CBSE Class 8 Maths Exemplar Chapter 5: Understanding Quadrilaterals NCERT Solutions

Let the four angles of the quadrilateral be $A$, $B$, $C$, and $D$. We are given that three of these angles are equal to $75^{\circ}$. Let $A = B = C = 75^{\circ}$.

The sum of the interior angles of any quadrilateral is always $360^{\circ}$. Therefore, we can write the equation:

$$A\; +\; B\; +\; C\; +\; D\; =\; 360^\{\backslash circ\}$$

Substituting the given values:

$$75^\{\backslash circ\}\; +\; 75^\{\backslash circ\}\; +\; 75^\{\backslash circ\}\; +\; D\; =\; 360^\{\backslash circ\}$$

First, sum the three known angles:

$$225^\{\backslash circ\}\; +\; D\; =\; 360^\{\backslash circ\}$$

To find the measure of the fourth angle, $D$, subtract $225^{\circ}$ from $360^{\circ}$:

$$D\; =\; 360^\{\backslash circ\}\; -\; 225^\{\backslash circ\}$$

$$D\; =\; 135^\{\backslash circ\}$$

Thus, the fourth angle of the quadrilateral is $135^{\circ}$.

The correct option is (B).

Let's examine the properties of the diagonals for each figure:

• Square: The diagonals of a square bisect each other and are equal and perpendicular.
• Kite: The diagonals of a kite are perpendicular, but only one diagonal is bisected by the other.
• Trapezium: In a general trapezium, the diagonals do not necessarily bisect each other.
• Quadrilateral: This is a general term. For a general quadrilateral, there is no guarantee that the diagonals bisect each other.

Based on these properties, the figure for which the diagonals bisect each other is a square.

The correct option is (A).

Let's consider the angles of each figure:

• Rectangle: All four interior angles of a rectangle are right angles, meaning they are all equal to $90^{\circ}$.
• Kite: A kite has two pairs of equal adjacent sides, but its angles are not necessarily equal. Typically, one pair of opposite angles is equal.
• Trapezium: In a trapezium, only one pair of opposite sides is parallel. The angles are not necessarily equal.
• Rhombus: A rhombus has all four sides equal. Opposite angles are equal, but all four angles are not necessarily equal (unless it is also a square).

Therefore, the figure for which all angles are equal is a rectangle.

The correct option is (A).

Let's review the properties of diagonals for each figure:

• Parallelogram: Diagonals bisect each other but are not necessarily perpendicular.
• Kite: The diagonals of a kite are perpendicular to each other.
• Trapezium: Diagonals are not necessarily perpendicular.
• Rectangle: Diagonals are equal and bisect each other, but are not necessarily perpendicular.

Thus, the figure whose diagonals are perpendicular to each other is a kite.

The correct option is (B).

We examine the lengths of the diagonals for each shape:

• Trapezium: Diagonals are not necessarily equal.
• Rhombus: Diagonals bisect each other at right angles but are not necessarily equal (unless it is a square).
• Parallelogram: Diagonals bisect each other but are not necessarily equal.
• Rectangle: The diagonals of a rectangle are always equal in length.

Therefore, the figure for which diagonals are equal is a rectangle.

The correct option is (D).

Let's analyze the given properties:

• All sides are congruent: This property is true for squares and rhombuses.
• All angles are right angles: This property is true for squares and rectangles.
• Opposite sides are parallel: This property is true for squares, rectangles, rhombuses, and parallelograms.

A figure that satisfies all three properties simultaneously is a square. Looking at the provided figures (assuming R represents a square based on typical geometric diagrams), figure R matches the description of a square.

The correct option is (C).

The property described is "Has two pairs of congruent adjacent sides." This is the defining characteristic of a kite.

Let's consider the typical shapes:

• A kite has two pairs of equal-length sides that are adjacent to each other (e.g., AB = AD and CB = CD).
• Other figures like squares, rectangles, rhombuses, and parallelograms have different side congruence properties (e.g., all sides equal, or opposite sides equal).

Assuming figure R represents a kite, it satisfies this property.

The correct option is (C).

The property "Only one pair of sides are parallel" is the definition of a trapezium (also known as a trapezoid in some regions).

Let's consider the options:

• Square, Rhombus, Rectangle, Parallelogram: These figures have two pairs of parallel sides.
• Trapezium: By definition, a trapezium has exactly one pair of parallel sides.

Therefore, the figure that satisfies this property is a trapezium.

The correct option is (C).