Question

the sum of four times a number and twice its complement is $270^0$. find the measure of the angle.
$\bigcirc$ $x = 35^0$
$\bigcirc$ $x = 40^0$
$\bigcirc$ $x = 55^0$
$\bigcirc$ $x = 45^0$

Explanation:

Step1: Define the angle and its complement

Let the angle be \( x \). The complement of an angle \( x \) is \( 90^\circ - x \) (since complementary angles add up to \( 90^\circ \)).

Step2: Set up the equation

The problem states that the sum of four times the number (angle) and twice its complement is \( 270^\circ \). So we can write the equation:
\( 4x + 2(90^\circ - x) = 270^\circ \)

Step3: Simplify and solve the equation

First, expand the equation:
\( 4x + 180^\circ - 2x = 270^\circ \)

Combine like terms:
\( 2x + 180^\circ = 270^\circ \)

Subtract \( 180^\circ \) from both sides:
\( 2x = 270^\circ - 180^\circ \)
\( 2x = 90^\circ \)

Divide both sides by 2:
\( x = \frac{90^\circ}{2} = 45^\circ \)? Wait, no, wait, let's check again. Wait, maybe I made a mistake. Wait, let's re - solve the equation:

Wait, \( 4x+2(90 - x)=270 \)

Expand: \( 4x + 180-2x=270 \)

Combine like terms: \( 2x+180 = 270 \)

Subtract 180: \( 2x=270 - 180=90 \)

Then \( x = 45 \)? But let's check the options. Wait, maybe I misread the problem. Wait, the problem says "the sum of four times a number and twice its complement is \( 270^\circ \)". Wait, maybe the angle is \( x \), complement is \( 90 - x \). Let's check each option:

For \( x = 45^\circ \): four times \( 45 \) is \( 180 \), twice its complement (complement of \( 45 \) is \( 45 \)) is \( 90 \), sum is \( 180 + 90=270 \). Yes, that works. Wait, but let's check other options:

For \( x = 55^\circ \): four times \( 55 \) is \( 220 \), complement is \( 35 \), twice complement is \( 70 \), sum is \( 220+70 = 290
eq270 \)

For \( x = 40^\circ \): four times \( 40 \) is \( 160 \), complement is \( 50 \), twice complement is \( 100 \), sum is \( 160 + 100=260
eq270 \)

For \( x = 35^\circ \): four times \( 35 \) is \( 140 \), complement is \( 55 \), twice complement is \( 110 \), sum is \( 140+110 = 250
eq270 \)

Wait, so \( x = 45^\circ \) is the solution. But let's re - do the equation:

\( 4x+2(90 - x)=270 \)

\( 4x + 180-2x=270 \)

\( 2x=270 - 180=90 \)

\( x = 45 \). So the correct answer is \( x = 45^\circ \)

Wait, but let's check again. Maybe the problem was about supplementary angles? No, the problem says complement. Wait, if it was supplementary (sum to \( 180 \)), but the problem says complement. So according to the calculation, \( x = 45^\circ \)

Answer:

\( \boldsymbol{x = 45^\circ} \) (the option with \( x = 45^\circ \))