prove corollary 4.3

[kiramarie]

prove a triangle is equilateral if and only if it is equiangular.
reasons
Definition of equilateral triangles
Isosceles triangle theorem
Definition of equiangular
MAY NOT BE IN CORRECT ORDER!!

 

[Aladdin]

.This can be proven by the Law of Sines, which states that sin A / a = sin B / b = sin C / c, where A, B and C are angles of a triangle and a, b and c are the opposing sides of their corresponding angles. If A = B = C, then sin A = sin B = sin C. Therefore for the equation to work out, a = b = c. Therefore the eqiangular triangle is equilateral

 

[Deleted member 4993]

prove a triangle is equilateral if and only if it is equiangular.
reasons
Definition of equilateral triangles
Isosceles triangle theorem
Definition of equiangular
MAY NOT BE IN CORRECT ORDER!!

This theorem is proven long before laws of Sines are introduced.

This is corrollary of "A triangle with two equal angle is isosceles" (the given reasons tell you so). "If and only if" needs to be proven in two directions. Start with an equiangular triangle - prove that it is equilateral. Then start with an equilateral triangle - prove that it is equiangular.

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[fasteddie65]

You have stated: prove a triangle is equilateral if and only if it is equiangular.
reasons
Definition of equilateral triangles
Isosceles triangle theorem
Definition of equiangular
MAY NOT BE IN CORRECT ORDER!!

--> If a triangle is equilateral, then it is equiangular.
Given: AB = BC = AC
Since AB = BC, then m<C = m<A. (Base angles of isosceles triangle are congruent.)
Similarly, since AC = BC, m<B = m<A, so m<A = m<B = m<C.
So the triangle is equiangular.

<-- If a triangle is equiangular, then it is equilateral.
Given: m<A = m<B = m<C.
Since m<A = m<B, then BC = AC. (If two angles of a triangle are congruent, then the opposite sides are congruent.)
Similarly since m<C = m<B, AB = AC, so AB = BC = AC.
So the triangle is equilateral.