Book I, Proposition 6

Problems

Back to Proposition 6.

1.  Complete the following with either "must be true," "must be false," or 1.  "may be true or false."

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1.  a)  In a valid argument, if the hypothesis is true, then the conclusion

must be true.

1.  b)  In a valid argument, if the conclusion is false, then the hypothesis

must be false.

2.  a)  What is meant by reductio ad absurdam?

Reduction to the absurd. This is how to disprove a statment by showing that, when carried to its logical conclusion, the statement leads to an absurdity.

 2.  b)  When do we use reductio ad absurdam, that is, what is the statement
 2.  b)  that we want to disprove?

The contradiction of what we want to prove !

3.  a)  State the hypothesis of Proposition 6

Two angles of a triangle are equal.

2.  b)  State the conclusion.

The sides opposite those angles are equal.

2.  c)  Practice Proposition 6.

4.   Name three ways of proving straight lines equal to one another.

They are radii of a circle.

They are corresponding sides of congruent triangles.

They are the sides of an isosceles triangle.

5.   Name three ways of proving angles equal to one another.

They are corresponding angles of congruent triangles.

They are the base angles of an isosceles triangle.

They are right angles. (Postulate 4.)

6.  In triangle ABC, angle B is equal to angle C, the point D falls
6.  on BC, and angle BAD is equal to angle CAD.  Prove that BD is equal
6.  to DC, and that AD is perpendicular to BC.

(Prove that triangles ADB, ADC are congruent.)

Since angle B is equal to angle C, then the sides opposite them, the sides AB, AC, are equal.
And since, by hypothesis, angle BAD is equal to angle CAD,
and side AD is common to the triangles ADB, ADC,
the remaining side BD is equal to the remaining side DC. (S.A.S.)
Further, angle ADB, opposite side AB, is equal to the adjacent angle ADC, opposite the equal side AC.
Therefore AD is perpendicular to BC. (Definition 3.)

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