The ratio of the measures of two supplementary angles is 2:7. How do you find the measures of the angles?

Question

Hazel Anglin · Accepted Answer

#40^@" and " 140^@# #color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(" the sum of 2 supplementary angles" = 180^@)color(white)(2/2)|)))# #"sum the parts of the ratio"# #rArr2+7=9" parts in total"# Find the value of 1 part by dividing #180^@" by " 9# #rArr180^@/9=20^@larrcolor(red)" value of 1 part"# #rArr"2 parts " =2xx20^@=40^@# #rArr"7 parts " =7xx20^@=140^@# #"Thus the supplementary angles are " 40^@" and " 140^@#

Eli Assouline · Answer

undefined To find the measures of the angles, you first need to determine the total measure of the two supplementary angles, which is 180 degrees. Then, set up the equation 2x + 7x = 180, where 2x represents one angle and 7x represents the other angle. Solve for x, then plug the value of x back into the expressions to find the measures of the angles.

Eva Adamson · Answer

undefined Let $x$ be the measure of the smaller angle and $y$ be the measure of the larger angle. Since the angles are supplementary, their sum is $180^\circ$.

Given that the ratio of the measures of the angles is 2:7, we can set up the following equation:

$$ \frac{x}{y} = \frac{2}{7} $$

To solve for the measures of the angles, we can use the fact that the sum of the measures of the angles is $180^\circ$.

Substituting $x = 2k$ and $y = 7k$ into the equation:

$$ \frac{2k}{7k} = \frac{2}{7} $$

$$ \frac{2}{7} = \frac{2}{7} $$

This means that the given ratio is satisfied.

Thus, the measure of the smaller angle ($x$) is $2k$ degrees, and the measure of the larger angle ($y$) is $7k$ degrees.

Since the sum of the measures of the angles is $180^\circ$, we have:

$$ 2k + 7k = 180 $$

$$ 9k = 180 $$

$$ k = \frac{180}{9} $$

$$ k = 20 $$

So, the measure of the smaller angle is $2k = 2 \times 20 = 40^\circ$, and the measure of the larger angle is $7k = 7 \times 20 = 140^\circ$.