Question

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

Answer 1

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 adjective complementary is associated with the verb complete, "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle.
The sum of two complementary angles is \[{90^ \circ }\].
let one of the angles be \[x\].
According to the question,
\[\begin{gathered}
  \;\dfrac{2}{3}{\text{ }}of{\text{ }}x{\text{ }} + x = {\text{ }}90\; \\
  \dfrac{{2x}}{3}{\text{ }} + x = 90\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }} \\
  \dfrac{{(2x + 3x)}}{3} = {\text{ }}90\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }} \\
  5x = {\text{ }}90 \times 3\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\;{\text{ }}\; \\
  x = {\text{ }}\dfrac{{270}}{5} \\
  x = {\text{ }}54\;\;\;\;\;\;\;\;\;\;\; \\
  \dfrac{2}{3} \times 54 = 36 \\
\end{gathered} \]
 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.