Question

The measure of the angles of a quadrilateral are in the ratio 2:3:2:3.What type of quadrilateral is it?


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Answer 1

Given:-

• Ratio of angles 2:3:2:3

To Find:-

• Measures of angles and which type of Quadrilateral is it .

Solution:-

Let the Ratio of angles be 2x, 3x ,2x,3x

As we know that sum of angles in Quadrilateral is 360°

Here ,

→ 2x+3x +2x+3x = 360°

→ 10x = 360°

→ x = 360°/10

→ x = 36°

∴ The value of x is 36°

Putting the value of x

→ 2x = 2×36° = 72°

→ 3x = 3×36° = 108°

→ 2x = 2×36° = 72°

→ 3x = 3×36° = 108°

Here, the opposite angles of Quadrilateral are equal.

∴ The given quadrilateral is Parallelogram .

Answer 2

Given :

• Ratio of the angles of the quadrilateral = 2 : 3 : 2 : 3

To Find :

The identity of the quadrilateral.

Solution :

To find the identity of the quadrilateral , first we have to find all the angles of the quadrilateral.

We know that the sum of angles of a quadrilateral sum up to 360°.

Let the angles be 2x , 3x , 2x and 3x.

Now , according to the information , we get the Equation as :

$\underline{\bf{2x + 3x + 2x + 3x = 360^{\circ}}}$

Now , by solving the above equation , we get :

$:\implies \bf{2x + 3x + 2x + 3x = 360^{\circ}}$

By adding we like terms in the LHS , we get :

$:\implies \bf{10x = 360^{\circ}}$

Now , dividing 10 on both the Sides , we get :

$:\implies \bf{\dfrac{10x}{10} = \dfrac{360^{\circ}}{10}}$

$:\implies \bf{x = \dfrac{360^{\circ}}{10}}$

$:\implies \bf{x = 36^{\circ}}$

$\boxed{\therefore \bf{x = 36^{\circ}}}$

Hence, the value of x is 36°.

Now , by substituting the value of x in the angles of the quadrilateral (in terms of x ), we get :

$:\implies \rm{\angle_{1} = 2x} \\ \\ :\implies \rm{\angle_{1} = 2 \times 36^{\circ}} \\ \\ :\implies \rm{\angle_{1} = 72^{\circ}} \\ \\ \boxed{\therefore \bf{\angle_{1} = 72^{\circ}}}$

Hence, the first angle is 72°.

$:\implies \rm{\angle_{2} = 2x} \\ \\ :\implies \rm{\angle_{2} = 3 \times 36^{\circ}} \\ \\ :\implies \rm{\angle_{2} = 108^{\circ}} \\ \\ \boxed{\therefore \bf{\angle_{2} = 108^{\circ}}}$

Hence, the second angle is 108°.

$:\implies \rm{\angle_{3} = 2x} \\ \\ :\implies \rm{\angle_{3} = 2 \times 36^{\circ}} \\ \\ :\implies \rm{\angle_{3} = 72^{\circ}} \\ \\ \boxed{\therefore \bf{\angle_{3} = 72^{\circ}}}$

Hence, the third angle is 72°.

$:\implies \rm{\angle_{4} = 2x} \\ \\ :\implies \rm{\angle_{4} = 3 \times 36^{\circ}} \\ \\ :\implies \rm{\angle_{4} = 108^{\circ}} \\ \\ \boxed{\therefore \bf{\angle_{4} = 108^{\circ}}}$

Hence, the fourth angle is 108°.

Thus , we get the angles of the quadrilateral as 72° , 108° , 72° and 108°.

Now , we know the property of a parallelogram that the opposite angles of a parallelogram are equal.

And according to the information also , we get that the angles are equal.

Hence, the identity of the quadrilateral is a parallelogram.