Question

Fill in the blanks:
$\left( 1 \right)$ A quadrilateral with one pair of opposite sides are parallel is called a $ \ldots $
$\left( 2 \right)$ The diagonals of a rhombus bisect each and other at $ \ldots $
$\left( 3 \right)$ When any two triangles are said to be congruent?
$\left( 4 \right)$ The angles of the triangle are ${\left( {x - 35} \right)^ \circ }{\left( {x - 20} \right)^ \circ }and{\left( {x - 40} \right)^ \circ }$ find the three angles.
$\left( 5 \right)$ Which is smaller: the perimeter of a square or the perimeter of a circle inscribed in it?

Answer 1

Hint:
We know that the square, rectangle, and rhombus have two pairs of opposite sides and in rhombus the adjacent sides are equal. Since we know for the congruence, the angles are equal. With the help of all of this, we can answer the question.

Complete step by step solution:
$\left( 1 \right)$ As we know that in a trapezium the only one pair of opposite sides are parallel and they are a little bit similar to a parallelogram.
Therefore the answer will be trapezium.
$\left( 2 \right)$ Since in a rhombus, the adjacent sides are equal and also the diagonals bisect each other so by using the SSS congruence rule we have the angle which will be equal to the ${90^ \circ }$ .
Hence, ${90^ \circ }$ the diagonals of a rhombus bisect each other.
$\left( 3 \right)$ So for the triangles to be congruent, their corresponding sides and the angles should have to be equal.
$\left( 4 \right)$ The angles of the triangle are ${\left( {x - 35} \right)^ \circ }{\left( {x - 20} \right)^ \circ }and{\left( {x - 40} \right)^ \circ }$ so we have to find the three angles.
As we know that the sum of angles of a triangle is ${180^ \circ }$, therefore on equating and solving it we get
$ \Rightarrow {\left( {x - 35} \right)^ \circ } + {\left( {x - 20} \right)^ \circ } + {\left( {x - 40} \right)^ \circ } = {180^ \circ }$
On adding the above equation, we get
$ \Rightarrow 3x - {95^ \circ } = {180^ \circ }$
And therefore on solving for the value of $x$ , we get
$ \Rightarrow x = {65^ \circ }$
Now on substituting these values in ${\left( {x - 35} \right)^ \circ }{\left( {x - 20} \right)^ \circ }and{\left( {x - 40} \right)^ \circ }$ , we get
The angles are ${30^ \circ },{45^ \circ },{105^ \circ }$.
$\left( 5 \right)$ From the question, we can say that when the circle is inscribed in a square then the diameter of the circle will be equal to the length of the square. So from this, we can say that the perimeter of the square will be smaller.

Note:
For solving this type of theoretical question, we must know that thing then only we can answer it. In the last question we can also know the answer by putting some values of the same length and then calculate the perimeter we will get that the circle has a higher perimeter than the square. So in this way, we can answer such types of questions.