Question

the acute angels of a right triangle are in the ratio 1:2. find the angels of a triangle​

Answer 1

$\huge \mapsto \: \: \huge \underbrace {\textrm{{{\color{blue}{Given}}}}}$

Acute angels of a right triangle are in the ratio 1:2.

$\:$

We know that , in right triangle one angle is always 90°.

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$\huge \mapsto \: \: \huge \underbrace{\textrm{{{\color{blue}{To \: Find}}}}}$

We have to find the angles of a triangle.

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$\huge\begin{gathered} {\underline{\boxed{ \rm {\red{Using \: \: Theorem}}}}}\end{gathered}$

$\large \star \underline {\sf {{{\color{orange}{Angle \: \: sum \: \: property \: \: of \: \: a \: \: triangle...}}}}} \star$

The sum of the angles of a triangle is always 180°.

$\bf \Large \hookrightarrow \: \angle \: 1 \: + \: \angle \: 2 \: + \: \angle \: 3 \: = \: 180 \degree$

⇨ Let the angles of a triangle be x.

$\bf \Large \hookrightarrow \: \angle \: 1 \: = 1x \\ \\ \bf \Large \hookrightarrow \: \angle \: 2 \: = \: 2x \\ \\ \bf \Large \hookrightarrow \: \angle \: 3 \: = \: 90 \degree$

$\huge \mapsto \: \: \huge \underbrace{\textrm{{{\color{blue}{Solution}}}}}$

$\bf \Large \rightarrow \: \angle \: 1 \: + \: \angle \: 2 \: + \: \angle \: 3 \: = \: 180 \degree$

$\bf \Large \rightarrow \: 1x \: + \: 2x + \: 90 \degree\: = \: 180 \degree$

$\bf \Large \rightarrow \: 3x \: + \: 90 \degree\: = \: 180 \degree$

$\bf \Large \rightarrow \: 3x \: \: \: = \: 180 \degree \: - \: 90 \degree$

$\bf \Large \rightarrow \: 3x \: \: \: = \: \: 90 \degree$

$\bf \Large \rightarrow \: x \: \: \: = \: \cancel\frac{90}{3} \: = \: 30 \degree$

$\bf \Large \rightarrow \: x \: \: \: = \: 30 \degree$

$\large \dag \: \: \underline {\bf {{{\color{indigo}{Substuting \: \: the \: \: values...}}}}} \: \: \dag$

$\bf \Large \implies \: \angle \: 1 \: = \: 1x \\ \\ \bf \Large \implies \: 1x \: = \: 1 \: \times \: 30 \degree \\ \\ \bf \Large \implies \: \angle \: 1 \: = \: 30 \degree$

$\bf \Large \implies \: \angle \: 2 \: = \: 2x \\ \\ \bf \Large \implies \: 2x \: = \: 2 \: \times \: 30 \degree \\ \\ \bf \Large \implies \: \angle \: 2 \: = \: 60 \degree$

$\bf \Large \implies \: \angle \: 3 \: = \: 90 \degree$

$\huge \mapsto \: \: \huge \underbrace{\textrm{{{\color{green}{Verification}}}}}$

$\bf \Large \rightarrow \: \angle \: 1 \: + \: \angle \: 2 \: + \: \angle \: 3 \: = \: 180 \degree$

$\bf \Large \rightarrow \: 30 \degree \: + \: 60 \degree \: + \: 90 \degree \: = \: 180 \degree$

$\bf \Large \rightarrow \: 90 \degree\: + \: 90 \degree \: = \: 180 \degree$

$\bf \Large \rightarrow \: 180 \degree \: = \: 180 \degree$

$\: \: \huge \underbrace{\textrm{{{\color{green}{LHS = RHS}}}}}$

☛ More Information : -

• A triangle cannot have more than one right angle.

$\:$

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

$\:$

• An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Answer 2

Answer

• Angles of the right angled triangle are 30°, 60° and 90°.

Given

• The acute angels of a right angled triangle are in the ratio 1 : 2.

To Do

• To find all the angles of the triangle.

Step By Step Explanation

Assumption :

Let us consider the two acute angles of the right angled triangle be x and 2x and the third angle will be 90° [ right angled triangle ].

By Angle Sum Property :

Angle sum property of triangle states that the sum of all the angles of a triangle = 180°.

Equation :

According to the given equation the equation will be ⤵

$\bigstar \: \: {\underline{ \boxed{ \bold{ \red{x + 2x + 90 = 180}}}}}$

Solution of the equation :

Let us solve the above equation to find the angels.

$\longmapsto \sf \: x + 2x + 90 = 180 \\ \\ \longmapsto \sf 3x + 90 = 180 \\ \\ \longmapsto \sf 3x = 180 - 90 \\ \\ \longmapsto \sf 3x = 90 \\ \\ \longmapsto \sf x = \cancel \cfrac{90}{3} \\ \\ \longmapsto {\underline{ \boxed{ \bold{ \green{x = 30}}}}}\: \: \bigstar$

Therefore, the all the angles of triangle are x = 30°, 2x = 60° and 90°.

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