Question

Suppose you wish to apply SSA to a triangle, in order to find an angle measure. Also suppose the given side of a triangle, opposite to a given angle, is greater than the given side. Which of the following statements is true?

Answer 1

Consider a triangle ΔABC, where sides AB, BC and angle ∡C are given.

Angle ∡C is the angle opposite of side AB.

It says that the given side opposite the given angle is less than the other given side. It means AB < BC.

It says that the ratio of the longer side to the shorter side, multiplied by the sine of the angle opposite the shorter side, is less than 1. It means

`(BC)/(AB)` · sin(C) < 1

We know about Law of Sines of Triangle is given by :-

`(AB)/(sin(C)) =(BC)/(sin(A)) =(AC)/(sin(B))`

`< br/ > Solving`
`(AB)/(sin(C)) =(BC)/(sin(A))`

`< br/ > Cross`
`Multiplying`

`< br/ > AB`
`sin(A)=BC`
`sin(C)`

`< br/ > sin(A)=(BC)/(AB)`
`sin(C)`

`< br/ > sin(A) < 1` \\

We got sin(A) < 1, and BC > AB.

Therefore, angle ∡A could be either acute or obtuse angle.

So, There will be two solutions for the angle ∡A.

⇒ option D is the final answer.