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Geometry: Euclidean Geometry: Euclidean MATH 3120, Spring 2017 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. All fifteen prior axioms are used to prove neutral geometry resulsts. Axioms 16 through 22 are needed to fully prove all Euclidean results for plane geometry. Note that the axioms and theorems of a neutral geometry allow us to prove that, for an arbitrary line and arbitrary external point, there exists at least one parallell line through the point parallel to the line. Moreover, the alternate interior angles of parallel lines cut by a transversal are congruent which leads directly to the fact that two lines perpendicular to the same line are parallel. The problem isn't having parallel lines in a neutral geometry. The problem is having too many. 1 Building Blocks of Euclidean Geometry SMSG Parallel Postulate. Through a given external point there is at most one line parallel to a given line. When we assume the strict Euclidean parallel postulate, we move from the possibility of infinitely many parallel lines to exactly one. This restriction immediately allows for the proof of several important theorems. Theorem 1. (E) The Euclidean parallel postulate is equivalent to the converse of the alternate interior angle theorem. (Proved in neutral geometry section.) Theorem 2. (E) If two parallel lines are cut by a transversal, then alternate interior angle are congruent. (Proved in neutral geometry section.) Theorem 3. (E) The sum of the measures of the interior angles of triangle is 180◦. Theorem 4.* The Euclidean parallel postulate is equivalent to Theorem 3: The angle sum for every triangle is 180◦. (Hint: for a neutral geometry, the angle sum for triangle was at most 180◦. What forces the equality in Euclidean space?) Corollary to Theorem 4. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Definition: Parallelogram. A quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel. Theorem 5. (E) The opposite sides of a prallelgoram are congruent. (Hint: use triangle congruence.) 1 Theorem 6. (E)* If a transversal intersects three parallel lines in such a way as to make congruent segments between the paralllels, then every transversal interesecting these parallel lines will do likewise. (Hint: given three parallel lines l1, l2, and l3 and a transversal t1 that cuts intersects it, create an arbitray t2 that also intersects it. Then, create a third transversal parallel to t1 and use similar triangles.) Corollary to Theorem 6. (E) If a transversal intersects n parallel lines (n > 2) in such a way as to make congruent segments between the paralllels, then every transversal interesecting these parallel lines will do likewise. (Hint: induction.) Theorem 7: Median Concurrence Theorem.* The three medians of a triangle are concurent at a point called the centroid. (Hint: consider 4DF G with median DM. Note M is the midpoint of FG. Call the centroid point C. Then the measure of DC is exactly twice the measure of CM. This is true - but must be proven. Theorem 6 and its corollary along with this fact provide an outline for the proof.) Corollary to the Median Concurrence Theorem. Medians of a triangle intersect at a point that is two-thirds the distance from any vertex to the midpoint of the opposite side. 2 Consequences of the Euclidean Parallel Postulate The following theorems are direct consequences of the Euclidean parallel postulate, and their proofs are reasonably elementary. Two key concepts are equivalent to the Euclidean parallel postulate: that a rectangle exists, and that all triangles have angle sum of 180◦. The theorems below establish many common notions in Euclidean geometry. Theorem 8. (E) Two lines parallel to another line are parallel to each other. Theorem 9. (E) If a line intersects one of two parallel llines, then it interesects the other. Theorem 10. Each diagonal of a parallelogram partitions the parallelogram into a pair of congruent triangles. Theorem 11. The diagonals of a parallelogram bisect each other. Theorem 12. The diagonals of a quadrilateral bisect each other if and only if the quadrilateral is a parallelogram. (Note that one direction has been proven in already Theorem 11.) Theorem 13. A quadrilateral has a pair of opposite sides that are congruent and parallel if and only if the quadrilateral is a parallelogram. (Note that one direction is true by definition of a parallelogram.) Definition. We call line segments parallel if they are contained in lines that are parallel. 2 Theorem 14. If a line segment has as its endpoints the midpoints of two sides of a triangle, then the segment is parallel to and one-half the length of the third side of the triangle. Definition. A rhombus is a quadrilateral in which all four sides are congruent. Theorem 15. The diagonals of a rhombus are perpendicular. Theorem 16. If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus. Definition. A median of a triangle is a line passing through a vertex and the midpoint of the side opposite that vertex. Theorem 17. In a right triangle, the median from the right angle to the hypotenuse is one-half the length of the hypotenuse. Theorem 18.* In a right triangle, one of the angles measures 30◦ if and only if the side opposite this angle is one-half thep length of the hypotenuse. (Note: this result along with the Pythagorean Theorem gives us the 1−2− 3 ratio of side lengths for 30−60−90 right triangles. Recall that the Pythagorean Theorem is equivalent to the Euclidean parallel postulate.) Theorem 19. The sum of the measures of the interior angles of a convex n-gon is (n − 2)(180◦). Theorem 20. The sum of the measures of the exterior angles of a convex n-gon is 360◦. Theorem 21.* The midpoints of the sides of a quadrilateral are the vertices of a convex parallelo- gram. 3 Area and Congruence We have discussed several issues with Euclid. We have arrived at another: How do we define area? If rectangles have an area that is defined as length times width, then we can calculate the area of triangles. Using triangles and rectangles, we can calcuate polygon areas. The existence of a rectangle is equivalent to Euclid's parallel postulate, so our standard notion of area only exists in Euclidean geometry. In any other geometry, calculating an area will prove troublesome. However, we're developing Euclidean geometry in this section. Please note that SMSG Axiom 20 states that \the area of a rectangle is the product of the length of its base and the length of its altitude." Theorem 22. (E) The area of a parallelogram is the product of the lengths of its base and height. Theorem 23. The area of a right triangle is one-half the product of the lengths of its legs. 3 Theorem 24. (E) The area of a triangle is one-half the product of the any base and the correspond- ing height. Definition: Trapezoid. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Note that there are two different definitions of a trapezoid: posssessing exactly one pair of parallel sides vs. possessing at least one pair of parallel sides. The first means no parallelograms are trapezoids while the second means all parallelograms are trapezoids. More common is the one defined above where parallelograms are not trapezoids. High school teachers should note carefully which definition is used in their curriculum materials especially when using classrooms resources and activities from the interwebs. Theorem 25. The area of a trapezoid is the product of its height and the arithmetic mean of its bases. Theorem 26. The area of a rhombus is one-half the product of its diagonals. 4 Similarity Definition: Similar Polygons. Two polygons are similar provided (i) corresponding sides of each are in the same proportion, and (ii) corresponding interior angles are congruent. Note that requirement (i) implies a constant ratio of proportionality for any two pairs of side lengths. Theorem 27: Basic Proportionality Theorem. (E)* A line parallel to one side of a triangle intersects the other two sides in two different points if and only if it divides these sides into segments that are proportional. (The \two different points" simply requires the point of intersection to not be the vertex.) Theorem 28: AAA Triangle Similarity. (E) If the interior angles of one triangle are congruent to corresponding angles of a second triangle, then the triangles are similar. Theorem 29: SAS Triangle Similarity. If an angle of one triangle is congruent to the correspond- ing angle of a second triangle, and the corresponding sides adjacent to these angles are in proportion, then the triangles are similar. Theorem 30: SAS Triangle Similarity. If the lengths of the sides of one triangle are proportional to the corresponding side lengths of a second triangle, then the triangles are similar. Theorem 31: Pythagorean Theorem. (E)* If a and b are the lengths of the legs of a right triangle the hypotenuse of which has length c, then a2 + b2 = c2. 4 5 Circles Definitions. A circle is a set of points all of which equidistant from a given point which is called the center. The common distance between the circle and center point is called the radius of the circle. Theorem 32. (E) In the Euclidean plane, three distinct points determine a line. (This proof requires both an existance proof and a uniqueness proof.
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