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Dados los ángulos [tex]a = 3x + 50[/tex] y [tex]f = 2x + 20[/tex], determinar su valor.

A. [tex]a = 100, f = 80[/tex]
B. [tex]a = 50, f = 20[/tex]
C. [tex]a = 150, f = 30[/tex]
D. [tex]a = 116, f = 64[/tex]

Answer :

We are given the angles expressed in terms of the variable [tex]$x$[/tex]:

[tex]$$
a = 3x + 50 \quad \text{and} \quad f = 2x + 20.
$$[/tex]

Assuming these angles are supplementary, we have:

[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]

Subtract [tex]$70$[/tex] from both sides to isolate the term with [tex]$x$[/tex]:

[tex]$$
5x = 180 - 70,
$$[/tex]

[tex]$$
5x = 110.
$$[/tex]

Now, solve for [tex]$x$[/tex]:

[tex]$$
x = \frac{110}{5} = 22.
$$[/tex]

With the value of [tex]$x$[/tex] found, substitute it back into the expressions for [tex]$a$[/tex] and [tex]$f$[/tex]:

For angle [tex]$a$[/tex]:

[tex]$$
a = 3(22) + 50 = 66 + 50 = 116.
$$[/tex]

For angle [tex]$f$[/tex]:

[tex]$$
f = 2(22) + 20 = 44 + 20 = 64.
$$[/tex]

Thus, the values of the angles are [tex]$a = 116$[/tex] and [tex]$f = 64$[/tex], which corresponds to option (d).

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