GSEB Class 8 Maths Solutions Chapter 3 Understanding Quadrilaterals Exercise 3.1

Gujarat Board Textbook Solutions Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 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
Answer:
(a) Simple curves include: (1), (2), (5), (6) and (7).
(b) Simple closed curves are: (1), (2), (5), (6) and (7).
(c) Polygons are: (1), (2) and (4).
(d) The convex polygon is: (2).
(e) Concave polygons are (1) and (4).
In simple words: Look at each drawing to decide if it's a simple line, a closed shape, a polygon (a closed shape with straight lines), or a polygon that bulges out (convex) or curves in (concave).

Exam Tip: Remember that a simple curve does not cross itself. A simple closed curve starts and ends at the same point without self-intersecting. Polygons are closed shapes made only of line segments.

 

Question 2. How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
Answer:
Note: The total number of diagonals in a polygon with \( n \) sides is given by the formula: \( \frac { n(n-1) }{ 2 } -n \).
(a) In a quadrilateral, the number of sides \( (n) = 4 \).
The number of diagonals \( = \frac { n(n-1) }{ 2 } -n \)
\( = \frac { 4(4-1) }{ 2 } - 4 \)
\( = \frac { 4 \times 3 }{ 2 } - 4 \)
\( = \frac { 12 }{ 2 } - 4 \)
\( = 6 - 4 = 2 \)
(b) In a regular hexagon, the number of sides \( (n) = 6 \).
The number of diagonals \( = \frac { n(n-1) }{ 2 } - n \)
\( = \frac { 6(6-1) }{ 2 } - 6 \)
\( = \frac { 6 \times 5 }{ 2 } - 6 \)
\( = \frac { 30 }{ 2 } - 6 \)
\( = 15 - 6 = 9 \)
(c) In a triangle, the number of sides \( (n) = 3 \).
The number of diagonals \( = \frac { n(n-1) }{ 2 } - n \)
\( = \frac { 3(3-1) }{ 2 } - 3 \)
\( = \frac { 3 \times 2 }{ 2 } - 3 \)
\( = \frac { 6 }{ 2 } - 3 \)
\( = 3 - 3 = 0 \)
In simple words: To find how many diagonals a shape has, use the formula: \( \frac { n(n-1) }{ 2 } -n \), where 'n' is the number of sides. For a quadrilateral, it has 2 diagonals. A hexagon has 9. A triangle has 0.

Exam Tip: Always remember that a diagonal connects two non-adjacent vertices. A triangle has no non-adjacent vertices, hence zero diagonals.

 

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!)
Answer: The sum of the angle measures of a convex quadrilateral equals \( 360° \). Yes, this characteristic stays true even if the quadrilateral is not convex.
In simple words: All the angles inside any four-sided shape (a quadrilateral) add up to 360 degrees, no matter if it's a regular shape or an irregular one.

Exam Tip: The sum of interior angles for any quadrilateral (convex or concave) is consistently \( 360° \). This is a fundamental property that you should commit to memory.

 

Question 4. Examine the table. (Each figure is divided into triangle 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
(d) n

Answer: From the table shown above, we can determine that the sum of the interior angles of a polygon with \( n \) sides equals \( (n - 2) \times 180° \).
(a) When \( n = 7 \): Substituting \( n = 7 \) into the formula, we have
The sum of interior angles for a polygon with 7 sides \( = (n - 2) \times 180° \)
\( = (7 – 2) \times 180° \)
\( = 5 \times 180° = 900° \)
(b) When \( n = 8 \): Substituting \( n = 8 \) into the formula, we have
The sum of interior angles for a polygon having 8 sides
\( = (8 – 2) \times 180° \)
\( = 6 \times 180° = 1080° \)
(c) When \( n = 10 \): Substituting \( n = 10 \) into the formula, we have
The sum of interior angles for a polygon having 10 sides
\( = (n - 2) \times 180° \)
\( = (10 – 2) \times 180° \)
\( = 8 \times 180° = 1440° \)
(d) When \( n = n \): The sum of interior angles for a polygon with \( n \) sides \( = (n - 2) \times 180° \).
In simple words: The table shows that you can find the total of all inside angles of a polygon by taking the number of sides, subtracting 2, and then multiplying that answer by 180 degrees. This works for any number of sides.

Exam Tip: This formula, \( (n-2) \times 180° \), is essential for calculating the sum of interior angles of any polygon. Make sure to apply it correctly by first determining the number of sides, \( n \).

 

Question 5. What is a regular polygon? State the name of a regular polygon of
(i) 3 sides
(ii) 4 sides
(iii) 6 sides
Answer: A polygon is considered a regular polygon if:
(a) The measures of its interior angles are all equal.
(b) The lengths of its sides are all equal.
The name of a regular polygon with:
(i) 3 sides is an 'equilateral triangle'.
(ii) 4 sides is a 'square'.
(iii) 6 sides is a 'regular hexagon'.
In simple words: A regular polygon is a shape where all its sides are the same length and all its inside angles are the same size. For example, a 3-sided regular polygon is an equilateral triangle, a 4-sided one is a square, and a 6-sided one is a regular hexagon.

Exam Tip: The key properties of a regular polygon are "equilateral" (equal sides) and "equiangular" (equal angles). Both conditions must be met.

 

Question 6. Find the angle measure x in the following figures.
(a) (Figure shows a quadrilateral with interior angles \( 50°, 130°, 120° \) and \( x \).)
(b) (Figure shows a quadrilateral with interior angles \( x, 60°, 70° \) and a right angle \( 90° \).)
(c) (Figure shows a pentagon with interior angles \( 30°, x, x \), and two exterior angles \( 70° \) and \( 60° \), implying interior angles \( 110° \) and \( 120° \) respectively.)
(d) (Figure shows a regular pentagon with all interior angles marked \( x \).)
Answer:
(a) The sum of the interior angles of a quadrilateral is \( 360° \).
So, \( x + 120° + 130° + 50° = 360° \)
\( \implies x + 300° = 360° \)
\( \implies x = 360° - 300° \)
\( \implies x = 60° \)
(b) The sum of the interior angles of a quadrilateral is \( 360° \).
So, \( x + 60° + 70° + 90° = 360° \)
\( \implies x + 220° = 360° \)
\( \implies x = 360° - 220° \)
\( \implies x = 140° \)
(c) The interior angles of the pentagon are: \( 30°, x°, x° \), and those formed by linear pairs with exterior angles \( (180° – 70°) \) and \( (180° – 60°) \).
Thus, the interior angles are \( 30°, x°, x°, 110° \) and \( 120° \).
The sum of the interior angles of a pentagon is \( 540° \).
So, \( 30° + x + x + 110° + 120° = 540° \)
\( \implies 2x + 260° = 540° \)
\( \implies 2x = 540° – 260° \)
\( \implies 2x = 280° \)
\( \implies x = \frac { 280° }{ 2 } \)
\( \implies x = 140° \)
(d) The figure is a regular pentagon.
The sum of all interior angles of a regular pentagon is \( 540° \).
Since it is regular, all its interior angles are equal.
So, \( x + x + x + x + x = 540° \)
\( \implies 5x = 540° \)
\( \implies x = \frac { 540° }{ 5 } \)
\( \implies x = 108° \)
In simple words: For any shape, all the inside angles add up to a specific total. For a 4-sided shape, it's 360 degrees. For a 5-sided shape, it's 540 degrees. Use this fact, along with any given angles, to find the missing angle 'x'. Remember that an exterior angle and its interior angle add up to 180 degrees.

Exam Tip: Always identify the type of polygon first to know the sum of its interior angles. Remember that a linear pair of angles adds up to \( 180° \), and the sum of interior angles of an n-sided polygon is \( (n-2) \times 180° \).

 

Question 7.
(a) Find \( x + y + z \)
(Figure shows a triangle with interior angles \( 90°, 30° \) and \( 60° \). Exterior angles \( x, y, z \) are linear pairs to these interior angles respectively.)
(b) Find \( x + y + z + w \)
(Figure shows a quadrilateral with interior angles \( 120°, 80°, 60° \) and \( 100° \). Exterior angles \( x, y, z, w \) are linear pairs to these interior angles respectively.)
Answer:
(a) \( \because x + 90° = 180° \) (Linear pair)
\( \implies x = 180° - 90° = 90° \)
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
\( \implies y = 30° + 90° = 120° \)
Similarly, \( z = 180° - 30° = 150° \)
Now, we calculate the sum: \( x + y + z = 90° + 120° + 150° = 360° \)
(b) \( \because \) The sum of interior angles of a quadrilateral equals \( 360° \).
Let the fourth interior angle be \( \angle 1 \).
So, \( \angle 1 + 120° + 80° + 60° = 360° \)
\( \implies \angle 1 + 260° = 360° \)
\( \implies \angle 1 = 360° – 260° = 100° \)
Now, we find the exterior angles using linear pairs:
\( x + 120° = 180° \) (Linear pair)
\( \implies x = 180° - 120° = 60° \)
\( y + 80° = 180° \) (Linear pair)
\( \implies y = 180° - 80° = 100° \)
\( z + 60° = 180° \) (Linear pair)
\( \implies z = 180° - 60° = 120° \)
\( w + 100° = 180° \) (Linear pair) (Using \( \angle 1 = 100^\circ \))
\( \implies w = 180° - 100° = 80° \)
Thus, the sum is \( x + y + z + w \)
\( = 60° + 100° + 120° + 80° = 360° \)
In simple words: The exterior angles of any polygon always add up to 360 degrees. For each interior angle, its exterior angle is found by subtracting the interior angle from 180 degrees.

Exam Tip: Remember the two critical properties: the sum of exterior angles of any convex polygon is always \( 360° \), and an interior angle and its corresponding exterior angle form a linear pair, summing to \( 180° \).