Question

How many degrees are there in an angle which equals two third of its complement?

Hint: Two angles are Complementary when they add up to ${90^ \circ }$ degrees, a Right Angle. These two angles $\left( {40^\circ {\text{ }}and{\text{ }}50^\circ } \right)$are Complementary Angles, because they add up to ${90^ \circ }$. Notice that together they make a right angle.
Complete step by step solution: Complementary angles are angle pairs whose measures sum to one right angle. If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is ${180^ \circ }$, and the right angle itself accounts for ${90^ \circ }$.
The sum of two complementary angles is ${90^ \circ }$.
let one of the angles be $x$.
$\begin{gathered} \;\dfrac{2}{3}{\text{ }}of{\text{ }}x{\text{ }} + x = {\text{ }}90\; \\ \dfrac{{2x}}{3}{\text{ }} + x = 90\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }} \\ \dfrac{{(2x + 3x)}}{3} = {\text{ }}90\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }} \\ 5x = {\text{ }}90 \times 3\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\; \\ x = {\text{ }}\dfrac{{270}}{5} \\ x = {\text{ }}54\;\;\;\;\;\;\;\;\;\;\; \\ \dfrac{2}{3} \times 54 = 36 \\ \end{gathered}$
So, one angle is${36^ \circ }$ which is two third of its complement ${54^ \circ }$. They are Complementary angles.
$\left( {36^\circ + 54^\circ = 90^\circ } \right)$Therefore, the angle will be ${36^ \circ }$ degree.
Note: When two angles add to ${90^ \circ }$, we say they "Complement" each other. Complementary comes from Latin complete meaning "completed". because the right angle is thought of as being a complete angle.