitution Property (Steps 2 and 3) 5. Division Prop. of  Equality 6. Substitution Property. (Steps 2 and 5) 26

The Midpoint Formula The Midpoint Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of segment AB has coordinates: 27

The Midpoint Formula Application: Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2). 28

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). 29

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 30

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 C B 31

Proof: Angle Bisector Theorem Given: 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 32

Reasons used in proofs Given Definitions Postulates Theorems

2.4: Special Pairs of Angles Page 50 Pre-AP Geometry 1

Angle Pair Relationships 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

Angle Pair Relationships Supplementary Angles Two angles that have a sum of 180º Each angle is a supplement of the other.

Angle Pair Relationships Example 1 Given that G is a supplement of H and mG is 82°, find mH. Given that U is a complement of V, and mU is 73°, find mV.

Angle Pair Relationships Example 2 T and S are supplementary. The measure of T is half the measure of S. Find mS. 38

Angle Pair Relationships Example 3 D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.

Theorem 2-3 Vertical angles are congruent Given: angle 1 and angle 2 are vertical angles Prove∠1≅ ∠2 3 2 1 Statement Reasons 1. 2. 3. 4.

Angle pair relationships Find x and the measure of each angle. ∠A 32° 2x + 10

2.5: Perpendicular Lines Page 56 Pre-AP Geometry 1

Perpendicular lines Two lines that intersect to form right angles We use the symbol ⊥ to show that lines are perpendicular. Line AB ⊥ Line CD C A B D

Perpendicular lines theorems 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.

p. 60 Pre-AP Geometry 1 September 11, 2008 Unit 2.6: Planning a proof p. 60 Pre-AP Geometry 1 September 11, 2008

Parts of a proof Statement of the theorem you are trying to prove A diagram to illustrate given information A list of the given information A list of what you are trying to prove A series of Statements and Reasons that lead from the given information to what you are trying to prove.

Example proof of theorem 2-7 If 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 supplementary Prove: ∠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).