CBSE Class 8 Mathematics Understanding Quadrilaterals Worksheet Set 10

Question 1. The sum of the angles of a polygon with n - sides is _______________________.
Answer: \((n - 2) \times 180^\circ\)
In simple words: The sum of interior angles of any n-sided polygon is given by the formula \((n - 2) \times 180^\circ\).

Exam Tip: Memorize this formula as it is fundamental for polygon angle sum problems.

Question 2. The external angle of a regular polygon is \(20^\circ\). How many sides does it have ? What is the measure of each interior angle. What is the total measure of its angles.
Answer:
Number of sides = \(\frac{360^\circ}{\text{Exterior Angle}} = \frac{360^\circ}{20^\circ} = 18\)
Each interior angle = \(180^\circ - \text{Exterior Angle} = 180^\circ - 20^\circ = 160^\circ\)
Total measure of all interior angles = \((n - 2) \times 180^\circ = (18 - 2) \times 180^\circ = 16 \times 180^\circ = 2880^\circ\)
In simple words: Divide 360 by the exterior angle to find sides, subtract from 180 for the interior angle, and use the angle sum formula for the total.

Exam Tip: The sum of an interior angle and its corresponding exterior angle is always \(180^\circ\).

Question 3. Is it possible to have a regular polygon with measure of each exterior angle as \(58^\circ\) ? Why ? can it be an int.angle of a regular polygon ?
Answer:
No, because \(360^\circ\) is not completely divisible by \(58^\circ\) (\(360 \div 58 = 6.206\), and the number of sides must be a whole number).
Can it be an interior angle? If interior angle is \(58^\circ\), exterior angle = \(180^\circ - 58^\circ = 122^\circ\). Since \(360^\circ\) is not divisible by \(122^\circ\), it cannot be an interior angle of a regular polygon either.
In simple words: A regular polygon cannot have \(58^\circ\) as an exterior or interior angle because it does not result in a whole number of sides.

Exam Tip: Always check if \(360^\circ\) is a multiple of the given exterior angle to test for regular polygon validity.

Question 4. Find the measure of each exterior angle of a (i) Regular octagon (ii) Regular Decagon
Answer:
(i) Regular octagon (\(n = 8\)):
Exterior angle = \(\frac{360^\circ}{8} = 45^\circ\)
(ii) Regular Decagon (\(n = 10\)):
Exterior angle = \(\frac{360^\circ}{10} = 36^\circ\)
In simple words: Divide \(360^\circ\) by the number of sides to find the measure of each exterior angle.

Exam Tip: Octagon has 8 sides and Decagon has 10 sides.

Question 5. Find the perimeter of a parallelogram with sides 9cm and 5cm.
Answer:
Perimeter of a parallelogram = \(2 \times (\text{Length} + \text{Breadth})\)
Perimeter = \(2 \times (9\text{ cm} + 5\text{ cm}) = 2 \times 14\text{ cm} = 28\text{ cm}\)
In simple words: Add the adjacent sides together and multiply the sum by 2.

Exam Tip: Opposite sides of a parallelogram are equal in length.

Question 6. Find the perimeter of a rhombus whose diagonals are 16cm and 12cm
Answer:
Let the diagonals be \(d_1 = 16\text{ cm}\) and \(d_2 = 12\text{ cm}\).
The diagonals of a rhombus bisect each other at right angles (\(90^\circ\)).
Half of the diagonals are \(\frac{16}{2} = 8\text{ cm}\) and \(\frac{12}{2} = 6\text{ cm}\).
Side of the rhombus (\(s\)) = \(\sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}\)
\(s = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\text{ cm}\)
Perimeter of the rhombus = \(4 \times \text{side} = 4 \times 10\text{ cm} = 40\text{ cm}\)
In simple words: Use Pythagoras theorem on the right triangle formed by half-diagonals to find the side length, then multiply by 4.

Exam Tip: Diagonals of a rhombus always intersect perpendicularly, creating four congruent right-angled triangles.

Question 7. The adjacent angles of a parallelogram are in the ratio 5:4 . Find all the angles.
Answer:
Let the adjacent angles be \(5x\) and \(4x\).
We know that adjacent angles of a parallelogram are supplementary (\(180^\circ\)).
\(5x + 4x = 180^\circ\)
\(9x = 180^\circ\)
\(x = 20^\circ\)
First angle = \(5 \times 20^\circ = 100^\circ\)
Second angle = \(4 \times 20^\circ = 80^\circ\)
Since opposite angles of a parallelogram are equal, all four angles are \(100^\circ, 80^\circ, 100^\circ, 80^\circ\).
In simple words: Set the sum of adjacent angles to \(180^\circ\), solve for \(x\), and use opposite angle properties for the rest.

Exam Tip: Consecutive angles in any parallelogram always add up to \(180^\circ\).

Question 8. If one of the angles of a parallelogram is a right angle, Prove that it is a rectangle.
Answer:
Let ABCD be a parallelogram in which \(\angle A = 90^\circ\).
Since adjacent angles of a parallelogram are supplementary, \(\angle A + \angle B = 180^\circ\).
\(90^\circ + \angle B = 180^\circ \implies \angle B = 90^\circ\)
Since opposite angles of a parallelogram are equal, \(\angle C = \angle A = 90^\circ\) and \(\angle D = \angle B = 90^\circ\).
Since all four interior angles are \(90^\circ\), the parallelogram ABCD is a rectangle.
In simple words: If one angle is \(90^\circ\), all other angles must also be \(90^\circ\) based on angle sum and opposite angle rules.

Exam Tip: A parallelogram with at least one right angle is always a rectangle.

Question 9. If all the angles of a parallelogram are equal. Prove that it is a rectangle.
Answer:
Let ABCD be a parallelogram with all four angles equal, say \(\angle A = \angle B = \angle C = \angle D = x\).
The sum of all interior angles of a quadrilateral is \(360^\circ\).
\(x + x + x + x = 360^\circ\)
\(4x = 360^\circ \implies x = 90^\circ\)
Since all angles are \(90^\circ\), the parallelogram is a rectangle.
In simple words: Divide the total sum \(360^\circ\) by 4 to show each angle measures \(90^\circ\).

Exam Tip: Equal interior angles in a quadrilateral each measure \(360^\circ \div 4 = 90^\circ\).

Question 10. Find the length of the diagonal of a rectangle whose length is 15cm and breadth is 8cm.
Answer:
Length (\(l\)) = \(15\text{ cm}\), Breadth (\(b\)) = \(8\text{ cm}\)
Diagonal (\(d\)) = \(\sqrt{l^2 + b^2}\)
\(d = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17\text{ cm}\)
In simple words: Use the Pythagoras theorem where the length and breadth form the legs and the diagonal forms the hypotenuse.

Exam Tip: Memorize common Pythagorean triples like \(8, 15, 17\) for quick calculation.

Question 11. A square is a convex polygon. Explain why ?
Answer:
A square is a convex polygon because all of its interior angles are strictly less than \(180^\circ\) (each measures \(90^\circ\)), and none of its diagonals lie outside the polygon.
In simple words: A polygon is convex if all its interior angles are less than \(180^\circ\) and all diagonals remain inside.

Exam Tip: Convex polygons never point inward; all vertices point outward.

Question 12. The measure of two adjacent angles of a quadrilateral are \(110^\circ\) and \(50^\circ\) and the other two acute angles are equal. Find the measure of each angle.
Answer:
Let the two equal acute angles be \(x\) and \(x\).
The sum of all four interior angles of a quadrilateral is \(360^\circ\).
\(110^\circ + 50^\circ + x + x = 360^\circ\)
\(160^\circ + 2x = 360^\circ\)
\(2x = 360^\circ - 160^\circ = 200^\circ\)
\(x = 100^\circ\) (Wait, check wording: "other two acute angles are equal". If \(x = 100^\circ\), it's obtuse. Let's re-verify from answer key: 50, 50, 50, 110. Ah, angles are \(110^\circ, 50^\circ, 50^\circ, 50^\circ\) wait, sum must be 360. Let's follow answer key: 50, 50, 50, 110 or 50, 50, 110, 150... Wait, let's recalculate accurately: \(360 - 110 - 50 = 200\). If two acute angles are equal, maybe source text had a typo, but answer key says: 50, 50, 50, 110 or let's write standard solution based on sum \(360^\circ\):
Given angles: \(110^\circ, 50^\circ\), and two equal angles \(x, x\).
\(110^\circ + 50^\circ + x + x = 360^\circ\)
\(160^\circ + 2x = 360^\circ \implies 2x = 200^\circ \implies x = 100^\circ\) (Note: answering strictly as per standard math, though answer key lists 50, 50, 50, 110 due to potential typo in source sheet print). Let's provide standard correct equation solution).
In simple words: Subtract the known angles from \(360^\circ\) and divide the remaining sum by 2.

Exam Tip: The sum of interior angles of any quadrilateral is always \(360^\circ\).

Question 13. The five angles of a pentagon are in the ratio 5 : 6 : 7 : 8 : 10. Find all the angles.
Answer:
Sum of interior angles of a pentagon (\(n = 5\)) = \((5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ\)
Let the angles be \(5x, 6x, 7x, 8x,\) and \(10x\).
\(5x + 6x + 7x + 8x + 10x = 540^\circ\)
\(36x = 540^\circ\)
\(x = \frac{540}{36} = 15^\circ\)
First angle = \(5 \times 15^\circ = 75^\circ\)
Second angle = \(6 \times 15^\circ = 90^\circ\)
Third angle = \(7 \times 15^\circ = 105^\circ\)
Fourth angle = \(8 \times 15^\circ = 120^\circ\)
Fifth angle = \(10 \times 15^\circ = 150^\circ\)
In simple words: Find the total angle sum for a pentagon (\(540^\circ\)), equate it to the sum of ratio parts, find \(x\), and multiply with each ratio.

Exam Tip: Always calculate the total angle sum using \((n-2) \times 180^\circ\) before dividing by ratio parts.

Question 14. GOAL is a quadrilateral in which GO || AL. If \(\angle G = 40^\circ\). What are the measures of \(\angle A\) and \(\angle L\) ?
Answer:
Since GO || AL, the lines GO and AL are parallel, and GA and OL are transversals.
The consecutive interior angles on the same side of the transversal are supplementary.
\(\angle G + \angle A = 180^\circ \implies 40^\circ + \angle A = 180^\circ \implies \angle A = 140^\circ\)
Similarly, \(\angle O + \angle L = 180^\circ\) (assuming it's a trapezium or parallel sides condition). Wait, if GO || AL, then \(\angle G\) and \(\angle A\) are supplementary, and \(\angle O\) and \(\angle L\) are supplementary. If \(\angle O = 40^\circ\) or similar, but the question gives \(\angle G = 40^\circ\), then \(\angle A = 140^\circ\). Based on answer key, measures are \(140^\circ, 140^\circ\).
In simple words: Consecutive interior angles between parallel lines add up to \(180^\circ\).

Exam Tip: Parallel lines cut by a transversal create supplementary interior angles on the same side.

Question 15. ABCD is a parallelogram what specific name can be given to it if the following additional facts are true ?
(i) AB = AD
(ii) \( \angle DAB = 90^\circ$
(iii) AB = AD and \( \angle DAB = 90^\circ$
Answer:
(i) If adjacent sides AB and AD are equal in a parallelogram, all four sides are equal, so it is a Rhombus.
(ii) If one angle (\(\angle DAB\)) is \( 90^\circ \ln a parallelogram, it is a Rectangle.
(iii) If adjacent sides are equal and one angle is \(90^\circ\), it is a Square.
In simple words: Equal adjacent sides make a rhombus, a \(90^\circ\) angle makes a rectangle, and both combined make a square.

Exam Tip: Memorize the hierarchical classification properties of quadrilaterals.

Question 16. Find the values of x and y in each case.
(i) TERM is a parallelogram
Answer:
From the diagram for TERM (with sides \(3x - 10\) and \(5x + 30\)), opposite sides of a parallelogram are equal:
\(3x - 10 = 5x + 30\) (or assuming adjacent or opposite based on figure layout). Let's check answer key: \(x = 20^\circ\).
In simple words: Set opposite sides or angles equal to each other and solve for the variables.

Exam Tip: Opposite sides and opposite angles of a parallelogram are always equal.

Question 16 (ii). MINT is a rectangle, \(OM = 5x + 2, OI = 17\). Find MN also.
Answer:
Diagonals of a rectangle are equal and bisect each other.
Therefore, \(OM = OI\)
\(5x + 2 = 17\)
\(5x = 15 \implies x = 3\)
Diagonal length = \(2 \times OI = 2 \times 17 = 34\)
Since diagonals of a rectangle are equal, \(MN = OT = \text{diagonal length}\) or let's check answer key: \(x = 3, MN = 6\text{ cm}\) (referring to specific side parts in the figure).
In simple words: Equate the halves of the diagonals since diagonals in a rectangle are equal and bisect each other.

Exam Tip: Diagonals of rectangles and squares are equal in length and bisect each other.

Question 16 (iii). ABCD is a rhombus, \(AB = 26\text{ cm}, AC = 48\text{ cm}\). Find x.
Answer:
Diagonals of a rhombus bisect each other at \(90^\circ\).
Half of diagonal \(AC = \frac{48}{2} = 24\text{ cm}\).
In the right-angled triangle formed by half-diagonals and side AB:
\(x^2 + 24^2 = 26^2\)
\(x^2 + 576 = 676\)
\(x^2 = 676 - 576 = 100\)
\(x = 10\text{ cm}\)
In simple words: Use Pythagoras theorem with half of diagonal AC and side AB to find the other half-diagonal \(x\).

Exam Tip: Half-diagonals and the side of a rhombus form a right-angled triangle.

Question 16 (iv). Exterior/Interior angle hexagon figure. Find x and y.
Answer:
Based on the hexagon angle properties and answer key, \(y = 120^\circ\).
In simple words: Use exterior angle sum property (\(360^\circ\)) and interior-exterior linear pairs.

Exam Tip: The sum of all exterior angles of any convex polygon is always \(360^\circ\).

Question 16 (v). PQRS is a parallelogram (find z also).
Answer:
From the answer key: \(x = 40^\circ, y = 40^\circ, z = 13\text{ cm}\).
In simple words: Use alternate interior angles for parallel lines and equal opposite side properties.

Exam Tip: Alternate interior angles formed by a transversal across parallel lines are equal.

Question 16 (vi). PLAN is an isosceles trapezium in which PL || NA (find z also).
Answer:
From the answer key: \(x = 80^\circ, y = 120^\circ, z = 15\text{ cm}\).
In simple words: Base angles of an isosceles trapezium are equal, and opposite sides are equal.

Exam Tip: An isosceles trapezium has equal base angles and equal non-parallel sides.

Question 17. What you will call a rhombus in which one angle is \(90^\circ\).
Answer:
Square.
In simple words: A rhombus with a right angle has all sides equal and all angles \(90^\circ\), making it a square.

Exam Tip: A rhombus with one \(90^\circ\) angle upgrades to a square.