Question

The difference of two complementary angles is $30°$. Then, the angles are:

• A. $60°,30°$
• B. $70°,40°$
• C. $20°,50°$
• D. $105°,75°$

Answer 1

The correct option is A

$60°,30°$

Step 1: Explanation of correct options.

Determine the pair of complementary angles with a difference of $30°$.

Option (A) : $60°,30°$

Assume that the angles are $x\mathrm{and}y$

The sum of complementary angle is $90°$.

So, $x+y=90°$.

Also, the difference of the angles is $30°$.

So, $x-y=30°$

Add both the obtained equations and simplify:

$\begin{array}{rcl}x+y+x-y& =& 90°+30°\\ & \Rightarrow & 2x=120°\\ & \Rightarrow & x=60°\end{array}$

Substitute $x=60°$ into $x+y=90°$ and solve:

$\begin{array}{rcl}60°+y& =& 90°\\ & \Rightarrow & y=90°-60°\\ & \Rightarrow & y=30°\end{array}$

So, the angles are $60°,30°$ and option (A) is correct.

Step 2: Explanation for incorrect options.

Option (B) : $70°,40°$

The sum of the angles is $70°+40°=110°$

The angles are not complementary.

So, option (B) is incorrect.

Option (C) : $20°,50°$

The sum of the angles is $20°+50°=70°$

The angles are not complementary.

So, option (C) is incorrect.

Option (D) : $105°,75°$

The sum of the angles is $105°+75°=180°$

The angles are not complementary.

So, option (D) is incorrect.

Hence, (A) is the correct option.