Question

show that if three angles in a convex polygon are equal to \(60^{\circ}\) then it must be an equilateral triangle

Answer 1

i'll try other form

Answer 2

assume x is 60

Answer 3

(if it works for quadrilateral then it works for any polygon ) geometric logic
so we need to show by contradiction that we cant construct a quadrilateral with 3 60 angle

Answer 4

yes that looks legit

Answer 5

that would make the other angle 180 seems more like not concave nor convex

Answer 6

i feel sleepy i might write it more need in morning,gn

Answer 7

Sum of all the exterior angles of a polygon is 360 degrees. The exterior angle corresponding to 60 degrees is 120 degrees. Since three angles are 60 degrees, the sum of those exterior angles is 120*3 = 360 degrees. Thus there can be no other non-zero exterior angle corresponding to a vertex. So, these are the only 3 vertices, and the polygon is an equilateral triangle.

Answer 8

more neat**

Answer 9

Nice :) More generally :

Any convex polygon with \(n\) angles equal to \(\dfrac{360}{n}\) must be a regular \(\text{n-gon}\)

Answer 10

hehehe ok that was direct

Answer 11

*

Any convex polygon with \(n\) angles equal to \(180-\dfrac{360}{n}\) must be a regular \(\text{n-gon}\)

Answer 12

for example, in a square we have \(4\) angles equal to \(180-\frac{360}{4} = 90\)