If [tex][tex]$\angle A$[/tex][/tex] and [tex][tex]$\angle B$[/tex][/tex] are supplementary and [tex][tex]$\angle A=4x-8$[/tex][/tex] and [tex][tex]$\angle B=2x+2$[/tex][/tex], what is the value of [tex][tex]$x$[/tex][/tex]?

A) 31
B) 22
C) 180
D) 43



Answer :

To find the value of [tex]\( x \)[/tex] for which [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are supplementary, given that [tex]\(\angle A = 4x - 8\)[/tex] and [tex]\(\angle B = 2x + 2\)[/tex], we need to follow these steps:

1. Recall the definition of supplementary angles. Supplementary angles are two angles whose sum is 180 degrees. Therefore, we can write the following equation:
[tex]\[ \angle A + \angle B = 180 \][/tex]

2. Substitute the expressions for [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] into the equation:
[tex]\[ (4x - 8) + (2x + 2) = 180 \][/tex]

3. Combine like terms on the left-hand side of the equation:
[tex]\[ 4x - 8 + 2x + 2 = 180 \][/tex]
[tex]\[ 6x - 6 = 180 \][/tex]

4. Isolate the term involving [tex]\( x \)[/tex] by adding 6 to both sides of the equation:
[tex]\[ 6x - 6 + 6 = 180 + 6 \][/tex]
[tex]\[ 6x = 186 \][/tex]

5. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 6:
[tex]\[ x = \frac{186}{6} \][/tex]
[tex]\[ x = 31 \][/tex]

Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{31} \)[/tex].

To verify, we can check the values of [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] using [tex]\( x = 31 \)[/tex]:

[tex]\[ \angle A = 4(31) - 8 = 124 - 8 = 116 \][/tex]
[tex]\[ \angle B = 2(31) + 2 = 62 + 2 = 64 \][/tex]

By adding them together:
[tex]\[ \angle A + \angle B = 116 + 64 = 180 \][/tex]

Since the sum is 180 degrees, the calculations confirm that the value of [tex]\( x = 31 \)[/tex] is correct.