Understanding Shapes-ii (quadrilaterals)
Class 8th Mathematics RD Sharma Solution
- Define the following terms: (i) Quadrilateral (ii) Convex Quadrilateral…
- In a quadrilateral, define each of the following: (i) Sides (ii) Vertices (iii)…
- Complete each of the following, so as to make a true statement: (i) A…
- In Fig. 16.19, ABCD is a quadrilateral. (i) Name a pair of adjacent sides. (ii)…
- The angles of a quadrilateral are 110, 72, 55 and x. Find the value of x.…
- The three angles of a quadrilateral are respectively equal to 110, 50 and 40.…
- A quadrilateral has three acute angles each measures 80. What is the measure of…
- A quadrilateral has all its four angles of the same measure. What is the…
- Two angles of a quadrilateral are of measure 65 and the other two angles are…
- Three angles of a quadrilateral are equal. Fourth angle is of measure 150.…
- The four angles of a quadrilateral are as 3 ; 5 : 7 : 9. Find the angles.…
- If the sum of the two angles of a quadrilateral is 180. What is the sum of the…
- In Fig. 16.20, find the measure of MPN.
- The sides of a quadrilateral are produced in order. What is the sum of the…
- In Fig. 16.21, the bisectors of A and B meet at a point P. If C =100 and D =…
- In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 :…
- In a quadrilateral ABCD, CO and DO are the bisectors of C and D respectively.…
- Find the number of sides of a regular polygon, when each of its angles has a…
- Find the numbers of degrees in each exterior angle of a regular pentagon.…
- The measure of angles of a hexagon are x, (x-5), (x-5), (2x-5), (2x-5),…
- In a convex hexagon, prove that the sum of all interior angle is equal to…
- The sum of the interior angles of a polygon is three times the sum of its…
- Determine the number of sides of a polygon whose exterior and interior angles…
- PQRSTU is a regular hexagon, Determine each angle of PQT.
Exercise 16.1
Question 1.Define the following terms:
(i) Quadrilateral
(ii) Convex Quadrilateral
Answer:
(i) Quadrilateral
A quadrilateral is a four sided enclosed figure.
(ii) Convex Quadrilateral
In a convex quadrilateral all the vertices are pointing outward.
Question 2.
In a quadrilateral, define each of the following:
(i) Sides
(ii) Vertices
(iii) Angles
(iv) Diagonals
(v) Adjacent angles
(vi) Adjacents sides
(vii) Opposite sides
(viii) Opposite angles
(ix) Interior
(x) Exterior
Answer:
Example:
(i) Sides: Sides are the edges of a quadrilateral. All the sides may have same of different length.
In the above figure AB, BC, CD and DA are sides.
Vertices are the angular points where two sides or edges meet.
In the above figure vertices are A, B, C and D
Angle is the inclination inclination between two sides of a quadrilateral.
In the above figure angles are: ABC, BCA, CDA and DAB
Diagonals are the lines joining two opposite vertices of a quadrilateral.
In the above figure diagonals are: BD and AC
Adjacent angles have one common arm.
In the above figure angles ABC, BCD are adjacent anges.
Adjacent sides make an angle.
In the above figure AB BC, BC CA, CD DA, DA AB are pairs of adjacent sides.
(vii) Opposite sides: Opposite sides don’t have anything in common like sides orv angles.
In the above figure AB CD, BC DA are the pairs of opposite sides.
Opposite angles are made by non adjacent sides.
In the above figure angles A and C, angles B and D are opposite angles.
Interior means within the quadrilateral.
Exterior means outside of a quadrilateral. For example point B is exterior of quadrilateral.
Question 3.
Complete each of the following, so as to make a true statement:
(i) A quadrilateral has ________ sides.
(ii) A quadrilateral has ________angles.
(iii) A quadrilateral has ________, no three of which are ________.
(iv) A quadrilateral has ________diagonals.
(v) The number of pairs of adjacent angles of a quadrilateral is ________.
(vi) The number of pairs of opposite angles of a quadrilateral is ________.
(vii) The sum of the angles of a quadrilateral is ________.
(viii) A diagonal of a quadrilateral is a line segment that joins two ________ vertices of the quadrilateral.
(ix) The sum of the angles of a quadrilateral is ________ right angles.
(x) The measure of each angle of a convex quadrilateral is ________ 180°.
(xi) In a quadrilateral the point of intersection of the diagonals lies in ________ of the quadrilateral.
(xii) A point os in the interior of a convex quadrilateral, if it is in the ________ of its two opposite angles.
(xiii) A quadrilateral is convex if for each side, the remaining ________ lie on the same side of the line containing the side.
Answer:
(i) A quadrilateral has Four sides.
(ii) A quadrilateral has Four angles.
(iii) A quadrilateral has Four vertices, no three of which are collinear.
(iv) A quadrilateral has two diagonals.
(v) The number of pairs of adjacent angles of a quadrilateral is two.
(vi) The number of pairs of opposite angles of a quadrilateral is two.
(vii) The sum of the angles of a quadrilateral is 360°.
(viii) A diagonal of a quadrilateral is a line segment that joins two opposite vertices of the quadrilateral.
(ix) The sum of the angles of a quadrilateral is four right angles.
(x) The measure of each angle of a convex quadrilateral is less than 180°.
(xi) In a quadrilateral the point of intersection of the diagonals lies in interior of the quadrilateral.
(xii) A point os in the interior of a convex quadrilateral, if it is in the interiors of its two opposite angles.
(xiii) A quadrilateral is convex if for each side, the remaining vertices lie on the same side of the line containing the side.
Question 4.
In Fig. 16.19, ABCD is a quadrilateral.
(i) Name a pair of adjacent sides.
(ii) Name a pair of opposite sides.
(iii) How many pairs of adjacent sides are there?
(iv) How many pairs of opposite sides are there?
(v) Name a pair of adjacent angles.
(vi) Name a pair of opposite angles.
(vii) How many pairs of adjacent angles are there?
(viii) How many pairs of opposite angles are there?
Answer:
(i) Name a pair of adjacent sides.
Adjacent sides are: AB, BC, CD and DA
(ii) Name a pair of opposite sides.
Adjacent sides are: AB CD and BC DA
(iii) How many pairs of adjacent sides are there?
Four pairs of adjacent sides.
AB BC, BC CD, CD DA and DA AB
(iv) How many pairs of opposite sides are there?
Two pairs of opposite sides.
AB DC and DA BC
(v) Name a pair of adjacent angles.
Four pairs of Adjacent angles are: DAB ABC, ABC BCA, BCA CDA and CDA DAB
(vi) Name a pair of opposite angles.
Pair of opposite angles are: DAB BCA and ABC CDA
(vii) How many pairs of adjacent angles are there?
Four pairs of adjacent angles. DAB ABC, ABC BCA, BCA CDA and CDA DAB
(viii) How many pairs of opposite angles are there?
Two pairs of opposite angles. DAB BCA and ABC CDA
Question 5.
The angles of a quadrilateral are 110°, 72°, 55° and x°. Find the value of x.
Answer:
Sum of angles of a quadrilateral is 360°
110° + 72° + 55° + x° = 360°
x° = 360° - 237°
x° = 23°
Question 6.
The three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.
Answer:
Sum of angles of a quadrilateral is 360°
Let the fourth angle is x°
110° + 50° + 40° + x° = 360°
x° = 360° - 200°
x° = 160°
Question 7.
A quadrilateral has three acute angles each measures 80°. What is the measure of the fourth angle?
Answer:
Sum of angles of a quadrilateral is 360°
Let the fourth angle is x°
80° + 80° + 80° + x° = 360°
x° = 360° - 240°
x° = 120°
Question 8.
A quadrilateral has all its four angles of the same measure. What is the measure of each?
Answer:
Sum of angles of a quadrilateral is 360°
Let each angle is x°
x° + x° + x° + x° = 360°
x° = 
x° = 90°
Question 9.
Two angles of a quadrilateral are of measure 65° and the other two angles are equal. What is the measure of each of these two angles?
Answer:
Sum of angles of a quadrilateral is 360°
Let each equal angle is x°
65° + 65° + x° + x° = 360°
2x° = 360° - 130°
x° = 
x° = 115°
Therefore other equal angles are 115° each.
Question 10.
Three angles of a quadrilateral are equal. Fourth angle is of measure 150°. What is the measure of equal angles.
Answer:
Sum of angles of a quadrilateral is 360°
Let each equal angle is x°
150° + x° + x° + x° = 360°
3x° = 
x° = 
x° = 
Therefore other equal angles are 70° each.
Question 11.
The four angles of a quadrilateral are as 3 ; 5 : 7 : 9. Find the angles.
Answer:
Sum of angles of a quadrilateral is 360°
Let angle is x°
Therefore each angle is 3x, 5x, 7x and 9x
3x° + 5x° + 7x° + 9x° = 360°
24x° = 
x° = 
x° = 
Therefore angles are: 3x = 3 × 15 = 45°
5x = 5 × 15 = 75°
7x = 7 × 15 = 105°
9x = 3× 15 = 45°
Question 12.
If the sum of the two angles of a quadrilateral is 180°. What is the sum of the remaining two angles?
Answer:
Sum of angles of a quadrilateral is 360°
Let the sum of remaining two angles is x°
180° + x° = 360°
x° = 
x° = 180°
Therefore the sum of other two angles is 180°
Question 13.
In Fig. 16.20, find the measure of ∠MPN.
Answer:
Sum of angles of a quadrilateral is 360°
In the quadrilateral MPNO
∠NOP = 45°, ∠OMP = ∠PNO = 90°,
Let angle ∠MPN is x°
∠NOP + ∠OMP + ∠PNO + ∠MPN = 360°
45° + 90° + 90° + x° = 360°
x° = 360° - 225°
x° = 135°
Therefore ∠MPN is 135°
Question 14.
The sides of a quadrilateral are produced in order. What is the sum of the four exterior angles?
Answer:
We know that, exterior angle + interior adjacent angle = 180° [Linear pair]
Applying relation for polygon having n sides
Sum of all exterior angles + Sum of all interior angles = n × 180°
Therfore sum of all exterior angles = n × 180° - Sum of all interior angles
Sum of all exterior angles = n × 180° - (n -2) × 180° [Sum of interior angles is = (n - 2) x 180°]
Sum of all exterior angles = n × 180° - n × 180° + 2 × 180°
Sum of all exterior angles = 180°n - 180°n + 360°
Sum of all exterior angles = 360°
Question 15.
In Fig. 16.21, the bisectors of ∠A and ∠B meet at a point P. If ∠C =100° and ∠D = 50°, find the measure of ∠APB.
Answer:
Sum of angles of a quadrilateral is 360°
In the quadrilateral ABCD
∠D = 50°, ∠C = 100°,
∠A + ∠B + ∠C + ∠D = 360°
∠A + ∠B + 100° + 50° =
∠A + ∠B =
∠A + ∠B = 210° ……….(i)
Now in Δ APB
[Sum of angles of a triangle is 180°]
…….(ii)
On substituting value of ∠A + ∠B = 210 from equation (i) in equation (ii)
Question 16.
In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angle of the quadrilateral.
Answer:
Sum of angles of a quadrilateral is 360°
Let angle is x°
Therefore each angle is x°, 2x°, 4x° and 5x°
x° + 2x° + 4x° + 5x° = 360°
12x° =
x° = 
x° = 
Therefore angles are: x = 30°
2x = 2 × 30° = 60°
4x = 4 × 30 = 120°
9x = 5 × 30 = 150°
Question 17.
In a quadrilateral ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD =
(∠A +∠B).
Answer:
Sum of angles of a quadrilateral is 360°
In the quadrilateral ABCD
∠A + ∠B + ∠C + ∠D = 360°
∠A + ∠B = 
…….(i)
Now in Δ DOC
[Sum of angles of a triangle is 180°]
……….(ii)
From above equations (i) and (ii) RHS is equal therefore LHS will also be equal.
Therefore 
Question 18.
Find the number of sides of a regular polygon, when each of its angles has a measure of
(i) 160°
(ii) 135°
(iii) 175°
(iv) 162°
(v) 150°
Answer:
(i) 160°
The measure of interior angle A of a polygon of n sides is given by 
Angle of quadrilateral is 160°
Therfore number of sides are 18
(ii) 135°
The measure of interior angle A of a polygon of n sides is given by
Angle of quadrilateral is 135°
Therfore numbers of sides are 8
(iii) 175°
The measure of interior angle A of a polygon of n sides is given by
Angle of quadrilateral is 175°
Therfore numbers of sides are 72
(iv) 162°
The measure of interior angle A of a polygon of n sides is given by
Angle of quadrilateral is 162°
Therfore numbers of sides are 20
(v) 150°
The measure of interior angle A of a polygon of n sides is given by
Angle of quadrilateral is 150°
Therfore numbers of sides are 12
Question 19.
Find the numbers of degrees in each exterior angle of a regular pentagon.
Answer:
The sum of exterior angles of a polygon is 360°
Measure of each exterior angle of a polygon is = 
where n is the number of sides
Number of sides in a pentagon is 5
Measure of each exterior angle of a pentagon is = 
Measure of each exterior angle of a pentagon is 
Question 20.
The measure of angles of a hexagon are x°, (x-5)°, (x-5)°, (2x-5)°, (2x-5)°, (2x+20)°. Find value of x.
Answer:
The sum of interior angles of a polygon = (n – 2) × 180°
where n = number of sides of polygon.Now, we know, a hexagon has 6 sides. So,
The sum of interior angles of a hexagon = (6 – 2) × 180° = 4 × 180° = 720°
therefore, we havex°+ (x-5)°+ (x-5)°+ (2x-5)°+ (2x-5)°+ (2x+20)° = 720°
x°+ x°- 5°+ x° - 5°+ 2x° - 5°+ 2x° - 5°+ 2x° + 20° = 720°
9x° = 720°
x = 80°
Question 21.
In a convex hexagon, prove that the sum of all interior angle is equal to twice the sum of its exterior angles formed by producing the sides in the same order.
Answer:
The sum of interior angles of a polygon = (n – 2) × 180°
The sum of interior angles of a hexagon = (6 – 2) × 180° = 4 × 180° = 720°
The Sum of exterior angle of a plygon is 360°
Therefoe sum of interior angles of a hexagon = twice the sum of interior angles.
Question 22.
The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sides of the polygon.
Answer:
The sum of interior angles of a polygon = (n – 2) × 180° …..(i)
The Sum of exterior angle of a plygon is 360°
According to the question:
Sum of interior angles = 3 × sum of exterior angles
Sum of interior angles = 3 × 360° = 1080°
Now applying relation as per equation (i)
(n – 2) × 180° = 1080°
n – 2 = 
n – 2 = 6
n = 6 + 2 = 8
Therfore numbers of sides in the polygon are 8.
Question 23.
Determine the number of sides of a polygon whose exterior and interior angles are in the ratio 1 : 5.
Answer:
The sum of interior angles of a polygon = (n – 2) × 180° ……..(i)
The Sum of exterior angle of a plygon is 360°
According to the question:
On cross multiplication we get
(n - 2) = 10
n = 12
Therefore the numbers of sides in the polygon are 12.
Question 24.
PQRSTU is a regular hexagon, Determine each angle of ΔPQT.
Answer:
The sum of interior angles of a polygon = (n – 2) × 180°
The sum of interior angles of a hexagon = (6 – 2) × 180° = 4 × 180° = 720°
Measure of each angle of hexagon = 
Proved above
In Δ PUT
[Angle sum property of a triangle]
[Since ΔPUT is isosceles triangle]
Similarly 
Therefore 
[Using angle sum property of triangle in ΔPQT]