Answer Happy

Need Help with 8.2, 8.4, 8.8, 8.9 and 8.10 pls
← 9:21 Foundations of Euclidean Geom... Definition 40 If ZX and Y are alternate interior angles, and ZY and ZZ form a vertical pair, then ZX and ZZ are corresponding angles. Theorem 34 Given two lines and a transversal. If a pair of alternate interior angles are congruent, then the lines are parallel. Theorem 35 Given two lines and a transversal. If a pair of corresponding angles are congruent, then the lines are parallel. called alternate interior angles. Note that we have only used the fact that the real number system is an ordered field. The following property of real numbers may be intuitively clear, however, it must be postulated since there do exist ordered fields which are not Archimedean. The Archimedean Postulate EXERCISE 8 Theorem 36 (Saccheri-Legendre Theorem) Exercise 8.3 Exercise 8.4 Let M and e be any two positive numbers. Then there is a positive integer n such that ne > M. Exercise 8.1 Prove Theorem 32. Exercise 8.5 Exercise 8.6 Exercise 8.2 Prove Theorem 33. The sum of the measure of the three angles in any triangle is less than or equal to 180. Exercise 8.8 Exercise 8.9 Prove Theorem 34. Prove Theorem 35 (Hint: First, prove that if a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent. Then, use Theorem 34.) Prove that the sum of the measures of any two angles of a triangle is less than 180. Given AABC, let D be the midpoint of BC and let E be a point on the ray AD such that A-D-E and ADDE. Prove that AAEC has the same angle sum as AABC and either mzEAC or mZAEC is less then or equal to mzBAC. Exercise 8.7 Given AABC, how would you construct a triangle which has the same angle sum as AABC, but with one angle having measure at most m/BAC? 19 Prove Theorem 36. Prove the following corollary of Theorem 36 (Exterior Angle Theorem II). The sum of the measures of two angles in a triangle is less than or equal to the measure of their remote exterior angle. Exercise 8.10 Prove the following statement. Let PQ 1 I, as in the diagram below. Then given any positive real number a, there is a point R on I such that mZQRP < a. In order to prove the above statement, you need to justify each of the following steps. You may need results in Part 8, the Archimedean Postulate and the Exterior Angle Theorem II in Exercise 8.9. (a) There exists a point R, on the line I such that PQ QR₁. (b) mzQR P ≤ 45. (c) There exists a point R₂ on the line I such that Q - R₁ - R₂ and PR₁ = R₁ R₂. (d) mzQR₂P ≤45. (e) Continuing this way, there exists R₁ on the line I such that Q - Rn-1-Rn and mzQR, P <a.
9:21 ← Foundations of Euclidean Geom... PART 8 PARALLEL LINES Definition 37 Two lines I and m are parallel if and only if they lie in the same plane and I n m = Ø. In this case, we will write || m. Theorem 32 If two lines lie in the same plane, and are perpendicular to the same line, then they are parallel. Theorem 33 Given a line and a point not on the line, there is always at least one line which passes through the given point and is parallel to the given line. Definition 38 If 1₁, 12, and t are three lines in the same plane, and t intersects 1₁ and 1₂ in two distinct points P and Q, respectively, then t is a transversal to 1₁ and 1₂. Definition 39 Let t is a transversal to 1₁ and 12, intersecting 1₁ and 1₂ at P and Q, respectively. Let A and D are points on 1₁ and 12, respectively, lying on opposite sides of t. Then ZAPQ and ZPQD are called alternate interior angles. Definition 40 If ZX and Y are alternate interior angles, and <Y and ZZ form a vertical pair, then ZX and ZZ are corresponding angles. Theorem 34 Given two lines and a transversal. If a pair of alternate interior angles are congruent, then the lines are parallel. Theorem 35 Given two lines and a transversal. If a pair of corresponding angles are congruent, then the lines are parallel. Note that we have only used the fact that the real number system is an ordered field. The following property of real numbers may be intuitively clear, however, it must be postulated since there do exist ordered fields which are not Archimedean. The Archimedean Postulate EXERCISE 8 Let M and e be any two positive numbers. Then there is a positive integer n such that ne > M. Theorem 36 (Saccheri-Legendre Theorem) The sum of the measure of the three angles in any triangle is less than or equal to 180. Exercise 8.1 Prove Theorem 32. Exercise 8.2 Prove Theorem 33. 18 Exercise 8.3 Prove Theorem 34. Buonging & 6 Cien AARC Exercise 8.4 Prove Theorem 35 (Hint: First, prove that if a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent. Then, use Theorem 34.) Exercise 8.5 Prove that the sum of the measures of any two angles of a triangle is less than 180. ho midnoint fRC lot F AD qugh that 19