We know that in a parallelogram, the sum of any two adjacent angles is [tex]$180^{\circ}$[/tex].

Let
[tex]\[
\begin{array}{l}
\angle A = (3x - 4)^{\circ} \\
\angle B = (2x - 1)^{\circ}
\end{array}
\][/tex]

Therefore,
[tex]\[
(3x - 4)^{\circ} + (2x - 1)^{\circ} = 180^{\circ}
\][/tex]



Answer :

Let's solve the given problem step-by-step.

1. Write down the given angles:
[tex]\[ \angle A = (3x - 4)^\circ \][/tex]
[tex]\[ \angle B = (2x - 1)^\circ \][/tex]

2. Use the property of a parallelogram that the sum of any two adjacent angles is [tex]\(180^\circ\)[/tex]:
[tex]\[ \angle A + \angle B = 180^\circ \][/tex]
Substitute the expressions for [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex]:
[tex]\[ (3x - 4)^\circ + (2x - 1)^\circ = 180^\circ \][/tex]

3. Combine the like terms:
[tex]\[ (3x - 4 + 2x - 1)^\circ = 180^\circ \][/tex]
[tex]\[ (5x - 5)^\circ = 180^\circ \][/tex]

4. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides of the equation:
[tex]\[ 5x - 5 + 5 = 180 + 5 \][/tex]
[tex]\[ 5x = 185 \][/tex]
Divide by 5:
[tex]\[ x = \frac{185}{5} \][/tex]
[tex]\[ x = 37 \][/tex]

5. Find the measures of [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] by substituting [tex]\(x = 37\)[/tex] into the expressions:

For [tex]\(\angle A\)[/tex]:
[tex]\[ \angle A = 3x - 4 \][/tex]
Substitute [tex]\(x = 37\)[/tex]:
[tex]\[ \angle A = 3(37) - 4 \][/tex]
[tex]\[ \angle A = 111 - 4 \][/tex]
[tex]\[ \angle A = 107^\circ \][/tex]

For [tex]\(\angle B\)[/tex]:
[tex]\[ \angle B = 2x - 1 \][/tex]
Substitute [tex]\(x = 37\)[/tex]:
[tex]\[ \angle B = 2(37) - 1 \][/tex]
[tex]\[ \angle B = 74 - 1 \][/tex]
[tex]\[ \angle B = 73^\circ \][/tex]

6. Summary of the solution:
[tex]\[ \angle A = 107^\circ \][/tex]
[tex]\[ \angle B = 73^\circ \][/tex]
[tex]\[ x = 37 \][/tex]

Thus, the measures of the angles are [tex]\(\angle A = 107^\circ\)[/tex], [tex]\(\angle B = 73^\circ\)[/tex], and the value of [tex]\(x\)[/tex] is 37.