In mathematics, we have **postulates** which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct **logical arguments** for new facts of mathematics.

Consider the Venn diagram:

Consider whether each conclusion is valid or not. Explain your reasoning.

A polygon is a parallelogram. Therefore, it is not a trapezoid.

A polygon has two sides that are not parallel. Therefore, it is a parallelogram.

If a shape is a rhombus, then it must be a parallelogram.

There are serveral laws that can be used to construct an argument:

Recall the **Law of Contrapositive**:

The **Law of Detachment** says if the hypothesis of a true conditional statement is true, then the conclusion is also true.

For example:

p: | A number is odd. |
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q: | It is the sum of an even and odd number. |

p \to q: | If a number is odd, then it is the sum of an even and odd number. |

Consider the number 11. Since p \to q is true and p is true since 11 is odd, by the law of detachment, 11 is the sum of an even number and an odd number.

The **Law of Syllogism** says if both p \to q \text{ and } q \to r are true, then p \to r is the logical conclusion."

For example:

p \to q: | \text{If a shape has three sides, it is a triangle.} |
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q \to r: | \text{If a shape is a triangle, the angles sum to } 360 \text{ degrees.} |

Since p \to q is true and q \to r is true, by the law of syllogism, p \to r is true which means "If a shape has three sides, the angles sum to 360 degrees".

Deductive reasoning uses facts, definitions, and the laws of logic to form an argument. Inductive reasoning uses observations and patterns to form a conjecture. Deductive reasoning can prove an argument is true, while we can only use inductive reasoning to prove a conjecture is false.

Determine the law of logic that is used in each argument.

a

If the sum of the angles in a polygon is 360 \degree, then it is a quadrilateral.

You see the following shape and know it is a quadrilateral.

Worked Solution

b

If you finish your homework, you will go to football game. If you go to the football game, you will sleep over at your friend's house.

If you finish your homework, then you will sleep over at your friends house.

Worked Solution

Which of the following conclusions can be drawn from these statements?

If we win the game, we will win the tournament. If we win the tournament, we will go out for pizza.

A

If we go out for pizza, we won the game.

B

If we win the game, we will go out for pizza

C

If we do not win the tournament, we will not go our for pizza.

Worked Solution

Write the logical argument symbolically, then determine whether the argument is valid. Justify using deductive reasoning.

If a quadrilateral has two sets of parallel sides, then it is a parallelogram. If a quadrilateral is a parallelogram, then its opposite angles are congruent. A quadrilateral does not have two sets of parallel sides. Therefore, its opposite angles are congruent.

Worked Solution

Idea summary

These laws can help determine the validity of arguments:

Law of contrapositive | \text{If } p \to q \text{ is true and } \sim q \text{ is true, then } \sim p \text{ is true.} |
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Law of syllogism | \text{If } p \to q \text{ is true and } q \to r \text{ is true, then } p \to r \text{ is true.} |

Law of detachment | \text{If } p \to q \text{ is true and } p \text{ is true, then } q \text{ is true.} |

In mathematics, we have **postulates** which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct **logical arguments** for new facts of mathematics. A new fact is called a **theorem** and its argument is called its **proof**.

We can use deductive reasoning to write proofs in a variety of ways. Formal proofs have a more rigid structure and can be helpful when we want to present our justification in a very organized way.

The structure of a two column proof follows:

Every statement in a proof must be show to be true. In geometry, an **undefined term** is a term or word that does not require further explanation or description. The include a point, set, line, and plane. A **defined term** is a term that has a formal definition and can be defined using other geometrical terms.

We have learned many properties and laws that can be used to make arguments. Recall the properties of equality:

Symmetric property of equality | \text{If } a=b, \text{then } b=a |
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Transitive property of equality | \text{If } a=b \text{ and } b=c, \text{then } a=c |

Addition property of equality | \text{If } a=b, \text{then } a+c=b+c |

Subtraction property of equality | \text{If } a=b, \text{then } a-c=b-c |

Multiplication property of equality | \text{If } a=b, \text{then } ac=bc |

Division property of equality | \text{If } a=b \text{ and } c \neq0, \text{then } \dfrac{a}{c}=\dfrac{b}{c} |

Substitution property of equality | \text{If } {a=b}, \text{ then } b \text{ may be substituted for } a \text{ in any expression} |

Additive identity | \text{If } {a=b}, \text{ then } a+0=b \text{ and } a=b+0 |

Multiplicative identity | \text{If } {a=b}, \text{ then } a\cdot 1=b \text{ and } a=b \cdot 1 |

We also have the properties of congruence:

**Properties of segment congruence** tell us that segment congruence is reflexive, symmetric, and transitive

Reflexive property of congruent segments | \text{Any segment is congruent to itself, so } \overline{AB} \cong \overline{AB} |
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Symmetric property of congruent segments | \text{If }\overline{AB} \cong \overline{CD}\text{, then } \overline{CD} \cong \overline{AB} |

Transitive property of congruent segments | \text{If }\overline{AB} \cong \overline{CD} \text{ and }\overline{CD}\cong \overline{EF}\text{, then }\overline{AB} \cong \overline{EF} |

**Properties of angle congruence** tell us that angle congruence is reflexive, symmetric, and transitive

Reflexive property of congruent angles | \text{Any angle is congruent to itself, so } \angle{A} \cong \angle{A} |
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Symmetric property of congruent angles | \text{If }\angle{A} \cong \angle{B}\text{, then} \angle{B} \cong \angle{A} |

Transitive property of congruent angles | \text{If }\angle{A} \cong \angle{B} \text{ and }\angle{B}\cong \angle{C}\text{, then }\angle{A} \cong \angle{C} |

Consider the **vertical angles theorem**, which states that vertical angles are congruent.

Use transformations to prove the following:

Given: \angle ABC and \angle DBE are vertical angles

Prove: \angle ABC \cong \angle DBE

Worked Solution

Consider the **congruent supplements theorem**, which states that if two angles are supplementary to the same angle then they are congruent.

Construct a two column proof to prove the following:

Given:

\angle A and \angle C are supplementary

\angle B and \angle C are supplementary

Prove: \angle A \cong \angle B

Worked Solution

Idea summary

Both paragraph and two column proofs are ways to use deductive reasoning to show an argument must be true. A proof shows that a conjecture is true for every case.