Question

Angles forming a linear pair can both be acute angles.
A) True
B) False
C) Ambiguous
D) Data Insufficient

Answer 1

Hint:
A linear pair of angles is formed when two lines intersect. The sum of two adjacent angles of intersecting lines is ${180}^{\circ }$ and it is called a linear pair.
The angle which measures less than ${90}^{\circ }$ is called acute angle, angle measure more than ${90}^{\circ }$ is called obtuse angle and angle which measures equal to ${90}^{\circ }$ is called right angle.

Complete step by step solution:
Let there be two intersecting straight lines at intersecting point O and suppose ‘x’ and ‘y’ are two adjacent angles which form a linear pair. So,
$x+y={180}^{\circ }$
Case-1: If $x<{90}^{\circ }$
Then ‘y’ must be greater than ${90}^{\circ }$. So, in this case, one angle (x) is acute and other is obtuse (y)

Case-2: If $x={90}^{\circ }$
Then ‘y’ must be equal to ${90}^{\circ }$. So, both are right angled triangles in this case.

Case-3: $x>{90}^{\circ }$
Then ‘y’ must be less than ${90}^{\circ }$. So, ‘x’ is obtuse and ‘y’ is an acute angle in this case.

As seen in all three cases, one angle is acute or both are acute angles. So there is no case in which both are acute angles.

So, option B is correct.

Note:
There are also two additional types of angles called straight angle and reflex angle. The straight angle measures ${180}^{\circ }$ and reflex angle measures between ${180}^{\circ }$ and ${360}^{\circ }$.
A polygon having interior angles less than ${180}^{\circ }$ is called a convex polygon and if a polygon has at least one angle more than ${180}^{\circ }$ then the polygon is called a concave polygon.