GSEB Class 8 Maths Solutions Chapter 3 ચતુષ્કોણની સમજ Exercise 3.1

Gujarat Board Textbook Solutions Class 8 Maths Chapter 3 ચતુષ્કોણની સમજ Ex 3.1

 

Question 1. આકૃતિઓ આપેલ છે. પ્રત્યેકનું નીચે દર્શાવેલ આધાર પ્રમાણે વર્ગીકરણ કરો:
(a) સરળ વક્ર
(b) સરળ બંધ વક્ર
(c) બહુકોણ
(d) બહિર્મુખ બહુકોણ
Answer:
(a) સરળ વક્ર: (1), (2), (5), (6) અને (7)
(b) સરળ બંધ વક્ર : (1), (2), (5), (6) અને (7)
(c) બહુકોણ : (1), (2) અને (4)
(d) બહિર્મુખ બહુકોણ : (2)
(e) અંતર્મુખ બહુકોણ (1) અને (4)
In simple words: Look at each picture and sort it into the right group. Some shapes are simple lines, some are simple closed lines, some are polygons, and some are specific types of polygons like convex or concave.

Exam Tip: Remember the basic definitions of curves, closed curves, polygons, convex, and concave polygons to correctly classify geometric figures.

 

Question 2. નીચે દર્શાવેલ પ્રત્યેકને કેટલા વિકર્ણ છે તે જણાવો:
(a) બહિર્મુખ ચતુષ્કોણ
(b) નિયમિત ષટ્કોણ
(c) ત્રિકોણ
Answer: Remember the formula: The number of diagonals in a polygon with n sides is given by:
\( \text{Number of diagonals} = \left[\frac{n(n-1)}{2}-n\right] \)
\( = \frac{n^{2}-n-2 n}{2} \)
\( = \frac{n^{2}-3 n}{2} \)
(a) For a convex quadrilateral, there are four sides, so \( n = 4 \).
The number of diagonals is: \( \left[\frac{4(4-1)}{2}-4\right] \)
\( = \left[\frac{4(3)}{2}-4\right] \)
\( = \left[\frac{12}{2}-4\right] \)
\( = [6-4] \)
\( = 2 \)
(b) For a regular hexagon, there are six sides, so \( n = 6 \).
The number of diagonals is: \( \left[\frac{6(6-1)}{2}-6\right] \)
\( = \left[\frac{6(5)}{2}-6\right] \)
\( = \left[\frac{30}{2}-6\right] \)
\( = [15-6] \)
\( = 9 \)
(c) For a triangle, there are three sides, so \( n = 3 \).
The number of diagonals is: \( \left[\frac{3(3-1)}{2}-3\right] \)
\( = \left[\frac{3(2)}{2}-3\right] \)
\( = [3-3] \)
\( = 0 \)
In simple words: Use the special formula to figure out how many diagonal lines you can draw inside each shape without going through its sides. A quadrilateral has 2, a hexagon has 9, and a triangle has 0.

Exam Tip: Always remember the formula for calculating the number of diagonals in an n-sided polygon. This formula is crucial for questions of this type.

 

Question 3. બહિર્મુખ ચતુષ્કોણના ખૂણાનાં માપનો સરવાળો કેટલો થાય? હવે જો, ચતુષ્કોણ બહિર્મુખ ના હોય, તો શું આ ગુણધર્મ લાગુ પડશે? (એક બહિર્મુખ ના હોય તેવો ચતુષ્કોણ બનાવો અને પ્રયત્ન કરો.)
Answer: The sum of all the angles in a convex quadrilateral is 360°. If the quadrilateral is not convex (meaning it's concave), yes, the sum of all its angles will still be 360°. Look at the concave quadrilateral ABCD shown here as an example.
\( m\angle A + m\angle B + m\angle C + m\angle D \)
\( = 40° + 55° + 35° + 230° = 360° \)

In simple words: The total degrees inside any quadrilateral, whether it's pushed inwards or not, always adds up to 360 degrees.

Exam Tip: Remember that the sum of interior angles of any quadrilateral (convex or concave) is always 360 degrees. This is a fundamental property of quadrilaterals.

 

Question 4. નીચેનું કોષ્ટક જુઓ. (અહીં પ્રત્યેક આકૃતિને ત્રિકોણમાં વિભાજિત કરેલ છે અને તેના પરથી ખૂણાનાં માપનો સરવાળો શોધેલ છે.) નિગ્નલિખિત સંખ્યા દર્શાવતી બાજુઓ ધરાવતા બહુકોણના ખૂણાનાં માપના સરવાળા વિશે શું કહી શકાય?
(a) 7
(b) 8
(c) 10
(d) n

Answer: From the table above, we can determine that the sum of all interior angles of an n-sided polygon is given by the formula \( (n – 2) \times 180° \).
(a) For a polygon with 7 sides, we use \( n = 7 \).
Sum of all interior angles \( = (n – 2) \times 180° \)
\( = (7 – 2) \times 180° \)
\( = 5 \times 180° = 900° \)
(b) For a polygon with 8 sides, we use \( n = 8 \).
Sum of all interior angles \( = (n – 2) \times 180° \)
\( = (8 – 2) \times 180° \)
\( = 6 \times 180° = 1080° \)
(c) For a polygon with 10 sides, we use \( n = 10 \).
Sum of all interior angles \( = (n – 2) \times 180° \)
\( = (10 - 2) \times 180° \)
\( = 8 \times 180° = 1440° \)
(d) For an n-sided polygon:
Sum of all interior angles \( = (n – 2) \times 180° \)
In simple words: The rule to find the total degrees inside any polygon is to take the number of sides, subtract 2, and then multiply that answer by 180 degrees. This works for any polygon, no matter how many sides it has.

Exam Tip: Memorize the formula \( (n-2) \times 180^\circ \) for the sum of interior angles of an n-sided polygon, as it is foundational for polygon problems.

 

Question 5. નિયમિત બહુકોણ એટલે શું? એવા નિયમિત બહુકોણનાં નામ આપો જેમાં:
(i) 3 બાજુ હોય
(ii) 4 બાજુ હોય
(iii) 6 બાજુ હોય
Answer: A regular polygon is a polygon where:
(1) All its internal angles have the same measure.
(2) All its sides have the same length.
Names of the requested regular polygons are:
(i) 3 sides: Equilateral triangle
(ii) 4 sides: Square
(iii) 6 sides: Regular hexagon
In simple words: A regular polygon has all its sides the same length and all its angles the same size. For 3 sides, it's an equilateral triangle; for 4 sides, it's a square; and for 6 sides, it's a regular hexagon.

Exam Tip: Understand the two key characteristics that define a regular polygon: equal sides and equal angles. Be ready to give examples for different numbers of sides.

 

Question 6. નીચેના દરેક આકૃતિઓમાં x (ખૂણાનું માપ) શોધો:
(a)
Answer: The sum of the measures of the four angles of a quadrilateral is 360°.
\( \implies x + 120° + 130° + 50° = 360° \)
\( \implies x + 300° = 360° \)
\( \implies x = 360° – 300° \)
\( \implies x = 60° \)

In simple words: For any four-sided shape, all the angles inside it must add up to 360 degrees. Subtract the known angles from 360 to find the missing angle 'x'.

Exam Tip: Always remember that the sum of interior angles in a quadrilateral is 360 degrees. Use this property to set up an equation and solve for unknown angles.

 

Question 6. (b)
Answer: The sum of the measures of the four angles of a quadrilateral is 360°.
Here, one angle is shown as a right angle, so its measure is 90°.
\( \implies x + 60° + 70° + 90° = 360° \)
\( \implies x + 220° = 360° \)
\( \implies x = 360° – 220° \)
\( \implies x = 140° \)

In simple words: Recognize that the square symbol means 90 degrees. Add up all the known angles of the quadrilateral and subtract from 360 degrees to find the missing angle 'x'.

Exam Tip: Always look for symbols indicating right angles (90°) in diagrams. For quadrilaterals, the sum of internal angles is consistently 360 degrees.

 

Question 6. (c)
Answer: The figure given here is a pentagon. First, we find the internal angles for the two corners based on their linear pairs.
One internal angle \( = 180° – 70° = 110° \)
Another internal angle \( = 180° – 60° = 120° \)
The sum of the measures of all the internal angles of a pentagon is:
\( = (n – 2) \times 180° = (5 – 2) \times 180° = 3 \times 180° = 540° \)
Therefore, \( 30° + x + 110° + 120° + 60° = 540° \)
\( \implies x + 320° = 540° \)
\( \implies x = 540° – 320° \)
\( \implies x = 220° \)

In simple words: This shape has five sides, making it a pentagon. First, figure out the unknown internal angles by using the idea that angles on a straight line add up to 180 degrees. Then, use the formula for the sum of angles in a pentagon (540 degrees) to solve for 'x'.

Exam Tip: For polygons with exterior angles, first find the corresponding interior angles using the linear pair property (angles on a straight line sum to 180°). Then apply the polygon angle sum formula.

 

Question 6. (d)
Answer: The given figure is a regular pentagon. In a regular pentagon, all interior angles have the same measure. Let's assume the measure of each angle is 'x'.
The sum of the measures of all the internal angles of a regular pentagon is:
\( (5 – 2) \times 180° = 3 \times 180° = 540° \)
Therefore, \( x + x + x + x + x = 540° \)
\( \implies 5x = 540° \)
\( \implies \frac{5 x}{5}=\frac{540^{\circ}}{5} \)
\( \implies x = 108° \)

In simple words: Since it's a regular pentagon, all five inside angles are equal. First, find the total sum of angles for a pentagon, which is 540 degrees. Then, divide this total by 5 to find the value of each 'x' angle.

Exam Tip: For regular polygons, all interior angles are equal. Calculate the total sum of interior angles and divide by the number of sides to find each individual angle.

 

Question 7. (a) x + y + z શોધો.
Answer: First, let's find the values of x, y, and z.
\( x + 90° = 180° \) (This is a linear pair of angles.)
\( \implies x = 180° - 90° \)
\( \implies x = 90° \)
For the triangle, the exterior angle \( y \) is equal to the sum of the two opposite interior angles (30° and 90°).
\( y = 30° + 90° \)
\( \implies y = 120° \)
\( z = 180° – 30° \) (This is a linear pair of angles.)
\( \implies z = 150° \)
Therefore, \( x + y + z = 90° + 120° + 150° = 360° \)

In simple words: First, find each missing angle (x, y, z) using rules like angles on a straight line adding up to 180 degrees, and the exterior angle of a triangle being the sum of the two opposite interior angles. Then, add these three angles together to get the final sum.

Exam Tip: Remember the properties of linear pairs (sum to 180°) and the exterior angle theorem (exterior angle equals sum of two opposite interior angles) to find unknown angles in triangles efficiently.

 

Question 7. (b) x + y + z + w શોધો.
Answer: First, we need to find the measure of angle 'a'. The sum of all interior angles of a quadrilateral is 360°.
Let the unknown interior angle be 'a'.
\( a + 120° + 80° + 60° = 360° \)
\( \implies a + 260° = 360° \)
\( \implies a = 360° – 260° \)
\( \implies a = 100° \)
Now, we find x, y, z, and w using the linear pair property.
\( x + 120° = 180° \) (Linear pair of angles)
\( \implies x = 180° - 120° \)
\( \implies x = 60° \)
\( y + 80° = 180° \) (Linear pair of angles)
\( \implies y = 180° – 80° \)
\( \implies y = 100° \)
\( z + 60° = 180° \) (Linear pair of angles)
\( \implies z = 180° – 60° \)
\( \implies z = 120° \)
\( w + 100° = 180° \) (Linear pair of angles)
\( \implies w = 180° - 100° \)
\( \implies w = 80° \)
Thus, \( x + y + z + w = 60° + 100° + 120° + 80° = 360° \)

In simple words: For this four-sided shape, first find the unknown inner angle 'a' by using the fact that all inside angles add up to 360 degrees. Then, for each outer angle (x, y, z, w), use the straight line rule (angles add to 180 degrees) to find its value. Finally, add all these outer angles together.

Exam Tip: The sum of exterior angles of any convex polygon (one at each vertex) is always 360 degrees. This property can be a shortcut for solving problems involving exterior angles.