Fundamental Concepts of Geometry — Statement I-II Type Questions

Statement 1: Two lines in a plane either intersect at a point or they are parallel to each other.

Statement 2: Exactly two lines in the same plane which do not intersect at any point even when further extended up to infinity are called parallel lines.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Two lines lying in the same plane either meet at a point or never meet; if they never meet, they are parallel. So, Statement 1 is true.

Two lines in the same plane which do not intersect at any point even when extended to infinity are called parallel lines. So, Statement 2 is true.

Hence, option 1 is the correct option.

Statement 1: If an angle is such that its supplement is 5 times its complement then the angle is $67\dfrac{1}{2}°$.

Statement 2: When two adjacent angles form a linear pair then they are supplementary.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Let the angle be x. Then its supplement = (180° − x) and its complement = (90° − x).

As the supplement is 5 times the complement,

⇒ 180° − x = 5(90° − x)

⇒ 180° − x = 450° − 5x

⇒ 5x − x = 450° − 180°

⇒ 4x = 270°

⇒ x = $\dfrac{270\degree}{4}$

⇒ x = $67\dfrac{1}{2}\degree$

So, Statement 1 is true.

When two adjacent angles form a linear pair, their sum is 180°, i.e. they are supplementary. So, Statement 2 is true.

Hence, option 1 is the correct option.