Question

One angle of a triangle is equal to the sum of the other two. If the ratio of the angles is 7:8, find the angles of the triangle.​

Answer 1

Given: The ratio of the angles is 7:8 & One angle of a triangle is equal to the sum of the other two.

Need to find: The angles of the triangle?

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¤ Let the "one angle" be ∠C & let the other two be ∠A & ∠B. Thus – ∠C = ∠A + ∠B.

« ∠C = 7x + 8x = 15x

• We will use –

$\qquad\:\star\;\underline{\boxed{\pmb{\sf{Sum\;of\;angles\; = \bigg \lgroup 180^\circ\bigg\rgroup }}}}$

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$⠀⠀⠀\begin{gathered}\underline{\bf{\dag} \:\mathfrak{Substituting\;values\;in\; formula\: :}}\\\\\end{gathered}$

$\begin{gathered}:\implies\sf 15x + 7x + 8x = 180\\\\\\:\implies\sf 30x = 180\\\\\\:\implies\sf x = \cancel{\dfrac{180}{30}} \\\\\\:\implies\underline{\boxed{\pmb{\frak{ \red{x =6}}}}}\;\bigstar\\\\\end{gathered}$

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Therefore,

• ∠A = 7x = 6(7) = 42°
• ∠B = 8x = 6(8) = 48°
• ∠C = 15x = 6(15) = 90°

Note :- They will add up to 180°

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${\therefore{\underline{ \sf{Hence, the \: angles \: of \: the \: triangle \: are \: \pmb{ 42^{\circ}, 48^{\circ}, 90^{\circ}} respectively.}}}}$

Answer 2

Given :-

• One angle of a traingle is equal to the sum of other 2 angles, Ratio of other 2 angles is 7:8

To find :-

• All angles of the traingles

Solution :-

Ratio of the 2 angles ↦7:8

let these angles be 7x and 8x

Then, the third angle ↦7x + 8x ( given )

∴ By angle sum property,

⇥7x + 8x + ( 7x + 8x ) = 180°

⇥30x = 180°

⇥x = 6

∴ The angles of traingle are :

7x = 7 x 6 = 42° ( 1st angle )

8x = 8 x 6 = 48° ( second angle )

and ( 7x + 8x ) = 42° + 48° = 90° ( third angle )

Hence, The angles of traingle are 42°, 48° and 90°