Show that the angle sum of any quadrilateral is 27.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:ometry
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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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