# A triangle is both isosceles and acute. If one angle of the triangle measures 36 degrees, what is the measure of the largest angle(s) of the triangle? What is the measure of the smallest angle(s) of the triangle?

Jan 31, 2016

The answer to this question is easy but requires some mathematical general knowledge and common sense.

Isosceles Triangle:-
A triangle whose only two sides are equal is called an isosceles triangle. An isosceles triangle also has two equal angels.

Acute Triangle:-
A triangle whose all angels are greater than ${0}^{\circ}$ and less than ${90}^{\circ}$, i.e, all angels are acute is called an acute triangle.

Given triangle has an angle of ${36}^{\circ}$ and is both isosceles and acute.

$\implies$ that this triangle has two equal angels.

Now there are two possibilities for the angels.

$\left(i\right)$ Either the known angel ${36}^{\circ}$ be equal and the third angel is unequal.

$\left(i i\right)$ Or the two unknown angels are equal and the known angel is unequal.

Only one of the two above possibilities will be correct for this question.

Let's verify the two possibilities one by one.

$\left(i\right)$

Let the two equal angels be of ${36}^{\circ}$ and the third angle be ${x}^{\circ}$

We know that the sum of all the three angels of a triangle is equal to ${180}^{\circ}$, i.e,

${36}^{\circ} + {36}^{\circ} + {x}^{\circ} = {180}^{\circ}$
$\implies {x}^{\circ} = {180}^{\circ} - {72}^{\circ}$
$\implies {x}^{\circ} = {108}^{\circ} > {90}^{\circ}$

In possibility $\left(i\right)$ the unknown angel comes to be ${108}^{\circ}$ which is greater than ${90}^{\circ}$ so the triangle becomes obtuse and hence this possibility is wrong.

$\left(i i\right)$

Let the two equal angels be of ${x}^{\circ}$ and the third angle be ${36}^{\circ}$. Then

${x}^{\circ} + {x}^{\circ} + {36}^{\circ} = {180}^{\circ}$

$\implies 2 {x}^{\circ} = {144}^{\circ}$

$\implies {x}^{\circ} = {72}^{\circ}$.

In this possibility the measures of the angels are ${36}^{\circ} , {72}^{\circ} , {72}^{\circ}$.

All the three angels are in the range of ${0}^{\circ}$ to ${90}^{\circ}$, therefore, the triangle is acute. and the two equal angels so the triangle is also isosceles. The two given conditions are verified therefore the possibility $\left(i i\right)$ is correct.

Hence, the measures of largest and smallest angels are ${36}^{\circ}$ and ${72}^{\circ}$ respectively.