To solve this problem, let's denote the smallest angle of the parallelogram as [tex]\( x \)[/tex].
Given that one angle is [tex]\( 20^\circ \)[/tex] less than twice the smallest angle, we can express the larger angle as [tex]\( 2x - 20^\circ \)[/tex].
We know that in a parallelogram, opposite angles are equal. Thus, we will have two angles each of [tex]\( x \)[/tex] and two angles each of [tex]\( 2x - 20^\circ \)[/tex].
In any parallelogram, the sum of the interior angles is always [tex]\( 360^\circ \)[/tex]. Therefore, we can set up the following equation:
[tex]\[ 2x + 2(2x - 20) = 360^\circ \][/tex]
First, simplify the equation:
[tex]\[ 2x + 2 \cdot (2x - 20) = 360 \][/tex]
[tex]\[ 2x + 4x - 40 = 360 \][/tex]
[tex]\[ 6x - 40 = 360 \][/tex]
Next, add [tex]\( 40 \)[/tex] to both sides of the equation to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[ 6x - 40 + 40 = 360 + 40 \][/tex]
[tex]\[ 6x = 400 \][/tex]
Now, divide both sides by [tex]\( 6 \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{400}{6} \][/tex]
[tex]\[ x \approx 66.67^\circ \][/tex]
Therefore, the smallest angle of the parallelogram is:
[tex]\[ 66.67^\circ \][/tex]
Now, let's determine the measure of the larger angle. Using [tex]\( x \)[/tex]:
[tex]\[ \text{Largest angle} = 2x - 20 \][/tex]
[tex]\[ 2(66.67) - 20 \][/tex]
[tex]\[ 133.34 - 20 \][/tex]
[tex]\[ \text{Largest angle} \approx 113.33^\circ \][/tex]
Thus, the angles of the parallelogram are approximately:
[tex]\[ 66.67^\circ, 66.67^\circ, 113.33^\circ, 113.33^\circ \][/tex]
