State whether the following statement is true or false.

A linear pair will have an obtuse angle if one of the angles is acute.

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True

False

answer is A.

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Detailed Solution

It is given that one of the angles of a linear pair is acute.
We know that two adjacent angles that have non-common sides that form opposite rays are called linear pairs.
A linear pair of angles are those angles that are adjacent and supplementary.
Therefore, the sum of the two adjacent angles formed when a ray stands on a line is

Consider that

y

and

z

are two adjacent angles forming a linear pair where

∠ z

is an acute angle and is equal to

90 − x °, x < 90 ° .

Therefore, their sum will be equal to

180 ° .

⇒ ∠ y + ∠ z = 180 °

⇒ ∠ y = 180 ° − (90 − x) ° [∵ ∠ z = (90 − x) °]

⇒ ∠ y = 180 ° − 90 ° + x °

⇒ ∠ y = (90 + x) °

This shows that the other angle

∠ y

is an obtuse angle when the first angle

∠ z

is an acute angle.
Therefore, a linear pair will have an obtuse angle if one of the angles is acute.
Thus, the given statement is true.
Hence, option (1) is correct.

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