**Understanding Quadrilaterals**

**Question 1.****Given here are some figures :**

Classify each of them on the basis of the following :

**(a)**Simple curve

**(b)**Simple closed curve

**(c)**Polygon

**(d)**Convex polygon

**(e)**Concave polygon

**Solution.**

The classification of the given figures is as under :**(a)** Simple curve : (1), (2), (5), (6), (7) and (8)**(b)** Simple closed curve : (1), (2), (5), (6) and (7)**(c)** Polygon: (1) and (2)**(d)** Convex polygon : (2)**(e)** Concave polygon : (1) and (4)

**Question 2.****How many diagonals does each of the following have?(a)** A convex quadrilateral

**(b)**A regular hexagon

**(c)**A triangle.

**Solution.****(a)** A convex quadrilateral has two diagonals.**(b)** A regular hexagon has nine diagonals.**(c)** A triangle has no diagonal.

**Question 3.****What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and j try!)**

**Solution.**

The sum of measures of the angles of a convex quadrilateral is 360°. Yes, this property holds in case of the quadrilateral is not convex.

**Question 4.****Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)**

What can you say about the angle sum of a convex polygon with number of sides?

**(a)**7

**(b)**8

**(c)**10

**(d)**n

**Solution.**

From the given table, dearly we observe that the sum of angles (interior angles) of a polygon with n sides = (n – 2) x 180°.**(a)** n = 7

∴ The sum of the angles of a polygon of 7 sides

Angle sum =

=

**(b)** n = 8

∴ The sum of the angles of a polygon of 8 sides

Angle sum =

=

**(c)** n = 10

∴ The sum of the angles of a polygon of 10 sides

Angle sum =

=

**(d)** Clearly from the given table it is observed that the number of triangles is two less them the number of sides in the polygon.

∴ If the polygon has n sides, the number of triangles formed will be (n – 2).

Since the sum of angles of a triangle = 180°

∴ The sum of angles of a polygon of n sides = (n – 2) x 180°.

**Question 5.****What is a regular polygon? State the name of a regular polygon of(i)** 3 slides

**(ii)**4 slides

**(iii)**6 slides

**Solution.****A polygon is said to be a regular polygon, if all its**

- interior angles are equal;
- sides are equal; and
- exterior angles are euqal.

**The name of a regular polygon of**

- 3 sides is equilateral triangle.
- 4 sides is square.
- 6 sides is regular hexagon.

**Question 6.****Find the angle measure x in the following figures.**

**Solution.**

**Question 7.****(a)** Find x + y + z**(b)** Find x + y + z + w.

**Solution.**