Angle Analysis of Multiple-Line Intersections

Overview

Objectives/Goals (1) To analyze the intersection of 'n' non-parallel lines locating on the same plane, and try to find: A) How many angles (less than 180 degree) that 'n' lines can form; B) What is the value of the smallest angle(s); C) How many right angles, acute angles, and obtuse angles that 'n' lines can be formed. (2) To extend the results that obtained in part (1) to the intersection of planes and the intersection of circles. Methods/Materials Begin with the simplest case: the intersection of 2 lines. Summarize the properties of the 2-line intersection including the number of intersecting points, the number of angles, the value of the smallest angle(s), and other findings. Then, gradually add more lines, one by one. Make the new line intersect with all previous lines respectively. Record the properties. When the number of lines is greater than 2, there are many possible ways for the lines to intersect. Now, rearrange the lines so that they intersect in as many ways as possible. Record the number of intersecting points and the number of angles that the lines form in each different case. Results When the number of lines 'n' is fixed: 1.the number of angles is always the same regardless of the number of intersecting points. 2.the value of the smallest angle(s) is also the same regardless of the number of intersecting points. 3.the number of acute angles is always the same as the number of obtuse angles. 4.the maximum number of intersecting points is the same as intersecting axes that formed by 'n' planes. Conclusions/Discussion I. Intersection of lines: 1)Find the maximum number of intersecting points when n lines intersect at different points, formula: n(n-1)/2 2)Find the number of angles when n lines intersect, formula: n(n-1) 3)The value of the smallest angle, formula:less or equal to 180 degree/n 4) The number of different types of angles When there are two lines perpendicular: Right angles: 4; Acute angles=Obtuse angles=n(n-1)-2 When no perpendicular lines: Right angles: 0; Acute angles=Obtuse angles=n(n-1) II.Intersection of Planes

Competition history

• CSEF 2015 Mathematics & Software · Entry S1420

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