Question

Find the value of $x$ for which the angles $(2x - 5)^\circ $ and $(x - 10)^\circ $ are the complementary angles.

Answer 1

Hint: Two angles are said to be complementary angles if they add up to $90$ degrees. Complementary angles form a right angle ($90$ deg) when they are put together. In other words, we can say that $\angle 1 + \angle 2 = 90^\circ $. So, in order to find the value of $x$ we equate the sum of the given two angles to $90^\circ $ and then after solving the equation we get our required result.

Complete step by step answer:
Complementary angles can be defined as the angles when the sum of the measure of the two angles is $90^\circ $, then the pair is said to be complementary angles. It is not necessary that the complementary angles are always adjacent to each other; they can be different; only their sum should be $90^\circ $.In other words, $\angle 1$ and $\angle 2$ are said to be complementary, if $\angle 1 + \angle 2 = 90^\circ $.

Now, it is given that $(2x - 5)^\circ $ and $(x - 10)^\circ $ are complementary angles.So, their sum should be equal to $90^\circ $. Therefore, we can form the equation as
$(2x - 5)^\circ + (x - 10)^\circ = 90^\circ $
Solving $x$ terms and constant terms on the left side of the equation. We get,
$ \Rightarrow 3x - 15 = 90^\circ $
Shifting $15$ to the right side of the equation. We get,
$ \Rightarrow 3x = 90 + 15$

Adding constant terms on the right side of the equation. We get,
$ \Rightarrow 3x = 105^\circ $
Shifting $3$ to the right side of the equation as the denominator. We get,
$ \Rightarrow x = \dfrac{{105}}{3}$
On dividing we get,
$ \therefore x = 35^\circ $

Hence, the value of $x$ is $35^\circ $.

Note: When the sum of two pairs of angles is equal to $90^\circ $, then we call that pair of angles complements of each other. So, we know that the sum of two complementary angles is $90^\circ $, and each of them is said to be a complement of each other. Thus, the complement of an angle is found by subtracting it from $90^\circ $. This means the complement of $x^\circ $ is $(90 - x)^\circ $. Note that three angles can never be complementary even though their sum is $90^\circ $ because complementary angles always occur in pairs. The definition of complementary angles holds true only for two angles.