GSEB Class 8 Maths Solutions Chapter 3 Understanding Quadrilaterals InText Questions

Try These (Page 43)

 

Question 1. Take a regular hexagon as shown in the figure:

1. What is the sum of the measures of its exterior angles x, y, z, p, q, r?
2. Is x = y = z = p = q = r? Why?
3. What is the measure of each?
Answer:
1. The sum of the exterior angles \( \angle x + \angle y + \angle z + \angle p + \angle q + \angle r = 360^{\circ} \). This is because the sum of all exterior angles of any polygon always equals \( 360^{\circ} \).
2. Yes, all these angles are equal. This polygon is a regular hexagon, meaning all its sides have the same length. Since the sides are equal, its interior angles must also be equal. Because each interior angle and its corresponding exterior angle add up to \( 180^{\circ} \), if the interior angles are equal, the exterior angles must also be equal. So, \( \angle x = \angle y = \angle z = \angle p = \angle q = \angle r \).
3. The measure of each exterior angle is \( \frac{360^{\circ}}{6} = 60^{\circ} \).
(i) interior angle
(ii) exterior angle
Since all exterior angles are equal, the measure of each exterior angle is \( 60^{\circ} \). For the interior angle, we know that an interior angle and its adjacent exterior angle form a linear pair, summing to \( 180^{\circ} \). Thus, the measure of each interior angle is \( 180^{\circ} - 60^{\circ} = 120^{\circ} \).
In simple words: The total of all outside angles of any polygon is always 360 degrees. For a regular hexagon, all its outside angles are the same. Each outside angle is 60 degrees. Each inside angle is 120 degrees because it adds up to 180 degrees with the outside angle.

Exam Tip: Remember that the sum of the exterior angles of any convex polygon is always 360 degrees, regardless of the number of sides. For a regular polygon, all exterior angles are equal.

 

Question 4. Repeat this activity for the cases of
(i) A regular octagon
(ii) a regular 20-gon
Answer:
(i) For a regular octagon:
The number of sides (n) in a regular octagon is 8.
Each exterior angle \( = \frac{360^{\circ}}{8} = 45^{\circ} \).
Each interior angle \( = 180^{\circ} - 45^{\circ} = 135^{\circ} \).
(ii) For a regular 20-gon:
The number of sides (n) in a regular 20-gon is 20.
Each exterior angle \( = \frac{360^{\circ}}{20} = 18^{\circ} \).
Each interior angle \( = 180^{\circ} - 18^{\circ} = 162^{\circ} \).
In simple words: To find the exterior angle of a regular polygon, just divide 360 degrees by the number of its sides. To find the interior angle, subtract the exterior angle from 180 degrees.

Exam Tip: Always state the formula you are using (e.g., sum of exterior angles = 360°) to clearly show your understanding of the concept.

 

Question 2. Find the number of sides of a regular polygon whose each exterior angle has a measure of \( 40^{\circ} \)?
Answer: Since the polygon given is a regular polygon, all its exterior angles are equal. We know that the sum of all exterior angles of any polygon is \( 360^{\circ} \).
To find the number of sides, we can divide the total sum of exterior angles by the measure of each exterior angle.
Number of exterior angles \( = \frac{360^{\circ}}{40^{\circ}} = 9 \).
This implies that the number of sides is also 9.
Therefore, it is a nonagon.
In simple words: Since all outside angles of a regular polygon add up to 360 degrees, if each outside angle is 40 degrees, you divide 360 by 40 to find there are 9 sides. A 9-sided polygon is called a nonagon.

Exam Tip: Remember that for a regular polygon, the number of sides is equal to the number of exterior angles. This relationship is key for such problems.

Try These (Page 47)

 

Question 1. Take two identical set squares with angles \( 30^{\circ} - 60^{\circ} - 90^{\circ} \) and place them adjacently to form a parallelogram as shown in Figure? Does this help to you to verify the above property?

Answer: Yes, the given figure helps to confirm that opposite sides of a parallelogram are of equal length. By placing two identical set squares, we visually see that the sides forming the parallelogram are equal to the corresponding sides of the set squares, thus demonstrating the property.
In simple words: Yes, by putting two exact same set squares next to each other, you can make a shape called a parallelogram. This helps you see that the sides that are opposite each other in this parallelogram are the same length.

Exam Tip: Visual demonstrations using geometry tools like set squares are effective for understanding fundamental properties of shapes. Always connect the visual to the geometric property.

Try These (Page 48)

 

Question 1. Take two identical \( 30^{\circ} - 60^{\circ} - 90^{\circ} \) set - squares and form a parallelogram as before. Does the figure obtained help you to confirm the above property?
Answer: Yes, the figure obtained by using two identical \( 30^{\circ} - 60^{\circ} - 90^{\circ} \) set squares also helps us to confirm that the opposite angles of a parallelogram are equal. When two identical right triangles form a parallelogram, the angles at the vertices can be easily measured or deduced, showing that opposite angles are congruent.
In simple words: Yes, using two of the same 30-60-90 set squares to make a parallelogram helps us confirm that the angles directly across from each other in a parallelogram are equal.

Exam Tip: When forming a parallelogram with two identical triangles, remember that the angles formed at each vertex will demonstrate angle properties, such as opposite angles being equal and consecutive angles summing to 180 degrees.

Try These (Page 50)

 

Question 1. After showing \( m\angle R = m\angle N = 70^{\circ} \), can you find \( m\angle I \) and \( m\angle G \) by any other method?

Answer: Yes, we can also find \( m\angle I \) and \( m\angle G \) without using the property of a parallelogram that opposite angles are equal. Given that \( m\angle R = m\angle N = 70^{\circ} \) and RG is parallel to IN. The transversal RI intersects these parallel lines.
So, \( m\angle R + m\angle I = 180^{\circ} \) because they are consecutive interior angles (sum of interior opposite angles on the same side of the transversal is \( 180^{\circ} \)).
\( 70^{\circ} + m\angle I = 180^{\circ} \)
\( m\angle I = 180^{\circ} - 70^{\circ} = 110^{\circ} \).
Similarly, \( m\angle G \) can be found by considering the transversal GI intersecting parallel lines RG and IN. Thus, \( m\angle G + m\angle N = 180^{\circ} \).
Since \( m\angle N = 70^{\circ} \), then \( m\angle G + 70^{\circ} = 180^{\circ} \), which gives \( m\angle G = 110^{\circ} \).
In simple words: Yes, you can find the other angles by remembering that angles next to each other in a parallelogram (consecutive interior angles) always add up to 180 degrees. Since angle R is 70 degrees, angle I must be 180 minus 70, which is 110 degrees. Angle G is also 110 degrees using the same idea with angle N.

Exam Tip: For parallelograms, remember that consecutive angles are supplementary (add up to 180°) and opposite angles are equal. You can use either property to find unknown angles.

 

Question 2. In the figure, ABCD is a parallelogram. Given that OD = 5 cm and AC is 2 cm less than BD. Find OA?

Answer: We know that the diagonals of a parallelogram bisect each other.
Given that \( OD = 5 \text{ cm} \). Since the diagonals bisect each other, \( OB = OD \).
So, \( OB = 5 \text{ cm} \).
Therefore, the length of diagonal \( BD = OD + OB = 5 \text{ cm} + 5 \text{ cm} = 10 \text{ cm} \).
We are given that \( AC \) is 2 cm less than \( BD \).
So, \( AC = BD - 2 \text{ cm} = 10 \text{ cm} - 2 \text{ cm} = 8 \text{ cm} \).
Since the diagonals bisect each other, \( OA = \frac{1}{2} AC \).
\( OA = \frac{1}{2} \times 8 \text{ cm} = 4 \text{ cm} \).
Thus, the length of \( OA \) is 4 cm.
In simple words: In a parallelogram, the lines that cross inside (diagonals) cut each other in half. Since OD is 5 cm, the full diagonal BD is 10 cm. Diagonal AC is 2 cm shorter than BD, so AC is 8 cm. Since OA is half of AC, OA is 4 cm.

Exam Tip: Always remember the key property that diagonals of a parallelogram bisect each other. This means they cut each other into two equal parts at their intersection point.

Try These (Page 56)

 

Question 1. A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?
Answer: A mason can ensure the concrete slab is rectangular using the following different methods:
1. By making sure opposite sides are of equal length.
2. By keeping each angle at the corners as \( 90^{\circ} \).
3. By ensuring the diagonals are of equal length.
4. By making opposite sides parallel and ensuring one angle is \( 90^{\circ} \).
5. By making all angles equal and ensuring the measure of one angle is \( 90^{\circ} \).
In simple words: A builder can check if a concrete slab is a rectangle by making sure its opposite sides are the same length, or that all its corners are perfect 90-degree angles, or that the lines connecting opposite corners (diagonals) are the same length.

Exam Tip: For questions about properties of shapes, remember the specific conditions that define each shape (e.g., a rectangle has four right angles and equal diagonals).

 

Question 2. A square was defined as a rectangle with all sides equal. Can we define it as a rhombus with equal angles? Explore this idea?
Answer: Yes, we can certainly define a square as a rhombus with equal angles. A rhombus is a quadrilateral where all four sides are of equal length. If a rhombus also has all its angles equal, then each angle must be \( 90^{\circ} \) (since the sum of angles in a quadrilateral is \( 360^{\circ} \), and \( 360^{\circ}/4 = 90^{\circ} \)). A figure with all four sides equal and all four angles equal to \( 90^{\circ} \) is precisely the definition of a square.
In simple words: Yes, a square can be thought of as a rhombus that has all its angles equal. A rhombus has four equal sides. If you make all its angles equal, they must all be 90 degrees, turning it into a square.

Exam Tip: Understanding the hierarchical relationship between quadrilaterals (e.g., a square is both a rectangle and a rhombus) helps in defining and identifying properties.

 

Question 3. Can a trapezium have all angles equal? Can it have all sides equal? Explain?
Answer: A trapezium (or trapezoid) is defined as a quadrilateral with at least one pair of parallel sides. 1. **Can a trapezium have all angles equal?** Yes, if a trapezium has all angles equal, then each angle must be \( 90^{\circ} \). This means it would be a rectangle. A rectangle is a special type of trapezium because it has two pairs of parallel sides (and thus at least one pair). 2. **Can it have all sides equal?** Yes, a trapezium can have all sides equal. If a trapezium has all its sides equal, it becomes a rhombus. A rhombus is also a special type of trapezium because it has two pairs of parallel sides (and thus at least one pair). If it has all sides equal AND all angles equal, it becomes a square, which is both a rhombus and a rectangle, and thus a trapezium.
In simple words: A trapezium can have all its angles equal, but only if it's a rectangle. It can also have all its sides equal, but then it becomes a rhombus (or a square, if angles are also equal). So, while it's still technically a trapezium, it takes on properties of other shapes.

Exam Tip: Always recall the base definition of a shape. A trapezium only requires *at least one* pair of parallel sides. This broad definition allows other quadrilaterals like rectangles and rhombuses to also be considered trapeziums under certain conditions.