Common Notions/Postulates
Finish this Common Notion:
2) If equals be ________ to equals, the wholes ______ _______.
2) If equals be added to equals, the wholes are equal.
What is an acute angle?
An angles less than a right angle.
Is it possible for the lights to be turned on and off at the same time?
No
Is this a valid or invalid syllogism:
1) Mr. DeBates loves all dogs,
2) Fido is a dog,
Therefore, Mr. DeBates loves Fido.
Valid
What are complementary angles? What are supplementary angles?
Complementary angles: Angles which add up to 90o
Supplementary angles: Angles which add up to 180o
Finish this Common Notion:
4) Things which _______ with one another are equal to one another.
4) Things which coincide with one another are equal to one another.
What is a rectilinear figure?
A figure contained by straight lines.
Is it possible for someone to be a pitcher and not a pitcher at the same time?
Yes
What is a syllogism?
A form of reasoning where you can arrive at a certain conclusion by first examining two premises.
Find the value of x.
x=55o
What is the difference between a Postulate and a Common Notion?
Postulate: An assumption made without and proof.
Common Notion: A notion that everyone can agree on.
What is a Reductio?
A proof by contradiction.
What is Aristotle's Law of Non-Contradiction?
1) “It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect”
2) “It is impossible to hold (suppose) the same thing to be and not to be”
3) “Opposite assertions cannot be true at the same time”
Is this a valid or invalid syllogism:
1) When a little boy becomes sad, he cries,
2) This boy is crying,
Therefore, the boy is sad.
Invalid
Finish this statement:
The exterior angle of a triangle is equal to _________.
The exterior angle of a triangle is equal to the two interior opposite angles.
Name 3 Postulates and 3 Common Notions.
Postulates:
1) To draw a straight line from any point to any point.
2) To produce a finite straight line continuously in a straight line,
3) To describe a circle with any centre and distance.
4) All right angles are equal to one another.
5) If a straight line falling on two other straight lines make the same side interior angles less than two right angles, then the two lines are not parallel.
Common Notions:
1) Things were are equal to the same thing are also equal to one another.
2) If equals be added to equals, the wholes are equal.
3) If equals be subtracted from equals, the remainders are equal.
4) Things which coincide with one another are equal to one another.
5) The whole is greater than the part.
What is a converse? Give two props which resemble a converse.
When the the hypothesis and conclusion switch places. Examples are prop 24 and 25.
If something is true, then the opposite of it is false. Is this a part of Aristotle's law of none contradiction?
Yes
Finish this syllogism:
1) People who have just ran a marathon sweat profusely,
2) You are sweating profusely,
Therefore,___________.
You may have ran a marathon.
Find the value of x and y and z.
x=120o
y=60o
z=1200
What does Postulate 5 mean?
If the same side interior angles on a transverse are less than 180o, than the two lines the transverse cuts are not parallel.
What is a circle?
A circle is a plane figure contained by one line so that all straight lines falling upon it from one point among those lying within the figure are equal to one another.
Are Aristotle's Laws of Non-Contradiction in Euclid? If so, give an example.
Yes,
Euclid uses this for Reductio props to prove his theory.
Is this a valid of invalid syllogism:
1) If today is a holiday, then we do not have school,
2) Today is not a holiday,
Therefore, we do have school.
Invalid
Angle 1 and angle 2 are complementary, and angle 2 and angle 3 form a linear pair. If angle 3 equals 140o, what is the measurement of angle 1? Explain.
Angle 1 equals 50o
Since angle 2 and angle 3 are linear pairs, then angle 2(40o) plus angle 3(140o) must equal 180o. And since angle 2 and angle 1 are equal to 90o(complementary), then subtract angle 2(40o) from 90o, which is 50o.