Answer :
We are given two angles in terms of the variable [tex]$x$[/tex]:
[tex]$$
a = 3x + 50 \quad \text{and} \quad f = 2x + 20.
$$[/tex]
If these two angles are supplementary, then their sum is [tex]$180^\circ$[/tex]. This gives us the equation:
[tex]$$
a + f = 180.
$$[/tex]
Substitute the expressions for [tex]$a$[/tex] and [tex]$f$[/tex]:
[tex]$$
(3x + 50) + (2x + 20) = 180.
$$[/tex]
Combine like terms:
[tex]$$
5x + 70 = 180.
$$[/tex]
Solve for [tex]$x$[/tex] by subtracting [tex]$70$[/tex] from both sides:
[tex]$$
5x = 180 - 70 = 110.
$$[/tex]
Divide both sides by [tex]$5$[/tex]:
[tex]$$
x = \frac{110}{5} = 22.
$$[/tex]
Now substitute [tex]$x = 22$[/tex] back into the expressions for [tex]$a$[/tex] and [tex]$f$[/tex].
For [tex]$a$[/tex]:
[tex]$$
a = 3(22) + 50 = 66 + 50 = 116.
$$[/tex]
For [tex]$f$[/tex]:
[tex]$$
f = 2(22) + 20 = 44 + 20 = 64.
$$[/tex]
Thus, the angles are:
[tex]$$
a = 116^\circ \quad \text{and} \quad f = 64^\circ.
$$[/tex]
The correct answer is option (a).
[tex]$$
a = 3x + 50 \quad \text{and} \quad f = 2x + 20.
$$[/tex]
If these two angles are supplementary, then their sum is [tex]$180^\circ$[/tex]. This gives us the equation:
[tex]$$
a + f = 180.
$$[/tex]
Substitute the expressions for [tex]$a$[/tex] and [tex]$f$[/tex]:
[tex]$$
(3x + 50) + (2x + 20) = 180.
$$[/tex]
Combine like terms:
[tex]$$
5x + 70 = 180.
$$[/tex]
Solve for [tex]$x$[/tex] by subtracting [tex]$70$[/tex] from both sides:
[tex]$$
5x = 180 - 70 = 110.
$$[/tex]
Divide both sides by [tex]$5$[/tex]:
[tex]$$
x = \frac{110}{5} = 22.
$$[/tex]
Now substitute [tex]$x = 22$[/tex] back into the expressions for [tex]$a$[/tex] and [tex]$f$[/tex].
For [tex]$a$[/tex]:
[tex]$$
a = 3(22) + 50 = 66 + 50 = 116.
$$[/tex]
For [tex]$f$[/tex]:
[tex]$$
f = 2(22) + 20 = 44 + 20 = 64.
$$[/tex]
Thus, the angles are:
[tex]$$
a = 116^\circ \quad \text{and} \quad f = 64^\circ.
$$[/tex]
The correct answer is option (a).
