Basic Theorems of Euclidean Geometry

We finally adopt the Euclidean Parallel Postulate and explore geometry under this assumption. The result is Euclidean Geometry and its major results are similarity, the Pythagorean Theorem, and trigonometry, although there is of course much more.

Axiom (The Euclidean Parallel Postulate): For every line l and for every point P that does not lie on l, there is exactly one line m such that P lies on m and m is parallel to l.

We now have as theorems all those statements that we proved equivalent to EPP earlier, including:

1. If two parallel lines are cut by a transversal, alternate interior angles are congruent.

2. (Euclid’s Postulate V) If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, then the lines will meet on that side of the transversal.

3. (Proclus’s Axiom) A third line intersecting one of two parallel lines intersects the other.

4. A line perpendicular to one of two parallel lines is perpendicular to the other.

5. The perpendicular bisectors of the sides of a triangle are concurrent.

6. There exists a circle passing through any three noncollinear points.

7. (Angle Sum Postulate) The sum of measures of the angles of a triangle is 180.

8. There exists one triangle such that the sum of measures of the angles is 180. 9. There exists a rectangle.

10. There exist two lines l and m such that l is equidistant from m.

11. There exists a pair of similar, noncongruent triangles.

12. There exists an acute angle such that every line intersecting and perpendicular to one ray of the angle intersects the other ray.

13. (Wallis’ Postulate) If ªABC is a triangle and is a segment, then there exists a point F such that ªABC is similar to ªDEF.

14. (Transitivity of Parallelism) If line l is parallel to line m and line m is parallel to line n, then either n = l or l is parallel to n.

There are also a flood of straightforward corollaries.

Theorem: If two lines in the same plane are cut by a transversal, then the lines are parallel iff

• alternate interior angles are congruent • alternate exterior angles are congruent • corresponding angles are supplementary • interior angles on the same side of the transversal are supplementary • exterior angles on the same side of the transversal are supplementary Note: Now we begin to strengthen some of the theorems of absolute geometry, beginning with the Exterior Angle Inequality.

Euclidean Exterior Angle Theorem: In any triangle, the measure of an exterior angle is the sum of the measures of the two remote interior angles.

~ We use the same construction as for the proofs of the exterior angle theorem and Saccheri - Legendre Theorem in absolute geometry. Begin with ªABC and point D with B*C*D. Find the midpoint M of and find point E with B-M-E and MB = ME. Connect E and C. Now by construction and SAS, ªBAM is congruent to ªECM, and it is easy to show that E is interior to pACD. Moreover, since pBAC –pECA, with as the transversal we have congruent alternate interior angles and thus . This makes pABC and pECD corresponding angles (with as the transversal), and hence congruent. Thus € Definitions:

• A convex quadrilateral is a parallelogram if the opposite sides are parallel. •A rhombus is a parallelogram having two adjacent sides congruent. •A square is a rhombus having two adjacent sides perpendicular.

A cascade of theorems, all of which rely on cutting things into triangles and using congruence theorems and transversal theorems:

• A diagonal of a parallelogram divides it into congruent triangles. • A convex quadrilateral is a parallelogram iff its opposite sides are congruent. • Opposite angles of a parallelogram are congruent • Adjacent angles of a parallelogram are supplementary • A convex quadrilateral is a parallelogram iff its diagonals bisect each other. • A parallelogram is a rhombus iff its diagonals are perpendicular. • A parallelogram is a rectangle iff its diagonals are congruent. • A parallelogram is a square iff its diagonals are both perpendicular and congruent. • Etc. These make good exercises. Try a few!!

Definition: A trapezoid is a convex quadrilateral with exactly one pair of opposite sides parallel. The parallel sides are called bases and the other two sides are called legs. The segment joining the midpoint of the legs is called the median.

Some books use a definition of trapezoid that omits the word “exactly.” This allows trapezoids to be parallelograms; that is, the two legs could also be parallel. For us, the set of all trapezoids and the set of all parallelograms are disjoint.

If a trapezoid is such that its two legs are congruent, it is called an isosceles trapezoid.

Can you prove that the base angles of an isosceles trapezoid are congruent?