Lesson 2.1 Conditional Statements You will learn to… * recognize and analyze a conditional statement * write postulates about points, lines, an planes using conditional statements

A conditional statement has two parts, a hypothesis and a conclusion. p  qp  q If p, then q.

hypothesis (p) conclusion ( q ) If the team wins the game, then they will win the tournament.

Write an if-then statement. 1. The intersection of two planes is a line. If two planes intersect, then their intersection is a line.

Write an if-then statement. 2. A line containing two given points lies in a plane if the two points lie in the plane. If two points lie in a plane, then the line containing them lies in the plane.

The converse is formed by switching the hypothesis and conclusion. The converse is q  p. If q, then p.

Write the converse of this if-then statement. Is it true or false? 3. If m  A = 125°, then  A is obtuse. If  A is obtuse, then m  A = 125°. False

The negation of a statement is formed by negating the statement. The negation is written ~ p.

Write the negation of this statement. 4. m  A = 125°  A is obtuse 5.  A is not obtuse m  A  125°

The inverse is formed by negating the hypothesis and the conclusion. The inverse is ~ p  ~ q. If ~ p, then ~ q.

Write the inverse of this if-then statement. Is it true or false? 6. If m  A = 125°, then  A is obtuse. If m  A  125°, then  A is not obtuse. False

The contrapositive is formed by negating the hypothesis and conclusion of the converse. The contrapositive is ~ q  ~ p. If ~ q, then ~ p.

Write the contrapositive of this if- then statement. Is it true or false? 7. If m  A = 125°, then  A is obtuse. If  A is not obtuse, then m  A  125°. True

Postulate 5 Through any two points there exists exactly one line.

Postulate 6 A line contains at least two points.

Postulate 7 If two lines intersect, then their intersection is exactly one point.

T Postulate 8 Through any three noncollinear points there exists exactly one plane. A B C

Postulate 9 A plane contains at least three noncollinear points.

Postulate 10 If two points lie in a plane, then the line containing them lies in the plane.

Postulate 11 If 2 planes intersect, then their intersection is ___________. a line

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Lesson 2.2 Biconditional Statements You will learn to… * recognize and use definitions * recognize and use biconditional statements

All definitions can be interpreted “forward” and “backward.” All definitions are biconditional.

For example, perpendicular lines are defined as two lines that intersect to form one right angle.

If two lines are perpendicular, then they intersect to form one right angle. If two lines intersect to form one right angle, then they are perpendicular.

A biconditional statement contains the phrase “if and only if.” Two lines are perpendicular if and only if they intersect to form one right angle.

A biconditional statement is true when the original if-then statement AND its converse are both true.

if-then statement: 1. Two angles are supplementary if and only if the sum of their measures is 180°. converse : If two angles are supplementary, then the sum of their measures is 180°. If the sum of the measures of two angles is 180°, then they are supplementary.

converse: If an angle is obtuse, then its measure is 135°. counterexample? 2. If an angle is 135˚, then it is an obtuse angle. Can we write a biconditional statement?

converse: If two angles are complementary, then the sum of their measures is 90°. Can we write a biconditional statement? Two angles are complementary if and only if the sum of their measures is 90°. 3. If two angle measures add up to 90˚, then they are complementary angles.

Workbook Page 25 (1-7)

Lesson 2.3 Deductive Reasoning You will learn to… * use symbolic notation to represent logical statements * form conclusions by applying laws of logic

p: m  B = 90 ˚ q:  B is a right angle Using these phrases, write the conditional statement. 1. p  q 2. q  p 3. ~ p  ~ q 4. ~ q  ~ p 5. p  q m  B = 90 ˚ if and only if  B is a right angle. If m  B = 90 ˚, then  B is a right angle. If  B is a right angle, then m  B = 90 ˚ If  B is not a right angle, then m  B ≠ 90 ˚ If m  B ≠ 90 ˚, then  B is not a right angle.

Deductive Reasoning uses facts to make a logical argument. definitions, properties, postulates, theorems, and laws of logic facts

Law of Detachment Therefore: Given facts You can use these symbols when asked to explain your reasoning. p  q p q hypothesis is true conclusion must be true

Law of Detachment If I learn my facts, then I will pass geometry. I learned my facts. p q p Therefore, I passed geometry. q

Law of Syllogism Given facts Therefore: You can use these symbols when asked to explain your reasoning. p  q q  r p  r

Law of Syllogism If I pass geometry, then my dad will be happy. If my dad is happy, then I will get a cell phone. p q r q Therefore, if I pass geometry, then I will get a cell phone. p r

Coplanar lines n and m are parallel. 6. Is this argument valid? Therefore, lines n and m do not intersect. If 2 lines in a plane are parallel, then they do not intersect. p  q p q VALID – Law of Detachment

If 2 angles form a linear pair, then they are supplementary. 7. Is this argument valid? Therefore, if 2 angles form a linear pair, then the sum of their measures is 180 ˚ If 2 angles are supplementary, then the sum of their measures is 180 ˚. p  q VALID – Law of Syllogism r  p r  q r  p p  q

m  1 + m  2 = 180 ˚ 8. Is this argument valid? Therefore,  1 and  2 are a linear pair. If 2 angles are a linear pair, then the sum of their measures is 180 ˚. p  q INVALID q p

If you live in South Carolina, then you live in North America. 9. Is this argument valid? Therefore, if you live in Canada, then you live in South Carolina If you live in Canada, then you live in North America. p  q INVALID r  q p  r r  q p  q

If you use this product, then you will have even-toned skin. If you have even-toned skin, then you will be beautiful. If you use this product, then you will be beautiful.

Lesson 2.4 Properties of Equality and Congruence You will learn to… * use properties from algebra * use properties of length and measure to justify segment and angle relationships

Equality Properties Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property

Reflexive Property

Symmetric Property

If XY = ST and ST = 10, then XY = 10 If m  A = m  B and m  B = 10°, then m  A = 10° Transitive Property

Division Property If 8x=16, then x=2.

Addition Property If x-7=5, then x=12.

Multiplication Property If ½ x = 7, then x=14.

Subtraction Property If x+3=7, then x=4.

Substitution Property If A=x 2 and x=6, then A=36. If 4 + 7x – 10 = 24, Then 7x - 6 = 24

Distributive Property If B=2(4x + 3), then B=8x + 6. If 4x + 7x = 24, Then 11x = 24

Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!! Let’s Practice…

4+2(3x+5)=11-x x+10=11-x 14+6x =11-x x = 11 7x = - 3 x = - 3 / 7 Distributive prop. Substitution Addition prop. Subtraction prop. Division Prop. Given

4. 1x + 20 = 10x / 5 x + 4 = 2x + 3 / 5 Multiplication Prop 17 = 9x Subtraction Prop20 = 9x / 9 = X Subtraction Prop Division Prop

Given 5. Given that MN-PQ, show that MP=NQ MP = MN + NP MN = PQ Segment Addition Postulate NQ = PQ + NP Substitution PropMP = PQ + NP MP = NQ Segment Addition Postulate Substitution Prop Q P N M

7. Given m  AQB=m  CQD,show that m  AQC=m  BQD m  AQB = m  CQD m  AQB + m  BQC = m  AQC Given Angle Addition Postulate Substitution m  CQD + m  BQC = m  BQD m  CQD + m  BQC = m  AQC m  AQC = m  BQD Angle Addition Postulate Q A B C D

m  RPS = m  TPV Given m  TPV = m  SPT m  RPS = m  SPT m  RPV= m  RPS+m  SPT+m  TPV m  RPV = 3(m  RPS) Given Angle Addition Postulate Substitution Distributive Prop Transitive Prop m  RPV= m  RPS+m  RPS+m  RPS P R S T V 8. Given m  RPS=m  TPV and m  TPV=m  SPT show that m  RPV=3(m  RPS)

Def. of right angles You can use definitions as reasons in proofs. 1)  A is a right angle 2) m  A = 90˚ 2) 1) Given Statements Reasons

2)  A is a right angle 1) m  A = 90˚ 2) 1) Given Def. of right angles Statements Reasons

Def. of  lines 1) AB  CD 2)  1 is a right angle 2) 1) Given Statements Reasons 1 BA C D

Def. of  lines 1)  1 is a right angle 2) AB  CD Statements Reasons 1) Given 2) 1 BA C D

Def. of vertical angles 1)  1 and  2 are vertical angles Statements Reasons 1) 1 2

Def. of linear pair 1)  1 and  2 are a linear pair Statements Reasons 1) 1 2

1) AB = CD 2) AB  CD 1) Given 2) Statements Reasons Def. of 

1) Given 1) AB  CD 2) AB = CD Statements Reasons 2)

1) m  1 = m  2 2)  1   2 2) 1) Given Statements Reasons Def. of 

2) m  1 = m  2 1)  1   2 2) 1) Given Statements Reasons Def. of 

Lesson 2.5 Proving Statements about Segments use practice sheet of proofs You will learn to… * justify statements about congruent segments * write reasons for steps in a proof

Proofs !! Memorize definitions, postulates, and theorems as we learn them. Write out entire proof each time one is in the assignment. Don’t give up!!!! You can do it!!!!

Reflexive Property of Congruence

Symmetric Property of Congruence

Transitive Property of Congruence

1. Given: EF = GH Prove: EG  FH E F G H (Proof is on next slide)

StatementsReasons 1) EF = GH1) 2) EF + FG = GH + FG2) 3) EG = EF + FG FH = GH + FG 3) 4) EG = FH4) 5) EG  FH 5) 1. Given Addition Prop. Segment Addition Postulate Substitution Def. of 

2. Given: RT  WY, ST = WX Prove: RS  XY R S T Y X W

StatementsReasons 1) RT  WY 1) 2) RT = WY2) 3) RT = RS + ST WY = WX + XY 3) 4) RS + ST = WX + XY4) 5) ST = WX5) 6) RS + ST = ST + XY6) 7) RS = XY7) 8) RS  XY 8) Given Subtraction Prop. Segment Addition Postulate Substitution Def. of  Given Substitution Def. of  2.

3. Given: X is the midpoint of MN and MX = RX Prove: XN = RX R X S N M

StatementsReasons 1) X is the midpoint of MN 1) 2) NX = MX2) 3) MX = RX3) 4) NX = RX4) Given Def. of midpoint 3. Given Transitive Prop.

Paragraph proof example for #1 Since EF = GH, EF + FG = GH + FG by the Addition Property. EG = EF + FG and FH = GH + FG by the Segment Addition Postulate. By Substitution, EG = FH. Therefore, EG  FH by the definition of congruent segments.

Paragraph proof example for #3 So, I was chillin’ with the homeboys and my homeboy Sherrod tells me, “Dave, x is the midpoint of MN, so NX = MX.” I said, “Sherrod, how do you figure?” Sherrod tells me “The definition of midpoint says so!” So I was like, “yo, Sherrod, did you know MX = RX,” and he said, “really, well then NX = RX Dawg. “Sherrod, my homie, I didn’t know you were so smart,” I said, “how did you figure that out?” He was like, “Substitution, my brother!” David Mathews

# 17 Statements 1) XY = 8, XZ = 8, 2) XY = XZ 3) XY  XZ 4) XY  ZY 5) XZ  ZY

# 18 Statements 1) NK  NL, NK = 13 2) NK = NL 3) NL = 13

Lesson 2.6 Proving Statements about Angles You will learn to… * use angle congruence * prove properties about special pairs of angles

Theorem 2.3 Right Angle Congruence Theorem right congruent All ________ angles are __________.

A is supplementary to 40° B is supplementary to 40° What do you know about A and B?

Theorem 2.4 Congruent Supplements Theorem If 2 angles are supplementary to the same angle, then they are _______________. congruent

Using the Congruent Supplements Theorem…  1 &  2 are supp.  1 &  3 are supp. Statements 1) 2)  2   3 2) Congruent Supplements Theorem Reasons

A is complementary to 50° B is complementary to 50° What do you know about A and B?

Theorem 2.5 Congruent Complements Theorem If 2 angles are complementary to the same angle, then they are _______________. congruent

Using the Congruent Complements Theorem…  1 &  2 are comp.  1 &  3 are comp. Statements 1) 2)  2   3 2) Congruent Compliments Theorem Reasons

Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are _______________. supplementary

Using the Linear Pair Postulate…  1 &  2 are a linear pair Statements 1) 2)  1 &  2 are supplementary 2) Linear Pair Postulate Reasons 1) Def. of linear pair 3) m  1 + m  2 = 180 3) Def. of supplementary

Theorem 2.6 Vertical Angles Theorem Vertical angles are _______________. congruent

Using the Vertical Angles Theorem…  1 &  2 are vertical angles Statements 1) 2)  1   2 2) Vertical Angles Theorem Reasons 1) Def. of vertical angles

1. Given:  1   2,  3   4,  2   3 Prove:  1  

1. StatementsReasons 1.  1   2,  2    1    3    1   4 4. Given Transitive Prop. Given Transitive Prop.

2. Given: m  1 = 63˚,  1   3,  3   4 Prove: m  4 = 63˚

2. StatementsReasons 1. m  1 = 63˚,  1   3,  3    1   m  1 = m  m  4 = 63˚ 4. Given Transitive Prop. Def of  Substitution

3. Given:  DAB &  ABC are right angles,  ABC   BCD Prove:  DAB   BCD D AB C

3. StatementsReasons 1.  DAB &  ABC are right angles  DAB   ABC  ABC   BCD  DAB   BCD 4. Given All right  s are  Given Transitive Prop.

4. Given: m  1 = 24˚,m  3 = 24˚  1 &  2 are complementary  3 &  4 are complementary Prove:  2  

4. StatementsReasons m  1 = 24˚, m  3 = 24˚ 1.  1 &  2 are comp.  3 &  4 are comp m  1 = m   1    2   4 4. Given Substitution Def of  Congruent Complements Theorem

5. StatementsReasons 1.  1 and  2 are a linear pair  2 and  3 are a linear pair  1 and  2 are supp.  2 and  3 are supp  1   3 3. Given Linear Pair Postulate Congruent Supplements Theorem

6. StatementsReasons 1.  QVW and  RWV are supplementary  QVW and  QVP are a linear pair  QVW and  QVP are supplementary  QVP   RWV 4. Given Def. of Linear Pair Postulate Congruent Supplements Theorem

#24 Statements 1)  3 and  2 are complementary 2) m  1 + m  2 = 90 3) m  3 + m  2 = 90 4) m  1 + m  2 = m  3 + m  2 5) m  1 = m  3 6)  1   3 #24 & #26 for homework

#26 Statements 1)  4 and  5 are vertical angles 2)  6 and  7 are vertical angles 3)  4   5,  6   7 4)  5   6 5)  4   7