Class 9 Maths Chapter 8 Question Answers - Quadrilaterals

Q1: What is a quadrilateral? Mention 6 types of quadrilaterals.
Sol: A quadrilateral is a 4 sided polygon having a closed shape. It is a 2-dimensional shape.
The 6 types of quadrilaterals include:

• Rectangle
• Square
• Parallelogram
• Rhombus
• Trapezium
• Kite

Q2: Identify the type of quadrilaterals:
(i) The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are perpendicular.
(ii) The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are congruent.
Sol: (i) The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are perpendicular is a rectangle.
(ii) The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are congruent is a rhombus.

Q3: In a rectangle, one diagonal is inclined to one of its sides at 25°. Measure the acute angle between the two diagonals.
Sol: Let ABCD be a rectangle where AC and BD are the two diagonals which are intersecting at point O.
Now, assume ∠BDC = 25° (given)
Now, ∠BDA = 90° – 25° = 65°
Also, ∠DAC = ∠BDA, (as diagonals of a rectangle divide the rectangle into two congruent right triangles)
So, ∠BOA = the acute angle between the two diagonals = 180° – 65° – 65° = 50°

Q4: Prove that the angle bisectors of a parallelogram form a rectangle.
Sol: LMNO is a parallelogram in which bisectors of the angles L, M, N, and O intersect at P, Q, R and S to form the quadrilateral PQRS.
LM || NO (opposite sides of parallelogram LMNO)
L + M = 180 (sum of consecutive interior angles is 180o)
MLS + LMS = 90
In LMS, MLS + LMS + LSM = 180
90 + LSM = 180
LSM = 90
RSP = 90 (vertically opposite angles)
SRQ = 90, RQP = 90 and SPQ = 90
Therefore, PQRS is a rectangle. 

Q5: Calculate all the angles of a parallelogram if one of its angles is twice its adjacent angle.
Sol: Let the angle of the parallelogram given in the question statement be “x”.
Now, its adjacent angle will be 2x.
It is known that the opposite angles of a parallelogram are equal.
So, all the angles of a parallelogram will be x, 2x, x, and 2x
As the sum of interior angles of a parallelogram = 360°,
x + 2x + x + 2x = 360°
Or, x = 60°
Thus, all the angles will be 60°, 120°, 60°, and 120°.

Q6: The diagonals of which quadrilateral are equal and bisect each other at 90°?
Sol: Square. The diagonals of a square are equal and bisect each other at 90°.

Q7: Find all the angles of a parallelogram if one angle is 80°.
Sol: For a parallelogram ABCD, opposite angles are equal.
So, the angles opposite to the given 80° angle will also be 80°.
It is also known that the sum of angles of any quadrilateral = 360°.
So, if ∠A = ∠C = 80° then,
∠A + ∠B + ∠C + ∠D = 360°
Also, ∠B = ∠D
Thus,
80° + ∠B + 80° + ∠D = 360°
Or, ∠B +∠ D = 200°
Hence, ∠B = ∠D = 100°
Now, all angles of the quadrilateral are found which are:
∠A = 80°
∠B = 100°
∠C = 80°
∠D = 100°

Q8: Is it possible to draw a quadrilateral whose all angles are obtuse angles?
Sol: It is known that the sum of angles of a quadrilateral is always 360°. To have all angles as obtuse, the angles of the quadrilateral will be greater than 360°. So, it is not possible to draw a quadrilateral whose all angles are obtuse angles.

Q9: In a trapezium ABCD, AB∥CD. Calculate ∠C and ∠D if ∠A = 55° and ∠B = 70°
Sol: In a trapezium ABCD, ∠A + ∠D = 180° and ∠B + ∠C = 180°
So, 55° + ∠D = 180°
Or, ∠D = 125°
Similarly,
70° + ∠C = 180°
Or, ∠C = 110°

Q10: Calculate all the angles of a quadrilateral if they are in the ratio 2:5:4:1.
Sol: As the angles are in the ratio 2:5:4:1, they can be written as-
2x, 5x, 4x, and x
Now, as the sum of the angles of a quadrilateral is 360°,
2x + 5x + 4x + x = 360°
Or, x = 30°
Now, all the angles will be,
2x =2 × 30° = 60°
5x = 5 × 30° = 150°
4x = 4 × 30° = 120°, and
x = 30°