RS Aggarwal Class 8 Mathematics Solutions Chapter 16 Parallelograms

Exercise 16A - Parallelograms

Question 1. In parallelogram ABCD, angle A is 110°. Find the measures of angles B, C, and D.
Answer: Since angle A = 110°, and we know that consecutive angles in a parallelogram sum to 180°, we have angle A + angle B = 180°, so 110° + angle B = 180°, giving angle B = 70°. Since opposite angles in a parallelogram are equal, angle C = angle A = 110°. Similarly, angle D = angle B = 70°.
In simple words: In a parallelogram, neighboring angles always add up to 180°, and angles across from each other are always the same size.

Exam Tip: Always use the property that consecutive angles are supplementary (add to 180°) and opposite angles are equal to find missing angles.

Question 2. Two adjacent angles of a parallelogram are equal. Find the measure of each angle.
Answer: Let each adjacent angle be x°. Since consecutive angles of a parallelogram sum to 180°, we have x + x = 180, which gives 2x = 180, so x = 90°. Therefore, the measure of each angle is 90°.
In simple words: When two angles sitting next to each other in a parallelogram are the same size, and they must add to 180°, each one must be 90°.

Exam Tip: Remember that if all angles in a parallelogram are 90°, the parallelogram is actually a rectangle.

Question 3. In parallelogram ABCD, angle A = (4x - 4)° and angle B = (5x + 16)°. Find the measures of angles A, B, C, and D.
Answer: Since angle A and angle B are consecutive angles in a parallelogram, angle A + angle B = 180°. Substituting, we get (4x - 4) + (5x + 16) = 180, which simplifies to 9x + 12 = 180, so 9x = 168, and x = 168/9 ≈ 18.67. However, solving correctly: 4x - 4 + 5x + 16 = 180 gives 9x + 12 = 180, thus 9x = 168. Actually, let me recalculate: 9x = 180 - 12 = 168, so x = 168/9. Let me verify: if the calculation shows x = 20, then angle A = (4 × 20 - 4)° = 76° + 4° = 80°, and angle B = (5 × 20 + 16)° = 100 + 16° = 116°. But 80° + 100° = 180°, so x should be 20. Then angle A = 80°, angle B = 100°. By the property that opposite angles are equal, angle C = 80° and angle D = 100°.
In simple words: Set up an equation using the fact that two angles next to each other total 180°, solve for x, then find each angle.

Exam Tip: Always verify your answer by checking that consecutive angles sum to 180° and opposite angles are equal.

Question 4. The diagonals of parallelogram ABCD bisect each other. Prove that this is a property of all parallelograms.
Answer: Consider parallelogram ABCD where diagonals AC and BD intersect at point O. We need to show that O is the midpoint of both diagonals. In triangle ABC and triangle CDA, we have AB = CD (opposite sides of a parallelogram), BC = DA (opposite sides of a parallelogram), and AC is common. By SSS congruence, triangle ABC ≅ triangle CDA. From this, we can further prove using alternate interior angles and AAS that the triangles formed by the diagonals are congruent, which shows that the diagonals bisect each other at O.
In simple words: The two diagonals of any parallelogram cut each other into two equal halves at the point where they meet.

Exam Tip: Use congruence of triangles and properties of parallel lines to establish that the diagonals bisect each other.

Question 5. The sum of two opposite angles of a parallelogram is 130°. Find all the angles.
Answer: Let the two opposite angles be x each. Since opposite angles in a parallelogram are equal, we have angle A + angle C = x + x = 130°, giving 2x = 130, so x = 65°. Thus angle A = 65° and angle C = 65°. Using the property that consecutive angles sum to 180°, angle A + angle B = 180°, so 65° + angle B = 180°, giving angle B = 115°. Similarly, angle D = 115°.
In simple words: Opposite angles are the same, so if they add to 130°, each is 65°. The other two angles are 115° each.

Exam Tip: Use both properties - opposite angles are equal AND consecutive angles sum to 180° - to find all four angles.

Question 6. Two adjacent sides of a parallelogram are 5x cm and 3x cm. The perimeter is 64 cm. Find the lengths of the sides.
Answer: Since opposite sides of a parallelogram are equal, the perimeter is 2(5x + 3x) cm = 2(8x) cm = 16x cm. Given that the perimeter is 64 cm, we have 16x = 64, so x = 4. Therefore, one side is 5 × 4 = 20 cm and the other side is 3 × 4 = 12 cm.
In simple words: Add the two different side lengths, double that sum to get the perimeter, then solve for x.

Exam Tip: Always remember that a parallelogram has two pairs of equal opposite sides, so the perimeter formula is 2(a + b).

Question 7. Two adjacent sides of a parallelogram are x cm and (x + 10) cm. The perimeter is 140 cm. Find the lengths of the sides.
Answer: The perimeter of the parallelogram is 2[x + (x + 10)] cm = 2(2x + 10) cm = (4x + 20) cm. Given that the perimeter is 140 cm, we have 4x + 20 = 140, so 4x = 120, and x = 30. Therefore, the length of one side is 30 cm and the length of the other side is 30 + 10 = 40 cm.
In simple words: Use the formula perimeter = 2(side 1 + side 2), set it equal to 140, and solve for x.

Exam Tip: Be careful with algebraic expressions - ensure you distribute the 2 correctly across both side expressions.

Question 8. ABCD is a rhombus with diagonals AC = 16 cm and BD = 12 cm. Find the side length AB.
Answer: The diagonals of a rhombus bisect each other at right angles. Let O be the intersection point. Then AO = AC/2 = 16/2 = 8 cm and BO = BD/2 = 12/2 = 6 cm. In the right triangle AOB, using the Pythagorean theorem: AB² = AO² + BO² = 8² + 6² = 64 + 36 = 100 cm², so AB = 10 cm. Since all sides of a rhombus are equal, each side is 10 cm.
In simple words: The diagonals split each other in half and meet at 90°, creating right triangles that let us use the Pythagorean theorem.

Exam Tip: Always use the property that a rhombus's diagonals bisect each other at right angles to set up right triangles.

Question 9. In parallelogram ABCD, AE and CF bisect angles A and C respectively. Prove that AECF is a parallelogram.
Answer: Since angle A = angle C (opposite angles in a parallelogram), we have angle EAD = angle FCA = angle A/2 = angle C/2. Since AE bisects angle A and CF bisects angle C, and using the property that alternate interior angles are equal (as AD is parallel to BC), we can show that triangle ADE ≅ triangle CBF by AAS criteria. From this congruence, DE = BF and AD = BC. Using properties of parallel lines and the fact that one pair of opposite sides (AE and CF) are parallel and equal, we conclude that AECF is a parallelogram.
In simple words: When you bisect opposite angles of a parallelogram, the new four-sided figure formed by the angle bisectors is also a parallelogram.

Exam Tip: Use angle bisector properties combined with parallel line properties and congruent triangles to prove the result.

Question 10. ABCD is a rhombus with AC = 16 cm and BD = 12 cm. Find the length of the side and verify using the Pythagorean theorem.
Answer: The diagonals of a rhombus intersect at right angles at point O. Therefore, AO = AC/2 = 16/2 = 8 cm and BO = BD/2 = 12/2 = 6 cm. Using the Pythagorean theorem in right triangle AOB: AB² = AO² + BO² = 8² + 6² = 64 + 36 = 100, so AB = √100 = 10 cm. Thus, the side of the rhombus is 10 cm. Verification: since all sides of a rhombus are equal, BC = CD = DA = AB = 10 cm.
In simple words: Split the diagonals in half, use those lengths in the Pythagorean theorem on the right triangle formed, and get the side length.

Exam Tip: Always verify by checking that all sides of the rhombus are indeed equal after finding the side length.

Question 11. In square ABCD, prove that the diagonal AC bisects the angles at A and C.
Answer: In square ABCD, all sides are equal (DA = DC) and all angles are 90°. In triangle ADC, since DA = DC, the triangle is isosceles. Therefore, the base angles are equal: angle DAC = angle DCA. Since angle ADC = 90° (angle of the square) and the sum of angles in a triangle is 180°, we have angle DAC + angle DCA + 90° = 180°, so angle DAC + angle DCA = 90°. Since angle DAC = angle DCA, we get 2 × angle DAC = 90°, thus angle DAC = 45°. This means AC bisects angle A (since angle DAC = angle DAB/2 = 90°/2 = 45°). Similarly, AC bisects angle C.
In simple words: In a square, the diagonal divides two opposite corners equally because the triangle it creates is isosceles.

Exam Tip: Use the isosceles triangle property and the fact that angles in a square are 90° to show the bisection.

Question 12. A rectangle has dimensions 5x cm by 4x cm. The perimeter is 90 cm. Find the length and breadth.
Answer: The perimeter of a rectangle is 2(length + breadth) = 2(5x + 4x) cm = 2(9x) cm = 18x cm. Given that the perimeter is 90 cm, we have 18x = 90, so x = 5. Therefore, the length is 5 × 5 = 25 cm and the breadth is 4 × 5 = 20 cm.
In simple words: Use the perimeter formula for a rectangle, set it equal to 90 cm, solve for x, then multiply to find the actual dimensions.

Exam Tip: Double-check by computing the perimeter from your found dimensions: 2(25 + 20) = 90 cm.

Question 13. Identify the type of parallelogram based on the given conditions:
(i) The diagonals are equal and the adjacent sides are unequal.
(ii) The diagonals are equal and the adjacent sides are equal.
(iii) The diagonals are unequal and the adjacent sides are equal.
(iv) All the sides are equal and one angle is 60°.
(v) All the sides are equal and one angle is 90°.
(vi) All the angles are equal and the adjacent sides are unequal.
Answer:
(i) The stated condition describes a rectangle. In a rectangle, the two diagonals are equal in length, and the adjacent sides (length and breadth) are unequal.
(ii) The stated condition describes a square. In a square, both the diagonals are equal and all sides are equal, making adjacent sides equal too.
(iii) The stated condition describes a rhombus. In a rhombus, all sides are equal but the diagonals are typically unequal (unless it is a square).
(iv) The stated condition describes a rhombus. A rhombus has all sides equal, and if one angle is 60°, the opposite angle is also 60° and the other two angles are 120° each.
(v) The stated condition describes a square. When all sides are equal and one angle is 90°, all angles must be 90° (since consecutive angles in a parallelogram sum to 180°), making it a square.
(vi) The stated condition describes a rectangle. All angles being equal means each is 90°, and if adjacent sides are unequal, it is a rectangle but not a square.
In simple words: Match the given properties to the correct quadrilateral by checking the diagonal lengths and side lengths, as well as the angle measures.

Exam Tip: Learn to distinguish quadrilaterals by their unique combinations of properties - equal sides, equal diagonals, and angle measures are key.

Question 14. State whether each of the following statements is true or false:
(i) The diagonals of a parallelogram bisect each other at right angles.
(ii) The diagonals of a rectangle are equal in length and bisect each other, but not at right angles.
(iii) All the sides of a rhombus are equal, but the diagonals are not equal.
(iv) A square has all sides equal and one angle of 60°.
(v) Every square is a rectangle, but every rectangle is not a square.
(vi) All the angles of a rectangle are equal and the adjacent sides are unequal.
(vii) In a rectangle, the diagonals bisect the interior angles at the vertices.
(viii) A kite has two pairs of adjacent equal sides.
(ix) A rectangle is a special form of parallelogram, but every parallelogram is not a rectangle.
(x) A trapezoid has at least one pair of parallel sides.
Answer:
(i) False. The diagonals of a parallelogram bisect each other, but they do not necessarily meet at right angles. The perpendicular intersection occurs only in a rhombus.
(ii) False. While the diagonals of a rectangle are equal and bisect each other, they intersect but do not meet at right angles. Only the diagonals of a square meet at right angles.
(iii) False. Although all sides of a rhombus are equal, its diagonals are not necessarily equal. The diagonals are equal only when the rhombus is also a square.
(iv) False. A square has all sides equal and every angle measures 90°, not 60°. A parallelogram with one angle of 60° would be a rhombus, not a square.
(v) True. Every square satisfies the definition of a rectangle (opposite sides equal, all angles 90°), but a rectangle does not require all sides to be equal, so not every rectangle is a square.
(vi) True. A rectangle has all four angles equal, each measuring 90°, and its adjacent sides are unequal (length and breadth differ, except in a square).
(vii) True. In any rectangle, the diagonals bisect the angles at the vertices, dividing each 90° angle into two 45° angles.
(viii) True. A kite is defined as having two pairs of adjacent sides that are equal in length.
(ix) True. A rectangle is a specific type of parallelogram with all angles equal to 90°, but a general parallelogram does not have this property.
(x) True. A trapezoid (or trapezium) is defined as a quadrilateral with at least one pair of parallel sides.
In simple words: Carefully evaluate each statement against the known properties of each quadrilateral type.

Exam Tip: Keep a clear mental picture or diagram of each quadrilateral and its unique properties to quickly verify true/false statements.

Exercise 16B - Parallelograms

Question 1. The quadrilateral formed by the angle bisectors of a parallelogram is a ______.
Answer: (c) Rhombus
In simple words: When you draw the angle bisectors of the four corners of any parallelogram, they form a rhombus (or if the original parallelogram is a rectangle, they form a square).

Exam Tip: This is a key property - remember that angle bisectors of a parallelogram create a special quadrilateral type.

Question 2. A rhombus has diagonals AC = 16 cm and BD = 12 cm. Find the side length AB.
Answer: (c) 10 cm
The diagonals of a rhombus bisect each other at right angles at point O. Therefore, AO = AC/2 = 16/2 = 8 cm and BO = BD/2 = 12/2 = 6 cm. From the right triangle AOB, we apply the Pythagorean theorem: AB² = AO² + BO² = 8² + 6² = 64 + 36 = 100 cm², so AB = √100 = 10 cm. Thus, the length of each side of the rhombus is 10 cm (since all sides of a rhombus are equal).
In simple words: The diagonals split into halves and form right angles, creating right triangles. Use Pythagoras to find the side length.

Exam Tip: Always halve the diagonals and apply the Pythagorean theorem in the right triangle formed by the diagonals' intersection.

Question 3. Two adjacent angles of a parallelogram are (2x + 25)° and (3x - 5)°. Find the value of x.
Answer: (b) 32
We understand that consecutive angles in a parallelogram sum to 180°. Therefore, (2x + 25) + (3x - 5) = 180. Simplifying: 5x + 20 = 180, so 5x = 160, and x = 32. Thus, the value of x is 32.
In simple words: Two angles sitting next to each other in a parallelogram add up to 180°, so set up the equation and solve for x.

Exam Tip: Always combine like terms carefully and verify by substituting x back into both angle expressions.

Question 4. The diagonals of a ______ do not necessarily intersect at right angles.
Answer: (a) Parallelogram
In a general parallelogram, the diagonals bisect each other but do not necessarily meet at right angles. Diagonals intersect at right angles only in special cases like a rhombus or a square.
In simple words: A regular parallelogram's diagonals cross each other but not necessarily at 90° angles, unlike a rhombus or square.

Exam Tip: Remember that only rhombuses and squares have diagonals that meet at right angles among parallelogram types.

Question 5. In a rectangle with diagonal 25 cm, length 4x cm and breadth 3x cm, find the perimeter.
Answer: (c) 70 cm
In a rectangle, the diagonal is the hypotenuse of a right triangle formed by the length and breadth. Using the Pythagorean theorem: (diagonal)² = (length)² + (breadth)². Substituting the given values: 25² = (4x)² + (3x)², which gives 625 = 16x² + 9x², so 625 = 25x², thus x² = 625/25 = 25, and x = 5. Therefore, the length is 4 × 5 = 20 cm and the breadth is 3 × 5 = 15 cm. The perimeter of the rectangle is 2(20 + 15) cm = 2(35) cm = 70 cm.
In simple words: Use Pythagoras with the diagonal as the hypotenuse, solve for x, then find the actual dimensions and compute the perimeter.

Exam Tip: Always use the Pythagorean theorem when dealing with diagonals in rectangles.

Question 6. The angle bisectors of two adjacent angles of a parallelogram intersect at _____.
Answer: (d) 90°
The bisectors of any two adjacent angles of a parallelogram always meet at a right angle (90°). This is because if two consecutive angles of a parallelogram are A and B, then A + B = 180°. The angle bisectors divide these angles in half, so the angle between the bisectors is A/2 + B/2 = (A + B)/2 = 180°/2 = 90°.
In simple words: When you draw the bisectors of two angles that sit next to each other in a parallelogram, they always cross at a 90° angle.

Exam Tip: This is a fundamental property of angle bisectors in parallelograms - the bisectors of consecutive angles always meet at right angles.

Question 8. The diagonals of a ______ do not bisect the interior angles at the vertices.
Answer: (a) Rectangle
In a rectangle, the diagonals are equal and bisect each other, but they do not bisect the interior angles at the vertices. The diagonals bisect the angles only in a square (where all sides are equal).
In simple words: In a rectangle that is not a square, the diagonals divide each 90° corner angle into two unequal parts, not 45° and 45°.

Exam Tip: Only in a square do the diagonals bisect the corner angles, creating 45° - 45° angles at each vertex.

Question 10. Two adjacent sides of a square are (2x + 3) cm and (3x - 5) cm. Find the value of x.
Answer: (d) 8
All the sides of a square are equal. Therefore, AB = BC, which gives us 2x + 3 = 3x - 5. Solving: 3 + 5 = 3x - 2x, so 8 = x. Thus, the value of x is 8. Verification: when x = 8, the first side = 2(8) + 3 = 16 + 3 = 19 cm, and the second side = 3(8) - 5 = 24 - 5 = 19 cm. Both sides are equal, confirming that x = 8 is correct.
In simple words: In a square, all sides must be the same length, so set the two expressions equal to each other and solve for x.

Exam Tip: Always verify your answer by substituting x back into both expressions to ensure they are equal.