Conditionals, Biconditionals, and Deductive Reasoning and algebraic properties

Be able to write a converse of a conditional statement (Geometry and Spatial Sense, GR9,1)

Be able to write a biconditional (Geometry and Spatial Sense, GR9,1)

Use law of detachment and law of syllogism to draw conclusions (Geometry and Spatial Sense, GR9,1)

Use algebraic properties to prove a statement (Geometry and spatial sense, GR9, 1)

Lets begin with an OGT question….

The graph as shown represents the amount of money Sarah can earn at her part-time job

Which of the following equations best represents the relationship between Sarah’s pay and the hours she works?

A. y = 4xB. y = 6.5xC. y = 4x + 10D. y = 6.5x + 10

If-Then Statements

Hypothesis – If statement

Conclusion – Then statement

Identify the Hypothesis and Conclusion:◦ If two lines are parallel, then they are coplanar

Identify the Hypothesis and Conclusion:◦ If two lines are parallel, then they are coplanar◦ Hypothesis: Two lines are parallel◦ Conclusion: They are coplanar

Write the statement as a conditional◦ An acute angle measures less than 90º

Identify the Hypothesis and Conclusion:◦ If two lines are parallel, then they are coplanar◦ Hypothesis: Two lines are parallel◦ Conclusion: They are coplanar

Write the statement as a conditional◦ An acute angle measures less than 90º◦ If an angle is acute, then it measures less than 90

The converse of a conditional statement if formed by exchanging the hypothesis and the conclusion in the conditional.

Example 3:◦ Conditional- If a figure is a triangle, then it has

three angles.

◦ Converse- If a figure has three angles, then it is a triangle.

Converse

• The converse of a conditional statement is formed by switching the hypothesis and the conclusion in the conditional.

– Conditional- If a figure is a triangle, then it has

three angles.

– Converse- If a figure has three angles, then it is a triangle.

If x = 5, then x + 15 = 20◦ Write the converse:

If x = 5, then x + 15 = 20◦ Write the converse:◦ If x + 15 = 20, then x = 5

◦ Write the Biconditional:

If x = 5, then x + 15 = 20◦ Write the converse:◦ If x + 15 = 20, then x = 5

◦ Write the Biconditional:◦ X = 5 if and only if x + 15 = 20

Write two statements that form this biconditional:◦ Lines are skew if and only if they are noncoplanar

Write two statements that form this biconditional:◦ Lines are skew if and only if they are noncoplanar

◦ If lines are skew, then they are noncoplanar◦ If lines are noncoplanar, then they are skew

If a conditional is true and its hypothesis is true – then its conclusion is true

If p implies q is true and p is true then q is true

Can you make a conclusion?…◦ Every High School student likes music. ◦ Ling likes music

If a conditional is true and its hypothesis is true – then its conclusion is true

If p implies q is true and p is true then q is true

Can you make a conclusion?…◦ Every High School student likes music. Ling

likes music◦ If you are a high school student, then you like

music

If p implies q is true and q implies r is true, then p implies r is true

P Q and QR, then PR

If a quadrilateral is a square, then it contains 4 right angles.If a quadrilateral contains 4 right angles, then it is a rectangle

A gardener knows that if it rains, the garden will be watered.It is raining.

If you want to build a skyscraper, start with a good foundation. The foundation is a concrete form that supports the rest of the structure.

The foundation of geometry is made of statements called postulates. These are accepted as true.

Addition Property

Subtraction Property

Multiplication Property

Division Property

Reflexive Property

Symmetric Property

Transitive Property

Substitution Property

Distributive Property

Fill in…

AB + BC = AC

3x + 2x + 1 = 36

5x + 1 = 36 Simplify

5x = 35

X =

3x 2x + 1 AC = 36

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