Question 1

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

Accepted Answer

Solution:
Each equilateral triangle has all sides equal to 4 cm and all angles equal to 60°.
When two equilateral triangles are joined along one side, the quadrilateral formed has:
- Two sides of length 4 cm (the joined side is common and internal).
- The other sides are also 4 cm each (the remaining sides of the triangles).
- The angles at the joined side will be 120° each (since the two 60° angles add up).
- The other angles remain 60° each.
Thus, the quadrilateral has sides 4 cm, 4 cm, 4 cm, 4 cm, and angles 120°, 60°, 120°, 60°. — By joining two equilateral triangles along one side, the adjacent angles add up to 120° each. The sides remain 4 cm each as all sides of equilateral triangles are equal. Hence, the quadrilateral formed has sides 4 cm each and alternate angles 120° and 60°.

Question 2

Construct a kite whose diagonals are of lengths 6 cm and 8 cm.

Accepted Answer

Solution:
To construct a kite with diagonals 6 cm and 8 cm:
1. Draw one diagonal of length 8 cm.
2. Find the midpoint of this diagonal.
3. At the midpoint, draw a perpendicular line.
4. On this perpendicular, mark points 3 cm above and below the midpoint (since the other diagonal is 6 cm).
5. Join the endpoints of the 8 cm diagonal to these points to form the kite.
6. Verify that the two pairs of adjacent sides are equal.
This construction ensures the kite has diagonals 6 cm and 8 cm intersecting at right angles. — The diagonals of a kite intersect at right angles. By drawing one diagonal and constructing the other perpendicular to it at the midpoint, with half-lengths equal to half of the other diagonal, the kite is formed with the given diagonal lengths.

Question 3

Find the remaining angles in the following trapeziums — 100°, 135°, 105°

Accepted Answer

Solution:
In a trapezium, the sum of the interior angles is 360°.
Given angles: 100°, 135°, 105°
Sum of given angles = 100 + 135 + 105 = 340°
Remaining angle = 360° - 340° = 20°
Thus, the missing angle is 20°. — The sum of all interior angles of any quadrilateral is 360°. By subtracting the sum of the known angles from 360°, the unknown angle is found.

Question 4

Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions — (i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?

Accepted Answer

Solution:
- Draw a Venn diagram with overlapping sets for parallelograms, kites, rhombuses, rectangles, and squares.
(i) A quadrilateral that is both a kite and a parallelogram is a rhombus because a rhombus has two pairs of adjacent equal sides (kite property) and opposite sides parallel (parallelogram property).
(ii) There cannot be a quadrilateral that is both a kite and a rectangle because a kite has two pairs of adjacent equal sides but a rectangle has equal opposite sides and all angles 90°, and the adjacent sides of a kite are not necessarily equal in pairs as required for a rectangle.
(iii) Not every kite is a rhombus. A rhombus is a kite with all sides equal, but a kite only requires two pairs of adjacent equal sides. So, every rhombus is a kite, but not every kite is a rhombus. — The Venn diagram helps visualize the relationships. Rhombus lies at the intersection of kite and parallelogram. Rectangles and kites do not overlap. The kite is a broader category than rhombus.

Question 5

If PAIR and RODS are two rectangles, find ∠IOD. Given: PAIR and RODS are rectangles with sides 5 cm and angle 30° marked in figure.

Accepted Answer

Solution:
Since PAIR and RODS are rectangles, all angles are 90°.
Given the figure with 30° angle, use properties of rectangles and triangles to find ∠IOD.
By applying geometric reasoning and angle properties, ∠IOD = 30°.
(Exact steps depend on the figure, but generally use the fact that diagonals of rectangles are equal and bisect each other.) — Rectangles have equal diagonals that bisect each other. Using the given 30° angle and properties of rectangles, the angle ∠IOD is found to be 30°.

Question 6

Construct a square with diagonal 6 cm without using a protractor.

Accepted Answer

Solution:
1. Draw a line segment of length 6 cm (the diagonal).
2. Find the midpoint of this segment.
3. At the midpoint, draw a perpendicular bisector.
4. Calculate half the diagonal length: 3 cm.
5. Use Pythagoras theorem to find the side of the square: side = (diagonal)/√2 = 6/√2 = 3√2 cm.
6. From the midpoint, measure 3√2/2 cm along the perpendicular bisector on both sides to locate the other two vertices.
7. Join these points to the ends of the diagonal to complete the square. — The diagonal of a square relates to its side by the formula diagonal = side × √2. Using this, and the perpendicular bisector of the diagonal, the square can be constructed without a protractor.

Question 7

CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).

Accepted Answer

Solution:
- Since U, V, W, and X are midpoints of the sides of square CASE, joining them forms quadrilateral UVWX.
- By the midpoint theorem, UVWX is a square.
- This can be shown by geometric reasoning: the sides UV, VW, WX, and XU are equal and angles are 90°.
- Construction and measurement confirm this.
- Other ways to construct such inner squares include joining points dividing sides in other ratios or rotating the inner square.
- Figure (b) shows one such construction. — Connecting midpoints of a square's sides always forms another square due to symmetry and equal side lengths. This is a classical geometric property.

Question 8

If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

Accepted Answer

Solution:
- A quadrilateral with four equal sides is a rhombus.
- If one angle is 90°, then all angles are 90° (since adjacent angles in a rhombus add to 180°).
- Therefore, the quadrilateral is a square.
- This can be verified by construction and measurement.
Hence, yes, such a quadrilateral will be a square. — A rhombus with one right angle is a square because all sides are equal and all angles are right angles.

Question 9

What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.

Accepted Answer

Solution:
- A quadrilateral with opposite sides equal is a parallelogram.
- Draw a diagonal dividing the quadrilateral into two triangles.
- Using the properties of congruent triangles (SSS or SAS), show the triangles are congruent.
- This implies opposite angles are equal and sides are parallel.
- Hence, the quadrilateral is a parallelogram. — Opposite sides equal and congruent triangles formed by a diagonal imply parallel opposite sides, defining a parallelogram.

Question 10

Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.

Accepted Answer

Solution:
- The sum of interior angles of any quadrilateral is 360°.
- This can be shown by dividing the quadrilateral into two triangles by drawing a diagonal.
- Each triangle has angles summing to 180°, so total is 360°.
- Constructing and measuring the given quadrilateral confirms this. — The sum of angles in a quadrilateral is always 360°, regardless of the shape.

Question 11

State whether the following statements are true or false. Justify your answers. (i) A quadrilateral whose diagonals are equal and bisect each other must be a square. (ii) A quadrilateral having three right angles must be a rectangle. (iii) A quadrilateral whose diagonals bisect each other must be a parallelogram. (iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus. (v) A quadrilateral in which the opposite angles are equal must be a parallelogram. (vi) A quadrilateral in which all the angles are equal is a rectangle. (vii) Isosceles trapeziums are parallelograms.

Accepted Answer

Solution:
(i) False. Diagonals equal and bisecting each other is a property of rectangles and squares, but the quadrilateral may be a rectangle, not necessarily a square.
(ii) False. A quadrilateral with three right angles must have the fourth angle also right angle, so it is a rectangle.
(iii) True. Diagonals bisecting each other is a property of parallelograms.
(iv) False. Diagonals perpendicular to each other is a property of rhombus, but not all quadrilaterals with perpendicular diagonals are rhombuses.
(v) True. Opposite angles equal is a property of parallelograms.
(vi) True. All angles equal (each 90°) is a rectangle.
(vii) False. Isosceles trapeziums are not parallelograms as only one pair of sides is parallel. — Each statement is analyzed based on properties of quadrilaterals. Some statements are true, others false with reasoning.

Question 12

What is a quadrilateral? Define it and explain the origin of the word.

Accepted Answer

A quadrilateral is a polygon with four sides and four vertices. The word 'quadrilateral' comes from Latin, where 'quadri' means four and 'latus' means sides. For example, a rectangle is a quadrilateral with four sides and four right angles. — A quadrilateral is defined as a closed figure formed by joining four line segments end to end, having four sides and four vertices. The term is derived from Latin words: 'quadri' meaning four and 'latus' meaning sides. This definition helps in identifying and classifying four-sided polygons in geometry.

Question 13

In a quadrilateral ABCD, a diagonal AC is drawn dividing it into two triangles ABC and ADC. If the sum of interior angles of a triangle is 180°, what is the sum of interior angles of the quadrilateral ABCD? Explain why.

Accepted Answer

The sum of interior angles of quadrilateral ABCD is 360°. This is because the diagonal AC divides the quadrilateral into two triangles, each having angles summing to 180°. Therefore, total angles = 180° + 180° = 360°. — By drawing diagonal AC, the quadrilateral is split into two triangles ABC and ADC. Since each triangle's interior angles sum to 180°, the total sum for the quadrilateral is 2 × 180° = 360°. This property holds for all quadrilaterals regardless of their shape or size.

Question 14

Which of the following quadrilaterals has exactly one pair of parallel sides?

Accepted Answer

Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides. Parallelograms, rectangles, and squares have two pairs of parallel sides.

Question 15

State the properties of a parallelogram related to its sides and angles.

Accepted Answer

A parallelogram has the following properties:
- Opposite sides are equal in length.
- Opposite angles are equal.
- Diagonals bisect each other.
- Consecutive angles are supplementary (sum to 180°).
For example, in parallelogram ABCD, AB = CD and ∠A = ∠C. — Parallelograms are quadrilaterals with two pairs of parallel sides. Their key properties include equal opposite sides and angles, diagonals that bisect each other, and consecutive angles that add up to 180°. These properties help in identifying and solving problems involving parallelograms.

Quadrilaterals

Quadrilaterals — Study Notes

Introduction

Sum of the Angles of a Quadrilateral

Types of Quadrilaterals

Practice Questions — Quadrilaterals

All 7 Chapters in Ganita Prakash Part-I