Question

Name the smallest angle of ABC. The diagram is not to scale.

Answer 1

The smallest angle would be angle C. The reason why is because angle C is opposite of the shortest side (AB). The smallest angle in a traingle is opposite of the shortest side.

Answer 2

The smallest angle of ABC is angle C.

Solution-

The sine law-

`(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)`

As,

`=8<9<10`

`=AB<AC<BC`

`=c<b<a`

If,
`(a)/(b)=(c)/(d),\ and\ b>d,\ then\ a>c`

If two fractions are equal in value and their denominator are decreasing, then the numerators must decrease in order to keep the value of the fraction same.

So, if
`(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)\ and\ c<b<a`, then

`\sin C<\sin B<\sin A`

As we know from the properties of trigonometry, if an angle is smaller than an another angle, then their sine will also smaller than the other.

i.e if
`A<B`, then
`\sin A<\sin B`

So applying the same,

if
`\sin C<\sin B<\sin A`, then
`C<B<A`

Therefore, the smallest angle of ABC is angle C.