Big Ideas Math High School Geometry Lessons 2.1 - 2.3

Hypothesis

"If" part of a conditional statement

Conclusion

"Then" part of a conditional statement

Converse

Exchange the hypothesis and the conclusion. Then/if form.

Is this statement true or false? If you are in math class, then you are in Geometry

False

Is this statement true or false? If you are not in Geometry, then you are not in math class

False

Is this statement true or false? M is the midpoint of line AB

False; the midpoint cannot be assumed unless line AM and line MB are marked as congruent

Solve 9 + x = 13 x = 4 Tell which algebraic property of equality you used.

Subtraction Property of Equality

Describe and correct the error in interpreting the statement. If a figure is a rectangle, then the figure has four sides. A trapezoid has four sides. Using the Law of Detachment, you can conclude that a trapezoid is a rectangle.

The Law of Detachment cannot be used because the hypothesis is not true. Using the Law of Detachment, since a square is a rectangle, you can conclude that it has four sides.

Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. Rational numbers can be written as fractions. Irrational numbers cannot be written as fractions. So, 1/2 is a rational number.

The conclusion is based on deductive reasoning. The conjecture is based on mathematical definitions and properties.

Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. Each time your mom goes to the store, she buys milk. So, the next time your mom goes to the store, she will buy milk.

The conclusion is based on inductive reasoning. The conjecture is based on the assumption that a pattern observed in specific cases will continue.

If-Then Form

The hypothesis and conclusion. If p, then q.

Write the negation of the statement: The lake is not cold

The lake is cold

Write the negation of the statement: The lake is cold

The lake is not cold

Negation

The opposite of the original statement.

Three Point Postulate

Through any three noncollinear points, there exists exactly one plane

Two Point Postulate

Through any two points, there exists exactly one line

Is this statement true or false? If you are in Geometry, then you are in math class

True

Is this statement true or false? If you are not in math class, then you are not in Geometry

True

Conjecture

Unproven statement based on observations

Deductive Reasoning

Using facts, definitions, accepted properties, and the laws of logic to form a logical argument

Equivalent Statements

When two statements are either both true or both false

Inductive Reasoning

When you find a pattern in specific cases and write a conjecture for the general case

Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible: If a figure is a rhombus, then the figure is a parallelogram. If a figure is a parallelogram, then the figure has two pairs of opposite sides that are parallel. OPTIONS: 2A) If a figure is a rhombus, then the figure has two pairs of opposite sides that are parallel. 2B) If a figure has two pairs of opposite sides that are parallel, then the figure is a parallelogram. 2C) If a figure is a parallelogram, then the figure is a rhombus. 2D) Not possible

2B

Line-Point Postulate

A line contains at least two points

Conditional Statement

A logical statement that has 2 parts: a hypothesis (p) and a conclusion (q). Written in if/then form

Plane-Point Postulate

A plane contains at least three noncollinear points

Counterexample

A specific case for which the conjecture is false

Biconditional Statement

A statement that contains the phrase "if and only if". The abbreviation is "iff".

Truth Table

A table that shows the truth values for the hypothesis (p) and conclusion (q).

Solve t − 6 = −4. t = 2 Tell which algebraic property of equality you used.

Addition Property of Equality

What types of statements are either both true or both false?

Conditional and Contrapositive; Converse and Inverse

Solve 3x = 21 x = 7 Tell which algebraic property of equality you used.

Division Property of Equality

Rewrite the conditional statement in if-then form: Today is Friday, and tomorrow is the weekend

If today is Friday, then tomorrow is the weekend

Write this statement as an inverse statement: Today is Friday, and tomorrow is the weekend

If today is not Friday, then tomorrow is not the weekend

Write this statement as a contrapositive statement: Today is Friday, and tomorrow is the weekend

If tomorrow is not the weekend, then today is not Friday

Write this statement as a converse statement: Today is Friday, and tomorrow is the weekend

If tomorrow is the weekend, then today is Friday

Perpendicular Lines

If two lines intersect to form a right angle

Line Intersection Postulate

If two lines intersect, then their intersection in exactly one point

Plane Intersection Postulate

If two planes intersect, then their intersection is a line

Plane-Line Postulate

If two point lie in a plane, then the line containing them lies in the plane

State the law of logic that is illustrated. If you miss practice the day before a game, then you will not be a starting player in the game. You miss practice on Tuesday. You will not start the game Wednesday.

Law of Detachment

State the law of logic that is illustrated. If ∠1 and ∠2 are vertical angles, then ∠1≅∠2. If ∠1≅∠2, then m∠1=m∠2. If ∠1 and ∠2 are vertical angles, then m∠1=m∠2.

Law of Syllogism

Solve x/7 = 5 x = 35 Tell which algebraic property of equality you used.

Multiplication Property of Equality

Inverse

Negate both the hypothesis and the conclusion. If not p, then not q.

Contrapositive

Negate the converse of a conditional statement. If not q, then not p.

How does the prefix "counter-" help you understand the term "counterexample"? OPTIONS: 1) The prefix counter- means "opposes". So, a counterexample opposes the truth of a statement. 2) The prefix counter- means "opposes". So, a counterexample opposes the false nature of a statement. 3) The prefix counter- means "opposes". So, a counterexample is an example that is the opposite of a converse. 4) The prefix counter- means "opposes". So, a counterexample is an example of the opposite of a statement.

Option 1

Use the Law of Detachment to determine what you can conclude from the given information, if possible: If you pass the final, then you pass the class. You passed the final. OPTIONS: 1A) You passed the class. 2A) You did not pass the class. 3A) Not possible.

Option 1A

Select the counterexample that shows that the conjecture is false: Conjecture: The product of two positive numbers is always greater than either number. OPTIONS: 1) 3 ⋅ 2 = 6 6 is greater than 3 2) 1 ⋅ 5 = 5 5 is not greater than 5 3) −2 ⋅ −1 = 2 2 is greater than −2 4) 6 ⋅ 2 = 12 6 is not greater than 12

Option 2

Use the Law of Detachment to determine what you can conclude from the given information, if possible: If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST has four right angles. OPTIONS: 1A) Quadrilateral QRST is a square. 2A) Not possible. 3A) Quadrilateral QRST is not a square.

Option 2A

Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. If a figure is a square, then the figure has four congruent sides. If a figure is a square, then the figure has four right angles. OPTIONS: 2A) Not possible 2B) If a figure is has four right angles, then the figure has four congruent sides. 2C) If a figure has four congruent sides, then the figure has four right angles. 2D) If a figure is a square, then the figure has four congruent sides.

Option 2A

Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. If a=3, then 5a=15 If 1/2a=1/2, then a=3. OPTIONS: 2A) If a=3, then 1/2a=1/2. 2B) If 5a=155a=15, then 12a=11212a=112. 5a=155a=1512a=11212a=112 2C) If 1/2a=1/2, then 5a=15. 2D) Not possible

Option 2C

Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. If x<−2, then |x|>2. If x>2, then |x|>2. OPTIONS: 2A) If x<−2, then x>2. 2B) If x>2, then x<−2 and |x|>2. 2C) If −2<x<2, then |x|>2. 2D)Not possible.

Option 2D

Your friend claims the statement "If I bought a shirt, then I went to the mall" can be written as a true biconditional statement. Your sister says you cannot write it as a biconditional. Who is correct? Explain your reasoning. OPTIONS: 1) Your friend because the converse is true. The mall is not the only place to buy shirts. 2) Your friend because the converse is true. You cannot buy a shirt if you never enter a mall. 3) Your sister because the converse is not true. It is possible to go to the mall without buying a shirt. 4) Your sister because the converse is not true. You can buy a shirt at a stand-alone clothing store.

Option 3

Use the Law of Detachment to determine what you can conclude from the given information, if possible: If your parents let you borrow the car, then you will go to the movies with your friend. You will go to the movies with your friend. OPTIONS: 1A) Your parents let you borrow the car. 2A) Your parents did not let you borrow the car. 3A) Not possible.

Option 3A

Select the counterexample that shows that the conjecture is false: Conjecture: If two angles are supplements of each other, then one of the angles must be acute. OPTIONS: 1) When one angle is 25º, the supplement is not acute. 2) Both angles could be straight angles. 3) When one angle is 135º, the supplement is not acute. 4) Both angles could be right angles.

Option 4

Rewrite the statements as a single biconditional statement: If an angle is obtuse, then it has a measure 90º between 180º. If an angle has a measure between 90º and 180º, then it is obtuse. OPTIONS: A) An angle has a measure between 90º and 180º if and only if it is obtuse. B) An angle is an obtuse angle if and only if it does not measure between 90º and 180º. C) An obtuse angle has a measure between 90º and 180º if and only if it is an angle. D) An obtuse angle is an angle if and only if it has a measure between 90º and 180º.

Option A

Question 1 The following numbers are the first nine Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Make a conjecture about each of the Fibonacci numbers after the first two. OPTIONS: AA) Each number in the sequence is the sum of the previous two numbers in the sequence. BB) The next consecutive integer, beginning with 1, is added to the previous number in the sequence. CC) Each number in the sequence is multiplied by 2, then 1 is subtracted from the product. DD) Find the difference between the square of each number and the next consecutive odd integer to get the next number.

Option AA

Rewrite the statements as a single biconditional statement: If a polygon has four sides, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. OPTIONS: A) A figure is a quadrilateral if and only if it is a polygon. B) A polygon is a quadrilateral if and only if it has four sides. C) A shape has four sides if and only if it is a polygon. D) A quadrilateral is a polygon if and only if it does not have four sides.

Option B

Determine which postulate is illustrated by the statement. AB+BC=AC OPTIONS: AA) Ruler Postulate BB) Segment Addition Postulate CC) Protractor Postulate DD) Angle Addition Postulate

Option BB

Your friend claims that even though two planes intersect in a line, it is possible for three planes to intersect in a point. Is your friend correct? Explain your reasoning. OPTIONS: AAA) Yes. Three planes will always intersect in a point because each pair of planes intersects in a line and the lines intersect in a point. BBB) Yes. For example, the ceiling and two walls of many rooms intersect in a point in the corner of the room. CCC) No. Three intersecting planes intersect in three possible points because there are three ways to pair the planes with one another.

Option BBB

Rewrite the definition of the term as a biconditional statement: Two angles are supplementary angles when the sum of their measures is 180º OPTIONS: A) Two angles are supplementary if and only if the sum of their measures is not 180º. B) Two angles cannot be supplementary if the sum of their measures is 180º. C) Two angles are supplementary angles if and only if the sum of their measures is 180º. D) Two angles can be supplementary if the sum of their measures is 180º.

Option C

Determine which postulate is illustrated by the statement. AD is the absolute value of the difference of the coordinates of A and D. OPTIONS: AA) Protractor Postulate BB) Segment Addition Postulate CC) Ruler Postulate DD) Angle Addition Postulate

Option CC

A way to solve a system of two linear equations that intersect is to graph the lines and find the coordinates of their intersection. Which postulate guarantees this process works for any two linear equations? OPTIONS: AAA) Two Point Postulate BBB) Line-Point Postulate CCC) Line Intersection Postulate

Option CCC

Rewrite the definition of the term as a biconditional statement: Two angles are vertical angles when their sides form two pairs of opposite rays OPTIONS: A) If an angle is vertical to another angle, then the angles do not form pairs of opposite rays. B) If two angles are vertical, then their sides form two pairs of opposite rays. C) An angle is vertical to another angle if and only if the angles do not form pairs of opposite rays. D) Two angles are vertical angles if and only if their sides form two pairs of opposite rays.

Option D

Determine which postulate is illustrated by the statement. m∠DAC = m∠DAE + m∠EAB OPTIONS: AA) Ruler Postulate BB) Segment Addition Postulate CC) Protractor Postulate DD) Angle Addition Postulate

Options DD