Class 8th // Math // Chapter - Understanding of Quadrilaterals

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1. What is polygon ?
Ans :- Plane surface which is bounded by three or more than three line segment is called polygon. 
          (i) △      (ii) ☐       (iii) ⬠    (iv)  ⬡

Smallest polygon is triangle. 
          △

.2. What is a regular polygon ? State the name of a regular polygon of a three sides , four sides and six sides .
Ans:- A polygon is said to be a regular polygon , if all its interior angles are equal , sides are equal and exterior angles are equal.
    The name of a regular polygon of three sides is equilateral triangle , four side is square and six side is regular hexagon.

3. Find the angle measure 'x' in the following figure .

Question 4.

(a) Find x + y + z
(b) Find x + y + z + w.
Solution.

 Question 5.
Find the measure of each exterior angle of a regular polygon of
(i) 9 sides
(ii) 15 sides
Solution.
(i) Each exterior angle of a regular polygon of 9 sides

(ii) Each exterior angle of a regular polygon of 15 sides

 Question 6.
How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Solution.
Since the number of sides of a regular polygon

Question 7.
How many sides does a regular polygon have if each of its interior angles is 165°?
Solution.
Let there be n sides of the polygon. Then, its each interior angle

Thus, there are 24 sides of the polygon.

Question 8.
Consider the following parallelograms. Find the values of the unknowns x, y, z.

Solution.

Question 9.
The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

Solution.
Since HOPE is a parallelogram, therefore, HE || OP and HO || EP.
Now, HE || OP and transversal HO intersects them.

Hence, x = 110°, y = 40° and z = 30°

Question 10.
The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

Solution.
(i) Since GUNS is a parallelogram, therefore, its opposite sides are equal.

(ii) In a parallelogram, diagonals bisect each other, therefore,

Question 11.

In the below figure both RISK and CLUE are parallelograms. Find the value of x.
Solution.

Question 12.
Identify all the quadrilaterals that have.
(a) four sides of equal length
(b) four right angles
Solution.
(a) The quadrilaterals having four sides of equal length is either a square or a rhombus. ,
(b) The quadrilaterals having four right angles is either a square or a rectangle.

Question 13.
Explain how a square is
(i) a quadrilateral
(ii)a parallelogram
(iii) a rhombus
(iv) a rectangle.
Solution.
(i) A square is 4 sided, so it is a quadrilateral.
(ii) A square has its opposite sides parallel, so it is a parallelogram.
(iii) A square is a parallelogram with all the four sides equal, so it is a rhombus.
(iv) A square is a parallelogram with each angle a right angle, so it is a rectangle.

Question 14.
Name the quadrilaterals whose diagonals :
(i) bisect each other
(ii) are perpendicular bisectors of each other
(iii) are equal.
Solution.
(i) The quadrilaterals whose diagonals bisect each other can be a parallelogram or a rhombus or a square or a rectangle.
(ii) The quadrilaterals whose diagonals are perpendicular bisectors of each other can be a rhombus or a square.
(iii) The quadrilaterals whose diagonals are equal can be a square or a rectangle.

Question 15.
Explain why a rectangle is a convex quadrilateral.
Solution.
Since the measure of each angle is less than 180° and also both the diagonals of a rectangle he wholly in its interior, so a rectangle is a convex quadrilateral.