Question

one angle of a triangle is 60. the other two angles are in ratio 6:4 . find the other two angles??​

Answer 1

Given:

One angle of a triangle is 60° and the other two angles are in the ratio of 6:4.

To find:

• The angles of triangle.

Solution:

❍ Let the two angles of the triangle be 6x and 4x

We know that,

$❍⠀⠀ { \pmb{Sum \: of \: the \: angles \: of \: the \: triangle \: = \: 180° \: [Angle \: sum \: property]}}$

$\dashrightarrow\sf \: \: \: \: 6x + 4x + 60^\circ = 180^\circ \\\\\dashrightarrow\sf \: \: \: \: \: \: \: \: \: \: 10x + 60^\circ = 180^\circ \\\\\dashrightarrow\sf \: \: \: \: \: \: \: \: \: 10x = 180^\circ - 60^\circ \\\\\dashrightarrow\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 10x = 120^\circ \\\\\dashrightarrow \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf x = \sf\cancel\dfrac{120}{10} \\\\\dashrightarrow \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \pmb x = \underline{\boxed{\pmb{\pink{12 {}^{ \circ} }}}}$

Hence,

• First angle = 6x = 6 × 12 = 72°

• Second angle = 4x = 4 × 12 = 48°

• Given, Third angle = 60°

⠀⠀⠀

${ \therefore{ \pmb{ The \: three \: angles \: of \: the \: given \: triangle \: are \: 72°, 48° and \: 60° respectively.}}}$

⠀

$\rule{300px}{.3ex}$

⠀⠀⠀

$\large \: ❍ \: \: { \underline{ \underline{ \pmb{ V E R I F I C A T I O N :}}}}$

$\: \: \: \: \: \: \: \: \: \:\implies\sf \Big(First\;angle\Big) +\Big(Second\;angle\Big)+\Big(Third\;angle\Big) = Sum \: of \: the \: angles \: of \: the \: triangle\\\\\\\implies\sf 48^\circ + 72^\circ + 60^\circ = 180^\circ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\\\\\\implies\sf 180^\circ = 180^\circ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$$\: \: \: \: \: \: \: \: \: \: {\implies\underline{\boxed{\pmb{{L.H.S = R.H.S}}}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$⠀⠀

⠀⠀⠀

⠀

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \underline{ \underline{ \pmb{{Hence\; Verified \: !}}}}}$

Answer 2

★ The two other angles of the triangle are 72° and 48°.

Step-by-step explanation

Analysis -

⠀⠀⠀In the question, it has been stated that the value of one angle of triangle is 60° and the two other angles are in the ratio 6 : 4. We've been asked to find the values of the second and the third angle of the triangle.

Solution -

⠀⠀⠀Since the values of second and third angle of the triangle is given in the form of ratio, let's consider them as:

• Second angle = 6k
• Third angle = 4k

Here, we have to use the angle sum property of a triangle to find the value of k.

According to a triangle, the sum of its three interior angles equals 180°, i.e.,

$\twoheadrightarrow{ \sf{ \quad \angle1 + \angle2 + \angle3 = 180 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 60\degree+ 6k + 4k = 180}}$

$\twoheadrightarrow{ \sf{ \quad 6k + 4k = 180 \degree - 60 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 6k + 4k = 120 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 10k = 120 \degree}}$

$\twoheadrightarrow{ \sf{ \quad k = \bigg(\frac{12 \cancel0}{1 \cancel0} \bigg) }}$

$\twoheadrightarrow{ \quad \boxed{ \frak{ \pmb{ k = 12}}}}$

As we have obtained the value of k, let's plug in its value in the expressions formed for second and third angle.

• Second angle = 6k = 6(12) = $\frak\red{\pmb{72 \degree}}$
• Third angle = 4k = 4(12) = $\frak\red{\pmb{48 \degree}}$

Verification -

$\twoheadrightarrow{ \sf{ \quad \angle1 + \angle2 + \angle3 = 180 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 60 \degree + 72 \degree + 48 \degree = 180 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 132 \degree + 48 \degree = 180 \degree}}$

$\twoheadrightarrow{ \sf{ \quad 180 \degree = 180 \degree}}$

The value of L.H.S. is equal to the value of R.H.S. Hence, the answer obtained is correct.

Additional information -

The properties of a triangle are listed below:

• The polygon with the least number of sides is a triangle.
• Sum of any two sides of a triangle is greater than its third side.
• Difference of any two sides of a triangle is less than the third side.
• The side opposite to the largest angle is the largest and vice versa.
• The side opposite to the smallest angle is the smallest ans vice versa.
• One exterior angle of a triangle is equal to the sum of two opposite interior angles.
• Sum of all exterior angles of a triangle is 360°.

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