Question

the angels of the triangle are in A.P. if the smallest angel is 36°,then the measure of the other angels are​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

The angles of the triangle are in A.P. If the smallest angle is 36°, then the measure of the other angless are ___

$\large\underline{\sf{Solution-}}$

• Since, angles of a triangle are in A. P.

So,

$\begin{gathered}\begin{gathered}\bf \:Let \: the \: angles \: be - \begin{cases} &\sf{(a - d) \degree \: } \\ &\sf{a\degree \:} \\ &\sf{(a + d)\degree \:} \end{cases}\end{gathered}\end{gathered}$

We know,

• Sum of angles of a triangle is 180°.

Therefore,

$\rm :\longmapsto\:a - \cancel d \: + \: a \: + a - \cancel d = 180$

$\rm :\longmapsto\:3a = 180$

$\bf\implies \:a \: = \: 60\degree \: - - (1)$

Now,

• Smallest angle of a triangle is 36°.

$\rm :\implies\:a - d = 36$

$\rm :\longmapsto\:60 - d = 36 \: \: \: \: \: \: \: \: \{using \: (1) \: \}$

$\rm :\longmapsto\:d = 60 - 36$

$\bf\implies \:d \: = \: 24\degree \:$

$\begin{gathered}\begin{gathered}\bf \:Hence, \: the \: angles \: are - \begin{cases} &\sf{a - d = 60 - 24 = 36 \degree \: } \\ &\sf{a = 60\degree \:} \\ &\sf{a + d= 60 + 24 = 84\degree \:} \end{cases}\end{gathered}\end{gathered}$

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\bigstar\:\:{\underline{{\boxed{\bf{{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

↝Sₙ (Sum of first n terms) of an arithmetic sequence is,

$\begin{gathered}\bigstar\:\:{\underline{{\boxed{\bf{{S_n\:= \: \dfrac{n}{2} \: (2\: a\:+\:(n\:-\:1)\:d)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.