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

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1. [tex]\large\underline{\bold{Given \:Question – }}[/tex]

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

   [tex]\large\underline{\sf{Solution-}}[/tex]

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

   So,

   [tex]\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}[/tex]

   We know,

   • Sum of angles of a triangle is 180°.

   Therefore,

   [tex]\rm :\longmapsto\:a – \cancel d \: + \: a \: + a – \cancel d = 180[/tex]

   [tex]\rm :\longmapsto\:3a = 180[/tex]

   [tex]\bf\implies \:a \: = \: 60\degree \: – – (1)[/tex]

   Now,

   • Smallest angle of a triangle is 36°.

   [tex]\rm :\implies\:a – d = 36[/tex]

   [tex]\rm :\longmapsto\:60 – d = 36 \: \: \: \: \: \: \: \: \{using \: (1) \: \}[/tex]

   [tex]\rm :\longmapsto\:d = 60 – 36[/tex]

   [tex]\bf\implies \:d \: = \: 24\degree \:[/tex]

   [tex]\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}[/tex]

   Additional Information :-

   ↝ nᵗʰ term of an arithmetic sequence is,

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

   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,

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

   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.

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