Question

Can two adjacent angles be complementary? Draw figure.

Answer 1

Hint: Draw a figure from a vertex of a triangle on a square. Show that the angles are adjacent as well as complementary, i.e. the sum of angle should be \[{{90}^{\circ }}.\]

Adjacent angles are angles that are adjacent to each other. Two angles are adjacent if they share a vertex and an edge with one angle on one side of the edge and the other on the other side. Two adjacent angles can be complimentary, if they add up to \[{{90}^{\circ }}\], i.e. the sum of two angles formed should be \[{{90}^{\circ }}.\]

Complete step by step answer:

From the figure \[\angle ABD\] and \[\angle CBD\] are adjacent angles from the same vertex. They are also complementary as the sum of the angles is \[{{90}^{\circ }}\], i.e. \[\angle ABD+\angle CBD={{30}^{\circ }}+{{60}^{\circ }}={{90}^{\circ }}.\]

We can take other examples of angles as \[{{45}^{\circ }}+{{45}^{\circ }}\]. In case of a right triangle, the altitude from a right adjacent angle vertex will split the right angle into 2 adjacent angles, e.g.:- \[{{30}^{\circ }}+{{60}^{\circ }},{{40}^{\circ }}+{{50}^{\circ }}\].

Thus we have proved that two adjacent angles are complementary with the help of a diagram.

Note: If we were asked to prove that 2 adjacent angles are supplementary then their sum should add up to \[{{180}^{\circ }}\]. Now their supplementary angle can be of \[{{30}^{\circ }}+{{150}^{\circ }},{{70}^{\circ }}+{{110}^{\circ }},{{90}^{\circ }}+{{90}^{\circ }}\] etc.

From the figure, a straight line intersecting forming adjacent angles.

\[\therefore {{60}^{\circ }}+{{120}^{\circ }}={{180}^{\circ }}\]

\[\therefore \]Supplementary angle.