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5. Two adjacent angles of a parallelogram measure [tex]\((2x + 3)^\circ\)[/tex] and [tex]\((3x + 7)^\circ\)[/tex]. Find their measures.

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GinnyAnswer
To solve for the measures of the two adjacent angles of the parallelogram, we need to use the property of a parallelogram stating that adjacent angles are supplementary. This means that the sum of their angles is [tex]\(180^\circ\)[/tex].

The given measures of the two adjacent angles are:
[tex]\[ (2x + 3)^\circ \quad \text{and} \quad (3x + 7)^\circ \][/tex]

Since these angles are supplementary, we can write the following equation:
[tex]\[ (2x + 3) + (3x + 7) = 180 \][/tex]

First, combine like terms:
[tex]\[ 2x + 3x + 3 + 7 = 180 \][/tex]
[tex]\[ 5x + 10 = 180 \][/tex]

Next, solve for [tex]\(x\)[/tex] by isolating it on one side of the equation. Start by subtracting 10 from both sides:
[tex]\[ 5x + 10 - 10 = 180 - 10 \][/tex]
[tex]\[ 5x = 170 \][/tex]

Then, divide both sides by 5:
[tex]\[ x = \frac{170}{5} \][/tex]
[tex]\[ x = 34 \][/tex]

Now that we have the value of [tex]\(x\)[/tex], we can find the measures of the two angles by substituting [tex]\(x = 34\)[/tex] back into the original expressions for the angles.

First angle:
[tex]\[ 2x + 3 = 2(34) + 3 = 68 + 3 = 71^\circ \][/tex]

Second angle:
[tex]\[ 3x + 7 = 3(34) + 7 = 102 + 7 = 109^\circ \][/tex]

Therefore, the measures of the two adjacent angles are:
[tex]\[ 71^\circ \text{ and } 109^\circ \][/tex]
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