1. Prove that given a line é and a point P not on e, there exists a linethrough P perpendicular to €.2. If in AABC we have ZB LC, then AB AC and AABC isisosceles.3. State and prove the Alternate Interior Angle Theorem.4. State and prove the Exterior Angle Theorem.5. Prove that in a triangle, the greater angle lies opposite the greaterside and the greater side lies opposite the greater angle.6. Prove that the angle sum of every triangle is 180° if we assumeHilbert's Euclidean parallel postulate.7. How many points and lines are there in the Fano's Geometry? Justifyyour answer.8. Draw a representation of the four-line geometry. Name all the pointsand lines.9. Draw a representation of the four-point geometry. Name all thepoints and lines.10. Draw a representation of the Young's Finite Geometry. Name all thelines and determine which lines are parallel to cach other.

Explanation of the answer is as follows

Answered: 1. Prove that given a line € and a…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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20. Consider lines ₁ and ₂. ₁: [x, y] = [1, 4] + [-7, 2] l₂: [x, y] = [-2, 6] + [k, 9] a) For what value of k are the lines parallel? b) For what value of k are the lines perpendicular? mine if each yvector equation € 3.
1. Every morming Neo packs his backpack and walks a distance L through a straight path in the forest from his home (A) to the unicorn sanctuary (B). One day he discovers someone has built an electric fence in the exact middle of his daily path (the dashed, red line in the picture): A• B The fence has length 2b - it extends for a distance b on each side of the path - and is perpendicular to the path. After a few days, Neo notices that the electric fence is turned on only half the time, but he does not know if the fence is on or off on any given day until he walks up to it and throws his cat at the fence to test it. If the fence is off, he can just quickly climb over it. Otherwise, he has to walk around it. He devises a plan: he will walk straight from his home to some point P in the fence; then, he will walk around it or climb over it depending on whether the fence is on or off. Which point P should he choose in order to minimize the average length of his trip?
5. Determine whether the following pair of lines is parallel, intersects at a single point, or is skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection: r(t) = (0, 1, 2) + t(1, 1, 1), R(s) = (-2, 1,0) + s(-1, -3, -1).
Given the lines L1 and L2: L1: (x,Y.z)=(-4,6,-2)+t(5,4,3), with tER. www L2: (x,V.z)=(7,16,6)+s(6,6,5), with sER. Consider the following statements about L1 and L2: I. The point P=(1,10,1) is a point of intersection of both lines. II. There does not exist a plane containing both straight lines. Of these, with certainty, which are true? a) None b) I AND II c) I d) II
is this true or false......When lines m and n intersect at point P , it follows that m ∩ n = P
QUESTION 8 Let m and n be two distinct lines. Suppose that Rn Rm (P)=P for some point P. Prove that Rn (P)=Rm (P). Attach File
28. Determine the point of intersection between the following pair of lines ở = (5,1,7) +s(2,0,5), s€R r = (-1,-1,3)+t(4,2,−1),t€R
9. If a projectivity from a point to itself fixes two lines through the point, then the projectivity sends every line in the pencil to itself.
7. On the plane shown, points A, B, C, and D lie on the same line. Select the set of words that correctly complete the sentence. A. greater than; greater than B. less than; less than B C. greater than; less than D D. equal to; equal to The slope of AB is horizontal change between points A and B is the horizontal change between points C and D. the slope of CD because the ratio of the vertical change to the the ratio of the vertical change to
2. Let l be the line with equation x = 2y = z +3 and let P be the point (1, 1, 1). Consider a line through P which intersects l in a right angle. Find the coordinates of the point of intersection, showing working.
Prove that the perpendicular line from a given point to a given line is unique. (Hint: There are twocases to consider: The point is not on the line and the point is on the line.)

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1. Prove that given a line € and a point P not on e, there exists a line through P perpendicular to e.