Chapter 5: Euclidean geometry x5.1 Basic Theorems of Euclidean Geometry x5.2 The Parallel Projection Theorem MTH 411/511 Foundations of Geometry MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Goals for today: • Begin our study of Euclidean geometry. • State and prove the Parallel Projection Theorem. It’s good to have goals MTH 411/511 (Geometry) Euclidean geometry Fall 2020 • Begin our study of Euclidean geometry. • State and prove the Parallel Projection Theorem. It’s good to have goals Goals for today: MTH 411/511 (Geometry) Euclidean geometry Fall 2020 • State and prove the Parallel Projection Theorem. It’s good to have goals Goals for today: • Begin our study of Euclidean geometry. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 It’s good to have goals Goals for today: • Begin our study of Euclidean geometry. • State and prove the Parallel Projection Theorem. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Pythagorean Theorem (Theorem 5.4.1) If 4ABC is a right triangle with right angle at vertex C, then (AC)2 + (BC)2 = (AB)2. Euclidean geometry We now add the Euclidean Parallel Postulate to our list of axioms. Our goal in this chapter will be to develop the theory sufficiently so as to prove the following fundamental results. Fundamental Theorem on Similar Triangles (Theorem 5.3.1) If 4ABC and 4DEF are two triangles such that 4ABC ∼ 4DEF , then AB DE = . AC DF MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Euclidean geometry We now add the Euclidean Parallel Postulate to our list of axioms. Our goal in this chapter will be to develop the theory sufficiently so as to prove the following fundamental results. Fundamental Theorem on Similar Triangles (Theorem 5.3.1) If 4ABC and 4DEF are two triangles such that 4ABC ∼ 4DEF , then AB DE = . AC DF Pythagorean Theorem (Theorem 5.4.1) If 4ABC is a right triangle with right angle at vertex C, then (AC)2 + (BC)2 = (AB)2. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Converse to the AIAT (Theorem 5.1.1) If two parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. Euclid’s Postulate V (Theorem 5.1.2) If ` and `0 are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less that 180◦, then ` and `0 intersect on that side of t. Angle Sum Theorem (5.1.3) If 4ABC is a triangle, then σ(4ABC) = 180◦. Wallis’ Postulate If 4ABC is a triangle and DE is a segment, then there exists a point F such that 4ABC ∼ 4DEF Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Euclid’s Postulate V (Theorem 5.1.2) If ` and `0 are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less that 180◦, then ` and `0 intersect on that side of t. Angle Sum Theorem (5.1.3) If 4ABC is a triangle, then σ(4ABC) = 180◦. Wallis’ Postulate If 4ABC is a triangle and DE is a segment, then there exists a point F such that 4ABC ∼ 4DEF Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Converse to the AIAT (Theorem 5.1.1) If two parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Angle Sum Theorem (5.1.3) If 4ABC is a triangle, then σ(4ABC) = 180◦. Wallis’ Postulate If 4ABC is a triangle and DE is a segment, then there exists a point F such that 4ABC ∼ 4DEF Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Converse to the AIAT (Theorem 5.1.1) If two parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. Euclid’s Postulate V (Theorem 5.1.2) If ` and `0 are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less that 180◦, then ` and `0 intersect on that side of t. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Wallis’ Postulate If 4ABC is a triangle and DE is a segment, then there exists a point F such that 4ABC ∼ 4DEF Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Converse to the AIAT (Theorem 5.1.1) If two parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. Euclid’s Postulate V (Theorem 5.1.2) If ` and `0 are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less that 180◦, then ` and `0 intersect on that side of t. Angle Sum Theorem (5.1.3) If 4ABC is a triangle, then σ(4ABC) = 180◦. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Converse to the AIAT (Theorem 5.1.1) If two parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. Euclid’s Postulate V (Theorem 5.1.2) If ` and `0 are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less that 180◦, then ` and `0 intersect on that side of t. Angle Sum Theorem (5.1.3) If 4ABC is a triangle, then σ(4ABC) = 180◦. Wallis’ Postulate If 4ABC is a triangle and DE is a segment, then there exists a point F such that 4ABC ∼ 4DEF MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Proclus’s Axiom (Theorem 5.1.5) If ` and `0 are parallel lines and t 6= ` is a line such that t intersects `, then t also intersects `0. (Theorem 5.1.6) If ` and `0 are parallel lines and t is a transversal such that t ⊥ `, then t ⊥ `0. (Theorem 5.1.7) If `, m, n and k are lines such that k k `, m ⊥ k, and n ⊥ `, then either m = n or m k n. Transitivity of Parallelism (Theorem 5.1.8) If ` is parallel to m and m is parallel to n, then either ` = n or ` k n. Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 (Theorem 5.1.6) If ` and `0 are parallel lines and t is a transversal such that t ⊥ `, then t ⊥ `0. (Theorem 5.1.7) If `, m, n and k are lines such that k k `, m ⊥ k, and n ⊥ `, then either m = n or m k n. Transitivity of Parallelism (Theorem 5.1.8) If ` is parallel to m and m is parallel to n, then either ` = n or ` k n. Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Proclus’s Axiom (Theorem 5.1.5) If ` and `0 are parallel lines and t 6= ` is a line such that t intersects `, then t also intersects `0. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 (Theorem 5.1.7) If `, m, n and k are lines such that k k `, m ⊥ k, and n ⊥ `, then either m = n or m k n. Transitivity of Parallelism (Theorem 5.1.8) If ` is parallel to m and m is parallel to n, then either ` = n or ` k n. Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Proclus’s Axiom (Theorem 5.1.5) If ` and `0 are parallel lines and t 6= ` is a line such that t intersects `, then t also intersects `0. (Theorem 5.1.6) If ` and `0 are parallel lines and t is a transversal such that t ⊥ `, then t ⊥ `0. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Transitivity of Parallelism (Theorem 5.1.8) If ` is parallel to m and m is parallel to n, then either ` = n or ` k n. Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry. Proclus’s Axiom (Theorem 5.1.5) If ` and `0 are parallel lines and t 6= ` is a line such that t intersects `, then t also intersects `0. (Theorem 5.1.6) If ` and `0 are parallel lines and t is a transversal such that t ⊥ `, then t ⊥ `0. (Theorem 5.1.7) If `, m, n and k are lines such that k k `, m ⊥ k, and n ⊥ `, then either m = n or m k n. MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Euclidean geometry Since we are assuming the Euclidean Parallel Postulate, the following statements which we proved equivalent are now all theorems in Euclidean Geometry.