Understanding Shapes — Statement I-II Type Questions

Statement I: A polygon is a simple closed curve bounded by line segments.

Statement II: All regular polygons are convex.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A simple closed curve made up entirely of line segments is called a polygon.

So, Statement I is true.

In a regular polygon, all the sides are of equal length and all the interior angles are of equal measure, and each of these interior angles is less than 180°. So, every regular polygon is convex.

So, Statement II is true.

Hence, option 3 is the correct option.

Statement I: Each interior angle of a regular heptagon is 120°

Statement II: Each exterior angle of a regular heptagon is 60°

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

A heptagon has 7 sides.

According to the properties of a polygon, if a regular polygon has $n$ sides, then each of its interior angles is $\dfrac{(n - 2) \times 180\degree}{n}$.

$\text{Each interior angle} = \dfrac{(7 - 2) \times 180\degree}{7} \\[1em] = \dfrac{5 \times 180\degree}{7} \\[1em] = \dfrac{900\degree}{7} \\[1em] = 128\dfrac{4}{7}^{\degree}$

As $128\dfrac{4}{7}^{\degree}$ ≠ 120°, Statement I is false.

According to the properties of a polygon, if a regular polygon has $n$ sides, then each of its exterior angles is $\dfrac{360\degree}{n}$.

$\text{Each exterior angle} = \dfrac{360\degree}{7} \\[1em] = 51\dfrac{3}{7}^{\degree}$

As $51\dfrac{3}{7}^{\degree}$ ≠ 60°, Statement II is false.

Hence, option 4 is the correct option.

Statement I: All rectangles are parallelograms.

Statement II: All rectangles are squares.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

In a rectangle, both the pairs of opposite sides are parallel and equal, so every rectangle is a parallelogram.

So, Statement I is true.

In a square, all the four sides are equal, but in a rectangle the adjacent sides need not be equal. So, every rectangle is not a square.

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: The opposite sides of a parallelogram are equal.

Statement II: The diagonals of a parallelogram are equal.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

The opposite sides of a parallelogram are of equal length.

So, Statement I is true.

The diagonals of a parallelogram bisect each other, but they are not equal in length. The diagonals are equal only in a rectangle and a square.

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: All rhombuses are parallelograms.

Statement II: All rhombuses are squares.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

In a rhombus, both the pairs of opposite sides are parallel and equal, so every rhombus is a parallelogram.

So, Statement I is true.

In a square, each angle is a right angle, but in a rhombus the angles need not be right angles. So, every rhombus is not a square.

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: The diagonals of a rectangle are also angle bisectors of the interior angles of the rectangle.

Statement II: The diagonals of a rectangle bisect each other.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

The diagonals of a rectangle bisect the angles of the rectangle only when the rectangle is a square. In a rectangle whose adjacent sides are unequal, a diagonal divides a right angle into two unequal parts.

So, Statement I is false.

A rectangle is a parallelogram and the diagonals of a parallelogram bisect each other, so the diagonals of a rectangle bisect each other.

So, Statement II is true.

Hence, option 2 is the correct option.