Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
ometry of N and M, not s line equals M. is unique. el to L through ed ASA ("angle shown in Fig- least one side y between L e point where etermined by But then an ence Land assumption that there is d M are in ollow from clid's par- it, so the , there is om," af- book in it does en need ate in- n such mply gle, 2.1 The parallel axiom To prove this property, draw a line L through one vertex of the trian- gle, parallel to the opposite side, as shown in Figure 2.3. L a a В Deduce from only for n = 3,4,6. Y 23 Y Figure 2.3: The angle sum of a triangle Then the angle on the left beneath L is alternate to the angle a in the triangle, so it is equal to a. Similarly, the angle on the right beneath Lis equal to y. But then the straight angle л beneath Lequals a+B+y, the angle sum of the triangle. Exercises The triangle is the most important polygon, because any polygon can be built from triangles. For example, the angle sum of any quadrilateral (polygon with four sides) can be worked out by cutting the quadrilateral into two triangles. Show that the angle sum of any quadrilateral is 27. A polygon P is called convex if the line segment between any two points in P lies entirely in P. For these polygons, it is also easy to find the angle sum. Explain why a convex n-gon can be cut into n - 2 triangles. Use the dissection of the n-gon into triangles to show that the angle sum of a convex n-gon is (n − 2). to find the angle at each vertex of a regular n-gon (an n-gon with equal sides and equal angles). that copies of a regular n-gon can tile the plane
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Transcribed Image Text:ometry of N and M, not s line equals M. is unique. el to L through ed ASA ("angle shown in Fig- least one side y between L e point where etermined by But then an ence Land assumption that there is d M are in ollow from clid's par- it, so the , there is om," af- book in it does en need ate in- n such mply gle, 2.1 The parallel axiom To prove this property, draw a line L through one vertex of the trian- gle, parallel to the opposite side, as shown in Figure 2.3. L a a В Deduce from only for n = 3,4,6. Y 23 Y Figure 2.3: The angle sum of a triangle Then the angle on the left beneath L is alternate to the angle a in the triangle, so it is equal to a. Similarly, the angle on the right beneath Lis equal to y. But then the straight angle л beneath Lequals a+B+y, the angle sum of the triangle. Exercises The triangle is the most important polygon, because any polygon can be built from triangles. For example, the angle sum of any quadrilateral (polygon with four sides) can be worked out by cutting the quadrilateral into two triangles. Show that the angle sum of any quadrilateral is 27. A polygon P is called convex if the line segment between any two points in P lies entirely in P. For these polygons, it is also easy to find the angle sum. Explain why a convex n-gon can be cut into n - 2 triangles. Use the dissection of the n-gon into triangles to show that the angle sum of a convex n-gon is (n − 2). to find the angle at each vertex of a regular n-gon (an n-gon with equal sides and equal angles). that copies of a regular n-gon can tile the plane
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