MODERN GEOMETRYSections 4.1, 5.1

THOMPSON

ANNOUNCEMENTS

Construction Project Part I Due Today

Exam 2 (Chapters 4, 5) on Tuesday 3/3

LOGIC…A VERY BRIEF INTRODUCTION

Proof by Direct Reasoning

LOGIC…A VERY BRIEF INTRODUCTION

Conditional Statement “if/then”

“p implies q”

p ⇒ q

Hypothesis Conclusion

If a triangle is equilateral, then it is equiangular.

THE LAW OF DETACHMENT

If a triangle has side lengths of 3, 4, 5, then it is a right triangle

The triangle has side lengths of 3, 4, 5.

So,…

Other examples…

p ⇒ q

p

∴ q

THE LAW OF SYLLOGISM

If a triangle has side lengths of 3, 4, 5, then it is a right triangle

If a triangle is right, then it has a right angle

So,…

Other examples…

p ⇒ q

q ⇒ r

∴ p ⇒ r

CONDITIONAL VARIATIONS

Conditional

Converse

Inverse

Contrapositive*

p ⇒ q

q ⇒ p

not p ⇒ not q

not q ⇒ not p

Other examples…

5.1 PARALLEL LINES

LINES CUT BY A TRANSVERSAL

1 2

34

5 6

7 8

• Corresponding

• Alternate interior

• Alternate exterior

• Same-side interior

• Same-side exterior

• Vertical

PARALLEL LINES CUT BY A TRANSVERSAL

1 2

3 4

5 6

7 8

If two parallel lines are cut by a transversal, then…

PARALLEL LINES CUT BY A TRANSVERSALCONVERSE

1 2

3 4

5 6

7 8

If …. , then the two lines cut by the transversal are parallel

PROBLEM SOLVING

EUCLID’S 5TH POSTULATETHE PARALLEL POSTULATE

If a line segment intersects two straight lines forming two interior angles on the

same side that sum to less than two right angles, then the two lines, if extended

indefinitely, meet on that side on which the angles sum to less than two right angles.

PLAYFAIR’S AXIOM

C

A B

In a plane, given a line and a point not on it, at most one line

parallel to the given line can be drawn through the point.

WHAT IF EUCLID’S 5TH POSTULATE IS WRONG?!

ELLIPTICAL (SPHERICAL) GEOMETRY

HYPERBOLIC GEOMETRY

HYPERBOLIC GEOMETRYPOINCARE DISC

HYPERBOLIC GEOMETRYPOINCARE DISC

HYPERBOLIC GEOMETRYMC ESCHER

HOMEWORK

4.1: 1 – 43 every other odd

5.1: 1 – 10, 29 – 32, 41

5.2: 21 – 29 all