Understanding Quadrilaterals - Solutions 1
CBSE Class –VIII Mathematics
NCERT Solutions
CHAPTER - 3
Understanding Quadrilaterals (Ex. 3.1)
NCERT Solutions
CHAPTER - 3
Understanding Quadrilaterals (Ex. 3.1)
1. Given here are some figures:
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image001.png)
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
Ans. (a) Simple curve
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image002.jpg)
(b) Simple closed curve
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image003.jpg)
(c) Polygons
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image004.jpg)
(d) Convex polygons
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image005.jpg)
(e) Concave polygon
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image006.jpg)
2. How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
Ans. (a) A convex quadrilateral has two diagonals.
Here, AC and BD are two diagonals.
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image007.jpg)
(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image008.jpg)
(c) A triangle has no diagonal.
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)
Ans. Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
=
1 +
6 +
5 +
4 +
3 +
2
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
= (
1 +
2 +
3) + (
4 +
5 +
6)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image010.jpg)
=
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image011.png)
[By Angle sum property of triangle]
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
Hence, the sum of measures of the triangles of a convex quadrilateral is ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image013.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image013.png)
Yes, if quadrilateral is not convex then, this property will also be applied.
Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles.
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image014.jpg)
Using angle sum property of triangle,
In
ABD,
1 +
2 +
3 =
……….(i)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image015.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image016.png)
In
BDC,
4 +
5 +
6 =
……….(i)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image015.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image016.png)
Adding eq. (i) and (ii),
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image009.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image012.png)
Hence proved.
4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
Figure
| ![]() | ![]() | ![]() | ![]() |
Side
|
3
|
4
|
5
|
6
|
Angle sum
| ![]() | ![]() | ![]() | ![]() |
What can you say about the angle sum of a convex polygon with number of sides?
Ans. (a) When
= 7, then
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image026.png)
Angle sum of a polygon = ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image028.png)
(b) When
= 8, then
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image026.png)
Angle sum of a polygon = ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image029.png)
(c) When
= 10, then
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image026.png)
Angle sum of a polygon = ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image030.png)
(d) When
=
then
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image026.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image031.png)
Angle sum of a polygon = ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
5. What is a regular polygon? State the name of a regular polygon of:
(a) 3 sides
(b) 4 sides
(c) 6 sides
Ans. A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.
(i) 3 sides
Polygon having three sides is called a triangle.
(ii) 4 sides
Polygon having four sides is called a quadrilateral.
(iii) 6 sides
Polygon having six sides is called a hexagon.
6. Find the angle measures
in the following figures:
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image032.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image033.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image034.png)
Ans. (a) Using angle sum property of a quadrilateral,
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image035.jpg)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image036.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image037.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image038.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image039.png)
(b) Using angle sum property of a quadrilateral,
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image040.jpg)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image041.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image042.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image043.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image044.png)
(a) First base interior angle
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image045.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image045.png)
Second base interior angle
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image046.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image046.png)
There are 5 sides,
= 5
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image026.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image047.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
=
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image050.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image049.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image050.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image051.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image052.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image053.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image054.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image044.png)
(b) Angle sum of a polygon = ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image027.png)
=
= ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image050.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image049.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image050.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image055.jpg)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image056.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image057.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image058.png)
Hence each interior angle is ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image059.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image059.png)
7. (a) Find
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image060.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image061.jpg)
(b) Find ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image062.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image062.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image063.jpg)
Ans. (a) Since sum of linear pair angles is ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image064.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image064.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image065.jpg)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image066.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image067.png)
And ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image068.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image068.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image069.png)
Also
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image070.png)
[Exterior angle property]
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image071.png)
(b) Using angle sum property of a quadrilateral,
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image072.jpg)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image073.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image074.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image075.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image076.png)
Since sum of linear pair angles is ![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image064.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image064.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image048.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image077.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image078.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image079.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image080.png)
Adding eq. (i), (ii), (iii) and (iv),
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image081.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image082.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image083.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image017.png)
![](https://media-mycbseguide.s3.amazonaws.com/images/static/ncert/08/maths/3_1/image084.png)
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