Understanding Quadrilaterals - 8th math

Understanding Quadrilaterals NCERT Exercise 3.1 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

(a) Simple curve: figure 1, 2, 5, 6 and 7

(b) Simple closed curve: figure 1, 2, 5, 6 and 7

(c) Polygon: figure 1 and 2

(d) Convex polygon: Figure 2

(e) Concave polygon: figure 1

Understanding Quadrilaterals NCERT Exercise 3.1 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

A convex quadrilateral has two diagonals.

(b) A regular hexagon

A regular hexagon has total 9 (nine) diagonals.

(c) A triangle

A triangle does not have any diagonals. That is a triangle has 0 (zero) diagonal.

Understanding Quadrilaterals NCERT Exercise 3.1 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 try!)

Solution

The sum of the measures of the angles of a convex quadrilateral is equal to 360⁰.

A quadrilateral which is not convex also holds the same property. This means the sum of the measures of the angles of a concave quadrilateral is equal to 360⁰ also.

For concave quadrilateral

A convex quadrilateral also has number of sides equal = 4

And we know that the sum of angles of a polygon

Where n = number of sides

Thus, in a quadrilateral which has 4 sides, the sum of angles And we know that the sum of angles of a polygon

= 2 × 180⁰

= 360⁰

Thus, sum of angles of a quadrilateral whether it is a convex or concave = 360⁰

Understanding Quadrilaterals NCERT Exercise 3.1 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 figures it can be deduced that angle sum of a polygon having n number of sides = (n–2) × 180⁰

(a) Number of sides = 7

Thus, angle sum of polygon having number of sides equal to 7 = (7 – 2) × 180⁰

= 5 × 180⁰

= 900⁰ Answer

(b) Number of sides = 8

Thus, angle sum of polygon having number of sides equal to 8 = (8 – 2) × 180⁰

= 6 × 180⁰

= 1080⁰ Answer

(c) Number of sides = 10

Thus, angle sum of polygon having number of sides equal to 10 = (10 – 2) × 180⁰

= 8 × 180⁰

= 1440⁰ Answer

(d) Number of sides = n

Thus, angle sum of polygon having number of sides equal to n

= (n – 2) × 180⁰ Answer

Understanding Quadrilaterals NCERT Exercise 3.1 Question (5) What is a regular polygon?

State the name of a regular polygon of

(i) 3 sides

(ii) 4 sides

(iii) 6 sides

Solution

Equiangular and equilateral polygons are known as regular polygons.

(i) 3 sides

Polygon with 3 sides is known as a triangle Answer

(ii) 4 sides

Polygon with 4 sides is known as a quadrilateral Answer

(iii) 6 sides

Polygon with 6 sides is known as a hexagon Answer

Understanding Quadrilaterals NCERT Exercise 3.1 Question (6) Find the angle measure x in the following figures.

(a)

Solution

Given, three angles of a quadrilateral = 50⁰, 130⁰ and 120⁰ respectively

Then fourth angle (x) = ?

We know that, angle sum of a quadrilateral = 360⁰

Therefore, The angle sum of the given quadrilateral

= 50⁰ + 130⁰ + 120⁰ + x = 360⁰

⇒ 300⁰ + x = 360⁰

⇒ x = 360⁰ – 300⁰

⇒ x = 60⁰

Therefore, fourth angle (x) of the given quadrilateral = 60⁰ Answer

(b)

Solution

Here, in the given quadrilateral,

Angle c = 70⁰

Angle B = 60⁰

And angle MAD = 90⁰

Thus, angle (x) = angle D = ?

Since, angle MAD = 90⁰

Thus, angle DAB will be = 90⁰ [∵ angle MAD and angle DAB both form a straight line]

Now, we know that, angle sum of a quadrilateral = 360⁰

Thus, in the given quadrilateral

∠ DAB + ∠ B + ∠ C + ∠D = 360⁰

⇒ 90⁰ + 60⁰ + 70⁰ + x = 360⁰

⇒ 220⁰ + x = 360⁰

⇒ x = 360⁰ – 220⁰

⇒ x = 140⁰

Thus, angle x of the given quadrilateral = 140⁰ Answer

(c)

Solution

In the given, pentagon

Angle G = 30⁰

Angle DAE = 70⁰

Angle CBF = 60⁰

And angle D = angle C = x

Thus, angle x = ?

In between angle DEA and angle CBF

Since angle DAE and angle DAB form a straight line and hence are supplemetary

Thus, angle DAE + angle DAB = 180⁰

⇒ 70⁰ + ∠ DAB = 180⁰

⇒ ∠ DAB = 180⁰ – 70⁰

⇒ &#DAB = 110⁰

In between angle ABC and angle CBF

Both the angles ABC and CBF form a straight line, and hence are supplementary

Thus, ∠ ABC + ∠ CBF = 180⁰

⇒ ∠ ABC + 60⁰ = 180⁰

⇒ ∠ ABC = 180⁰ – 60⁰

⇒ ∠ ABC = 120⁰

Now, we know that, angle sum of a pentagon = 540⁰

Thus, in the given pentagon,

∠ G + ∠ D + ∠ DAB + ∠ ABC + ∠C = 540⁰

⇒ 30⁰ + x + 110⁰ + 120⁰ + x = 540⁰

⇒ 260⁰ + 2x = 540⁰

⇒ 2x = 540⁰ – 260⁰

⇒ 2x = 280⁰

⇒ x = 140⁰

Thus, the unknown angle x of given pentagon = 140⁰ Answer

(d)

Solution

In the given pentagon all sides are equal,

Thus all the angles will also be equal and will be = x

Thus, angle x = ?

Now, we know that, angle sum of a pentagon = 540⁰

Thus, in the given pentagon,

x + x + x + x + x = 540⁰

⇒ 5x = 540⁰

⇒ x = 108⁰

Thus, the unknown angle x of given pentagon = 108⁰ Answer

Understanding Quadrilaterals NCERT Exercise 3.1 Question (7)

(a) Find x + y + x.

Solution

Now, we know that sum of exterior angles of a polygon = 360⁰

Thus, in the given triangle angle x + y + x = 360⁰ Answer

Alternate method

Given, two angles of the given triangle = 30⁰ and 90⁰

Thus, sum of exterior angles x + y + z = ?

Between angles x and 90⁰

Both the angles form a straight line, and hence are supplementary

Thus, angle x + 90⁰ = 180⁰

⇒ x = 180⁰ – 90⁰

⇒ x = 90⁰

Between angles z and 30⁰

Both the angles form a straight line, and hence are supplementary

Thus, angle z + 30⁰ = 180⁰

⇒ z = 180⁰ – 30⁰

⇒ z = 150⁰

Now we know that, sum of all the interior angles of a triangle = 180⁰

Thus, in the given triangle,

∠ CAE + ∠ ABF + ∠ ACB = 180⁰

⇒ 30⁰ + ∠ ABF + 90⁰ = 180⁰

⇒ ∠ ABF + 120⁰ = 180⁰

⇒ ∠ ABF = 180⁰ – 120⁰

⇒ ∠ ABF = 60⁰

Now between angles ABF and ABE

Both the angles ABF and ABE forms a straight line and hence are supplementary

Thus, ∠ ABF + ∠ ABE = 180⁰

⇒ 60⁰ + y = 180⁰

⇒ y = 180⁰ – 60⁰

⇒ y = 120⁰

Now, x + y + z = 90⁰ + 120⁰ + 150⁰

⇒ x + y + z = 360⁰ Answer

(b) Find x + y + x + w

Solution

Given, x, y , z and w are exterior angles of the given quadrilateral.

Thus, x + y + x + w = ?

We know that, sum of exterior angles of a polygon = 360⁰

Thus, in the given, quadrilateral,

x + y + x + w = 360⁰ Answer

Alternate Method

Given, three angles of a quadrilateral are 120⁰, 80⁰ and 60⁰

Then find the sum of exterior angles of given quadrilateral,

i.e. x + y + z + w = ?

In between angles z and 60⁰

Both the angles form a straight line, and thus are supplementary.

Thus, z + 60⁰ = 180⁰

⇒ z = 180⁰ – 60⁰

⇒ z = 120⁰

In between angles y and 80⁰

Both the angles form a straight line, and thus are supplementary.

Thus, y + 80⁰ = 180⁰

⇒ y = 180⁰ – 80⁰

⇒ y = 100⁰

In between angles x and 120⁰

Both the angles form a straight line, and thus are supplementary.

Thus, x + 120⁰ = 180⁰

⇒ x = 180⁰ – 120⁰

⇒ y = 60⁰

Now, we know that, sum of angle of a quadrilateral = 360⁰

Thus, In the given quadrilateral,

∠ HAB + 120⁰ + 80⁰ + 60⁰ = 360⁰

⇒ ∠ HAB + 260⁰ = 360⁰

⇒ ∠ HAB = 360⁰ – 260⁰

⇒ ∠ HAB = 100⁰

In between angles w and ∠ HAB

Both the angles form a straight line, and thus are supplementary.

Thus, w + ∠ HAB = 180⁰

Thus, w + 100⁰ = 180⁰

⇒ w = 180⁰ – 100⁰

⇒ w = 80⁰

Now, in the given quadrilateral,

w + x + y + x

= 80⁰ + 60⁰ + 100⁰ + 120⁰

= 360⁰

Thus, w + x + y + x = 360⁰ Answer