GSEB Class 8 Maths Solutions Chapter 3 ચતુષ્કોણની સમજ InText Questions

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબ 43)

એક નિયમિત પકોણ લો.

 

Question 1. તેના બહિષ્કોણ x, y, z, p, q તથા નાં માપનો સરવાળો કેટલો છે?
Answer: બહુકોણના બધા બહિષ્કોણોનાં માપનો સરવાળો \( = 360^{\circ} \).
\( \implies x + y + z + p + q + r = 360^{\circ} \)
In simple words: The total of all outside angles of any polygon is always 360 degrees. So, when you add up angles x, y, z, p, q, and r, they will come to 360 degrees.

Exam Tip: Remember that the sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides. This is a fundamental property to recall.

 

Question 2. x = y = z = p = q = r છે? ડેમ?
Answer: આ નિયમિત બહુકોણ છે. તેથી તેની બધી બાજુઓની લંબાઈ સરખી છે. તે નિયમિત પણ છે, તેથી તેના બધા જ અંતઃકોણોનાં માપ પણ સરખાં છે. તેથી:
\( x = (180^{\circ} - a) \), \( y = (180^{\circ} - a) \), \( z = (180^{\circ} - a) \)
\( p = (180^{\circ} - a) \), \( q = (180^{\circ} - a) \), \( r = (180^{\circ} - a) \)
\( \implies x = y = z = p = q = r \)
In simple words: Yes, all these angles are equal because it is a regular polygon. Regular polygons have sides of the same length and all interior angles are also the same. Since exterior angles are formed by extending sides, if interior angles are equal, then their supplementary exterior angles must also be equal.

Exam Tip: For a regular polygon, all interior angles are equal, and consequently, all exterior angles are also equal. This helps simplify calculations for such shapes.

 

Question 3. નીચેના પ્રત્યેકનું માપ કેટલું હશે?
(i) બહિષ્કોણ
(ii) અંતઃકોણ
Answer:
(i) \( x + y + z + p + q + r = 360^{\circ} \) જો \( x = y = z = p = q = r \) છે.
\( \implies \) ષટ્કોણના પ્રત્યેક બહિષ્કોણનું માપ \( = \frac{360^{\circ}}{6} = 60^{\circ} \)
(ii) ષટ્કોણના પ્રત્યેક અંતઃકોણનું માપ \( = 180^{\circ} - \) બહિષ્કોણનું માપ \( = 180^{\circ} - 60^{\circ} = 120^{\circ} \)
In simple words: For a regular hexagon, each of its six exterior angles measures 60 degrees. To find each interior angle, simply subtract the exterior angle from 180 degrees, which gives 120 degrees.

Exam Tip: Remember the relationship between interior and exterior angles: they are supplementary (sum to 180°). This is crucial for calculating one if the other is known.

 

Question 4. આ પ્રવૃત્તિ નીચે આપેલ સ્થિતિ માટે ફરીથી કરોઃ
(i) નિયમિત અષ્ટકોણ
(ii) નિયમિત 20-કોણ
Answer:
(i) નિયમિત અષ્ટકોણમાં આઠ બાજુઓ છે. તેથી \( n = 8 \)
\( \implies \) નિયમિત અષ્ટકોણના પ્રત્યેક બહિષ્કોણનું માપ \( = \frac{360^{\circ}}{n} = \frac{360^{\circ}}{8} = 45^{\circ} \)
\( \implies \) નિયમિત અષ્ટકોણના પ્રત્યેક અંતઃકોણનું માપ \( = 180^{\circ} - 45^{\circ} = 135^{\circ} \)
(ii) નિયમિત 20-કોણમાં વીસ બાજુઓ છે. તેથી \( n = 20 \)
\( \implies \) નિયમિત 20-કોણના પ્રત્યેક બહિષ્કોણનું માપ \( = \frac{360^{\circ}}{n} = \frac{360^{\circ}}{20} = 18^{\circ} \)
\( \implies \) નિયમિત 20-કોણના પ્રત્યેક અંતઃકોણનું માપ \( = 180^{\circ} - 18^{\circ} = 162^{\circ} \)
In simple words: For a regular octagon (8 sides), each outer angle is 45 degrees, and each inner angle is 135 degrees. For a regular 20-sided polygon, each outer angle is 18 degrees, and each inner angle is 162 degrees.

Exam Tip: The formula for each exterior angle of a regular n-sided polygon is \( \frac{360^{\circ}}{n} \). Once you have this, the interior angle is easily found by subtracting from 180°.

 

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબ 47)

Question. 30° - 60°- 90° ના ખૂણા ધરાવતા બે કાટખૂણિયાં લો. હવે તેમને એ પ્રમાણે ગોઠવો કે જેથી સમાંતરબાજુ ચતુષ્કોણ બને. શું આ પ્રવૃત્તિ તમને ઉપરોક્ત ગુણધર્મને ચકાસવામાં મદદ કરશે?

Answer: હા, ઉપરની આકૃતિ પરથી સ્પષ્ટ થાય છે કે સમાંતરબાજુ ચતુષ્કોણના સામસામેના ખૂણાનાં માપ સરખાં હોય છે. (લંબચોરસ એ સમાંતરબાજુ ચતુષ્કોણ પણ છે.)
In simple words: Yes, the picture above clearly shows that the opposite angles of a parallelogram are equal. (A rectangle is also a type of parallelogram.)

Exam Tip: Visual demonstrations using set-squares can powerfully illustrate geometric properties like the equality of opposite angles in a parallelogram. Always connect such activities back to the core definitions.

 

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબ 48)

Question. 30° - 60° - 90°નાં માપ ધરાવતાં બે કાટખૂણિયાં લઈને અગાઉની જેમ એક સમાંતરબાજુ ચતુષ્કોણ બનાવો. શું આ રીતે બનેલ આકૃતિ ઉપરોક્ત ગુણધર્મની પુષ્ટિ કરે છે?

Answer: Yes, this picture also helps to show that opposite angles of parallelograms are the same. When you put two 30°-60°-90° set-squares to make a parallelogram, you can clearly see that the angles at opposite corners are identical. This setup gives more visual evidence for what makes parallelograms special.
In simple words: Yes, by arranging two set-squares this way, it clearly demonstrates that opposite angles in the parallelogram are equal, confirming the property.

Exam Tip: Understanding how geometric shapes can be constructed from basic tools like set-squares reinforces your understanding of their properties. Note how the angles of the set-squares combine.

 

વિચારો, ચર્ચા કરો અને લખો (પાઠ્યપુસ્તક પાન નંબ 50)

Question. mzR = m<N = 70° દર્શાવ્યા બાદ, બીજી કોઈ રીતે mzl અને mzGનું માપ શોધી શકાય?

Answer: Yes, even without using the properties of a parallelogram, we can determine m∠I and m∠G using these steps: In quadrilateral INGR, \( \overline{RG} \parallel \overline{IN} \) and \( \overleftrightarrow{RI} \) is their transversal line. Therefore, \( m\angle R + m\angle I = 180^{\circ} \) (because they are consecutive interior angles on the same side of the transversal). So, \( 70^{\circ} + m\angle I = 180^{\circ} \) (since \( m\angle R = m\angle N = 70^{\circ} \)). Therefore, \( m\angle I = 180^{\circ} - 70^{\circ} \).
\( \implies m\angle I = 110^{\circ} \). Similarly, we can find that \( m\angle G = 110^{\circ} \).
In simple words: Yes, you can find the other angles by using the rule that angles on the same side of a transversal line add up to 180 degrees. Since \( \angle R \) is 70 degrees, \( \angle I \) will be 180 minus 70, which is 110 degrees. \( \angle G \) will also be 110 degrees.

Exam Tip: Remember the properties of parallel lines intersected by a transversal: consecutive interior angles are supplementary. This is an alternative way to find angles in a parallelogram if you don't initially assume it's a parallelogram, but rather a trapezoid with parallel sides.

 

વિચારો, ચર્ચા કરો અને લખો (પાઠ્યપુસ્તક પાન નંબ 56)

 

Question 1. કડિયો કોંક્રિટનો એક ‘સ્લેબ' બનાવે છે. તે તેને લંબચોરસ બનાવવા માગે છે. કેટલા અલગ અલગ પ્રકારથી, તે આ ‘સ્લેબ’ લંબચોરસ જ છે તેવી ચકાસણી કરી શકશે?
Answer: The mason can check that the slab is rectangular in the following ways:
1. He will make sure the opposite sides of the slab are the same length.
2. He will make sure each angle of the slab is 90° (a right angle).
3. He will make sure the lengths of both diagonals of the slab are equal.
4. He will make sure the opposite sides (length and width) of the slab are parallel.
In simple words: To check if a slab is a rectangle, a builder can make sure its opposite sides are equal, all its corners are 90 degrees, its diagonals have the same length, and its opposite sides run parallel to each other.

Exam Tip: To prove a quadrilateral is a rectangle, you can show: (1) all angles are 90°, (2) opposite sides are equal and one angle is 90°, or (3) diagonals are equal and bisect each other (implying it's a parallelogram first).

 

Question 2. સમાન લંબાઈની બાજુઓ ધરાવતા લંબચોરસ તરીકે ચોરસને વ્યાખ્યાયિત કરવામાં આવ્યો હતો. આપણે તેને સમાન ખૂણા ધરાવતાં સમબાજુ ચતુષ્કોણ તરીકે વ્યાખ્યાયિત કરી શકીએ?
Answer: Yes, we can define a square this way. If all four angles of a rhombus are made 90°, then it is defined as a square. This provides a clear explanation.
In simple words: Yes, if a rhombus has all its angles as 90 degrees, it then becomes a square.

Exam Tip: A square is a special type of rectangle (all sides equal) and a special type of rhombus (all angles equal). Understanding these hierarchical relationships is key to geometric definitions.

 

Question 3. સમલંબ ચતુષ્કોણના બધા જ ખૂણા સમાન હોઈ શકે? તેની દરેક બાજુઓ, સમાન હોઈ શકે? સ્પષ્ટતા કરો.
Answer: If all angles of a trapezium are equal, it turns into a rectangle. If all sides of a trapezium become equal, it turns into a square. As a result, not all angles of a trapezium can be equal, nor can all its sides be equal.
In simple words: No, a trapezium cannot have all angles equal or all sides equal, because if it did, it would no longer be a trapezium; it would become a rectangle or a square instead.

Exam Tip: A trapezium (or trapezoid) is defined by having only one pair of parallel sides. If all angles or all sides were equal, it would satisfy the conditions for a more specific quadrilateral (rectangle or square), not a general trapezium.