Question

If angle $A$ and $B$ are complementary and $A=5x+8$ and $B=x+4$ what are the measurements of each angle?

Answer 1

Hint: Here we have to find the measure of the two angles given. Firstly as it is given that the two angles are complementary which means the sum of them will be equal to ${{90}^{\circ }}$ . So add the values of the given two angles and put it equal to ${{90}^{\circ }}$ . Then simplify the equation and get the value of the unknown variable.Finally substitute the value of the unknown variable in each angle and get the desired answer.

Complete step by step answer:
We have been given that two angles $A$ and $B$ are complementary and their values is given as follows:
$A=5x+8$….$\left( 1 \right)$
$B=x+4$…..$\left( 2 \right)$
We have to find the measure of the two angles given which have an unknown variable in them.

Now by the definition of complementary angles if two angles are complementary their sum is equal to ${{90}^{\circ }}$. So we will add equation (1) and (2) and put it equal to ${{90}^{\circ }}$ as follows:
$\Rightarrow \left( 5x+8 \right)+\left( x+4 \right)={{90}^{\circ }}$
$\Rightarrow 6x+12={{90}^{\circ }}$
Simplifying further we get,
$\Rightarrow 6x=90-12$
$\Rightarrow x=\dfrac{78}{6}$
So we get,
$\Rightarrow x=13$

Substitute above value in equation (1) as follows:
$\Rightarrow A=5\times 13+8$
$\Rightarrow A=65+8$
So, $A={{73}^{\circ }}$
Next substitute $x=13$ in equation (2) as follows:
$\Rightarrow B=13+4$
$\therefore B={{17}^{\circ }}$
So we got the two values as $A={{73}^{\circ }}$ and $B={{17}^{\circ }}$ .

Hence the two complementary angles are ${{73}^{\circ }}$ and ${{17}^{\circ }}$.

Note: When the sum of two angles is equal to ${{90}^{\circ }}$ they are known as complementary angles and they form a right angle together. We can cross check our answer by seeing whether the sum of two angles obtained is equal to ${{90}^{\circ }}$ as follows:
We got the two angles as ${{73}^{\circ }}$ and ${{17}^{\circ }}$ .
On adding the above two angles we get,
$\Rightarrow {{73}^{\circ }}+{{17}^{\circ }}$
$\Rightarrow {{90}^{\circ }}$
So our answer is correct.