Chapter 2

Reasoning and Proof

2.1 Inductive Reasoning and Conjecture Conjecture- an educated guess

based on known information Inductive reasoning- reasoning

that uses a number of specific examples to come to a plausible prediction/generalization

Counterexample- a false example(an example that shows how a conjecture is false)

Make a conjecture about the next number based on the pattern.

2, 4, 12, 48, 240

Make a conjecture and draw a figure to illustrate your conjecture.

Given: points L, M, and N; LM = 20, MN = 6, and LN = 14. Examine the measures of the

segments. Since LN + MN = LM, the points can be collinear with point N between points L

and M.

Answer:

Conjecture: L, M, and N are collinear.

UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people.

Answer: Maverick has a population of 50,436 people in its population, and it has a higher rate of unemployment than El Paso, which has 713,126 people in its population.

Examine the data in the table. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. El Paso has a greater population than Maverick while El Paso has a lower unemployment rate than Maverick.

DRIVING The table on the next screen shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, which two states could be used as a counterexample for the following statement? The greater the population of a state, the lower the number of drivers per 1000 residents.

2.3 Conditional Statements

Conditional statement- a statement that can be written in if-then form

A. Identify the hypothesis and conclusion of the following statement.

If a polygon has 6 sides, then it is a hexagon.

Answer: Hypothesis: a polygon has 6 sidesConclusion: it is a hexagon

B. Identify the hypothesis and conclusion of the following statement.

Tamika will advance to the next level of play if she completes the maze in her computer game.

B. Identify the hypothesis and conclusion of the given conditional.

To find the distance between two points, you can use the Distance Formula.

A. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form.

Distance is positive.

Sometimes you must add information to a statement. Here you know that distance is measured or determined.

Answer: Hypothesis: a distance is measuredConclusion: it is positiveIf a distance is measured, then it is positive.

B. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form.

A five-sided polygon is a pentagon.

Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal.

A. Yukon rests for 10 days, and he still has a hurt ankle.

Since the result is not what was expected, the conditional statement is false.

B. Yukon rests for 3 days, and he still has a hurt ankle.

In this case, we cannot say that the statement is false. Thus, the statement is true.

reminders

Write the converse by switching the hypothesis and conclusion of the conditional.

A. Write the converse of the conditional, then determine the truth value of each set of conditions.

If it rains today, then Michael will not go skiing.

Write the converse of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample.

Hint: First, write the conditional in if-then form.

Write the converse of the statement The sum of the measures of two complementary angles is 90. Determine if each statement is true or false, if false write a counter example.

2.5 Postulates and Paragraph Proofs

Postulate- (also called an axiom) a statement that is accepted as true

Theorem- a statement or conjecture that has been shown/proven to be true

A. Determine whether the following statement is always, sometimes, or never true. Explain.

If plane T contains contains point G, then plane T contains point G.

B. Determine whether the following statement is always, sometimes, or never true. Explain.

For if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R.

Given:

Prove: ACD is a plane.

Proof: and must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on . Points A, C, and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line.

2.6 Algebraic Proof

Solve the equation- write the property used beside each step.

Solve 2(5 – 3a) – 4(a + 7) = 92.

Write a two-column proof. If

Statements ReasonsProof:

SEA LIFE A starfish has five arms. If the length of arm 1 is 22 centimeters, and arm 1 is congruent to arm 2, and arm 2 is congruent to arm 3, prove that arm 3 has length 22 centimeters.

Given: arm 1 arm 2, arm 2 arm 3

m arm 1 = 22 cm

Prove: m arm 3 = 22 cm

Proof:Statements Reasons

2.7 Proving Segment Relationships

Prove the following.

Given: PR = QSProve: PQ = RS

Proof:Statements Reasons

Prove the following.

Given: AC = ABAB = BXCY = XD

Prove: AY = BD

Statements Reasons

Prove the following.

Given:

Prove:

Statements Reasons

2.8 Proving Angle Relationships

QUILTING The diagram below shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD.

Statements Reasons

If 1 and 2 are vertical angles and m1 = d – 32 and m2 = 175 – 2d, find m1 and m2.