2 1 Conditional Statements Note there will be

2. 1 Conditional Statements Note, there will be a notes handout for 2. 1 given in class.

A Conditional Statement has TWO parts, the HYPOTHESIS and CONCLUSION In if-then form If -> HYPOTHESIS p Then -> CONCLUSION q (it’s not necessarily these letters all the time)

Writing a statement in if-then form. People who go to USC need help. If you go to USC, then you need help A Troy student is a Warrior. If you are a Troy student, then you are a Warrior. Write the statement in if-them form. All mountains have rocks. If it is a mountain, then it has rocks. A lover of potatoes is a lover of French fries. If you love potatoes, then you love French fries.

It just takes one COUNTEREXAMPLE to prove something is wrong. If you go to USC, then you are smart. False, virtually every USC student If x 2 = 25, then x = 5 False, x = -5 All triangles are equilateral False, some triangles have different side lengths. All even numbers can be divided by 2. True

A CONVERSE is when you reverse the original conditional statement. If you are in room 302, then the room is cold. If it is cold, then you are in room 302. The converse is NOT always true!!!!

Write the converse and say whether it is true, or state a counterexample. If a student is in room 302, the student is in math class. If a student is in math class, the student is in room 302. Mr. Booze’s class.

A statement can be changed by negation, which is writing the negative of a statement. Conditional If you are in room 302, you are in Mr. Kim’s class. Converse If you are in Mr. Kim’s class, then you are in room 302. If you are not in room 302, you are not in Mr. Kim’s class. Negation symbol Contrapositive If you are not in Mr. Kim’s class, you are not in room 302.

• Conditional If you go to Troy, then you are a student. • Converse If you are a student, then you go to Troy. • Inverse If you don’t go to Troy, then you’re not a student. • Contrapositive If you are not a student, then you don’t go to Troy.

THE Conditional statement and the contrapositive are EQUIVALENT (if one is true the other is true, if one is false, the other is false. THE CONVERSE and INVERSE are EQUIVALENT (if one is true the other is true, if one is false, the other is false.

Postulate 5: Through any Postulate 6: A line two points there is contains at least two EXACTLY one line points. Postulate 7: If two lines intersect, then they intersect in exactly one point.

Postulate 8: Through any three noncollinear points there is EXACTLY one plane. Postulate 9: A plane contains at least three noncollinear points.

Postulate 10: If two points are in a plane, then the line that contains the points is in that plane. If not, it’d be like this. Postulate 11: If two planes intersect, then their intersection is a line.

I P A Z Name two lines on plane AZHU that are not drawn. T H U One plane If two points are on a plane, the line containing them is on the plane. (Post) Postulate 8

2. 2 – Definitions and Biconditional Statements

Definition of Perpendicular lines (IMPORTANT): Two lines that intersect to form RIGHT ANGLES! A line perpendicular to a plane is a line that intersects the plane in a point that is perpendicular to every line in the plane that intersects it. All definitions work forwards and backwards If two lines are perpendicular, then they form a right angle. If two lines intersect to form right angles, then they are perpendicular.

All definitions work forwards and backwards If two lines are perpendicular, then they form a right angle. If two lines intersect to form right angles, then they are perpendicular. If a conditional statement and its converse are both true, it is called biconditional, and you can combine them into a “if and only if” statement Two intersecting lines are perpendicular if and only if they form right angles.

Z W T True or false? Why? (Check some hw) Y WVT and YVX are X complementary. V S U R WVZ and RVS form a linear pair. YVU and TVR are supplementary Y, V, and S are collinear

Write the conditional statement and the converse as a biconditional and see if it’s true. If two segments are congruent, then their lengths are the same. If the lengths of the segments are the same, then they are congruent. Two segments are congruent if and only if their lengths are the same. TRUE!

Write the conditional statement and the converse as a biconditional and see if it’s true. If B is between A and C, then AB + BC = AC If AB + BC = AC, then B is between A and C if and only if AB + BC = AC TRUE!

Write the converse of the statement, then write the biconditional statement. Then see if the biconditional statement is true or false. (Check more hw) If x = 3, then x 2 = 9 If x 2 = 9, then x = 3 if and only if x 2 = 9 False, x = -3 is a counterexample If two angles are a linear pair, then they are supplementary angles. If two angles are supplementary, then they form a linear pair Two angles are a linear pair if and only if they are supplementary. False, they don’t have to be on the same line.

Split up the biconditional into a conditional statement and its converse. Pizza is healthy if and only if it has bacon. Students are good citizens if and only if they follow the ESLRs.

2. 4 – Reasoning with Properties from Algebra

Addition Prop. = Subtraction Prop. = Multiplication Prop. = Division Prop. = Substitution Prop. = Reflexive Prop. = Symmetric Prop. = Transitive Prop. =

Reasons Given Equation Addition Prop = Division Prop = Reasons Given Equation Multiplication Prop = Subtraction Prop =

Reflexive Prop. Of equality Symmetric Prop. Of equality Transitive Prop. Of equality

We will fill in the blanks M A T H 1) 1) 2) 2) 3) 3) 4) 4) 5) 5)

U D 1 2 C K 1) 1) 2) 2) 3) 3) 4) 4) 5) 5)

A L N G E S

2. 6 – Proving Statements about Angles

Copy a segment 1) Draw a line Use same radius for both circles, so segments are congruent. Center of compass Writing end GREEN DOT DRAWN PART (compass and ruler) 2) Choose point on line 3) Set compass to original radius, transfer it to new line, draw an arc, label the intersection.

_____ Property Symmetric Property _____ Property Right Angle Congruence Thrm - All ______ angles are _______

Congruent Supplements Theorem If two angles are ______ to the same angle (or to congruent angles), then they are congruent. If _____ + _____ = 180 and _____ + ____ = 180, then ____ Congruent Complements Theorem If two angles are ______ to the same angle (or to congruent angles), then they are congruent. If _____ + _____ = 90 and _____ + ____ = 90, then ____

Vertical Angles Thrm - _____ angles are ______ Linear Pair Postulate – If two angles form a linear pair, then they are _____

L W R I O A S Z

Given E T R Prove A Given

Given Prove Given

Given P N J K Prove M Given 2. 5 -9 Number 2

Given T V R Prove P Q Given