Reasoning and Proof

Conditional Statements

100

Find the hypothesis of the conditional statement: If the Tigers make the playoffs, then they will be in the World Series.

The Tigers make the playoffs

200

Show the statement is false by finding a counterexample. If I am passing the class, then my parents give me $5.

My parents don't gave me $5.

300

Write a conditional from the following statement: All squares are rectangles.

If the figure is a square then it is a rectangle.

Biconditionals

100

Given the following conditional write the converse. Are both statements true? If two lines are perpendicular then they intersect at a 90 degree angle.

If two lines intersect at a 90 degree angle then they are perpendicular. Yes, both statements are true. We can write a biconditional.

200

Write the biconditional of the two statements if possible. If it is sunny, then I'm at the beach. If I'm at the beach, then it's sunny.

Not possible, counterexample: I'm at the beach and it's cloudy.

300

Given the definition write it's conditional, converse, and if possible biconditional. A number is prime when its only factors are 1 and itself.

If a number is prime then its only factors are 1 and itself. If a numbers only factors are 1 and itself then it is prime. Both are true. A number is prime if and only if its only factors are 1 and itself.

Deductive Reasoning

100

What is the Law of Syllogism?

If p -> q is true. And q->r is true. Then, p->r is true.

200

A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?

Using the Law of Detachment we can conclude that the garden will be watered.

300

Use the Laws of Detachment and Syllogism to draw a possible conclusion. If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman. The circus is in town.

Using the Law of Syllogism first we can say: If the circus is in town, then Paul is working as a night watchman. Now using the Law of Detachment we can conclude: Paul is working as a night watchman.

Reasoning in Algebra

100

Given the following, name the three properties: 1) If a=b and b=c then a=c. 2) If a=b then a-c = b-c. 3) a (b+c)= ab +ac.

1) Transitive Property of Equality 2) Subtraction Property of Equality 3) Distributive Property of Equality

200

Justify each step used to solve 5x-12=32+x for x. Given: 5x-12=32+x

1) 5x=44+x : Addition Property of Equality 2) 4x=44 : Subtraction Property of Equality 3) x=11 : Division Property of Equality

300

Name the property that justifies the statement. If x=y and y+4=3x, then x+4=3x.

The conclusion of the conditional statement is the same as the equation y+4=3x (given) after x has been substituted for y (given). The property used is the Substitution Property of Equality.

Proving Angles Congruent

100

A pair of vertical angles have the equations: 4x-101 and 2x+3 Apply the Vertical Angles Theorem to find x.

4x-101=2x+3 : Vertical Angles Theorem 4x=2x+104 : Addition Property of Equality 2x=104 : Subtraction Property of Equality x=52 : Division Property of Equality

200

Given: angle 1 and angle 2 are supplementary angle 3 and angle 2 are supplementary Prove: angle 1 is congruent to angle 3

By the definition of supplementary angles, measure an. 1 + measure an. 2 =180 and measure an. 3 + measure an. 2=180. By substitution m.an. 1 + m.an. 2 = m.an. 3 + m.an. 2. Subtract m.an.2 from each side. You get m.an.1 = m.an.3

300

Given that two angles are vertical angles we know the following: angle 1 = 4x angle 2= 3x+35 x=35. How do we know this? What are the measures of the two angles?

We know this because the vertical angles are congruent so we set their measures equal to each other then solve for x. Then we just plug the value for x into each equation to get: angle 1 = 140 angle 2 = 140