Question
The angle measurements in the diagram are represented by the following expressions.
\[\qquad \blueD{\angle A} = \blueD{6x +18 ^\circ} \]
\[\qquad \green{\angle B} = \green{x +93^\circ} \]
Two parallel lines with a third line intersecting each line. Where the third line crosses the top parallel line, the top, right most angle measure has a curved blue line and is labeled A. Where the third line crosses the bottom parallel line, the bottom, left angle measure has a curved green line and is labeled B.
\[A\]
\[B\]
Solve for
\[x\] and then find the measure of
\[\greenD{\angle B}\]:
\[\greenD{\angle B} = \]
\[^\circ\]
\[\qquad \blueD{\angle A} = \blueD{6x +18 ^\circ} \]
\[\qquad \green{\angle B} = \green{x +93^\circ} \]
Two parallel lines with a third line intersecting each line. Where the third line crosses the top parallel line, the top, right most angle measure has a curved blue line and is labeled A. Where the third line crosses the bottom parallel line, the bottom, left angle measure has a curved green line and is labeled B.
\[A\]
\[B\]
Solve for
\[x\] and then find the measure of
\[\greenD{\angle B}\]:
\[\greenD{\angle B} = \]
\[^\circ\]
Answers
Answered by
GPT-4o mini
To solve for \( x \) and then find the measure of \( \angle B \), we first need to recognize the relationship between the angles formed by the intersection of a transversal with parallel lines. In this case, angles \( \angle A \) and \( \angle B \) are consecutive interior angles, which means they are supplementary. Therefore, we can set up the following equation:
\[
\angle A + \angle B = 180^\circ
\]
Substituting in the expressions for \( \angle A \) and \( \angle B \):
\[
(6x + 18^\circ) + (x + 93^\circ) = 180^\circ
\]
Now, combine like terms:
\[
6x + x + 18^\circ + 93^\circ = 180^\circ
\]
\[
7x + 111^\circ = 180^\circ
\]
Next, we solve for \( x \) by isolating it. First, subtract \( 111^\circ \) from both sides:
\[
7x = 180^\circ - 111^\circ
\]
\[
7x = 69^\circ
\]
Now, divide both sides by \( 7 \):
\[
x = \frac{69^\circ}{7}
\]
\[
x = 9.85714285714 \quad \text{(approximately)} \quad \text{or } \frac{69}{7} \text{ in fraction form.}
\]
Now that we have \( x \), we can substitute it back into the expression for \( \angle B \):
\[
\angle B = x + 93^\circ
\]
\[
\angle B = \frac{69}{7} + 93^\circ
\]
To add \( \frac{69}{7} \) and \( 93 \), we first convert \( 93 \) into a fraction:
\[
93 = \frac{651}{7}
\]
Now we can add:
\[
\angle B = \frac{69}{7} + \frac{651}{7} = \frac{69 + 651}{7} = \frac{720}{7}
\]
Now we can convert \( \frac{720}{7} \) into degrees:
\[
\angle B \approx 102.85714285714 \quad \text{(approximately)}
\]
For a more precise answer:
\[
\angle B = \frac{720}{7} \text{ degrees, which is approximately } 102.86^\circ.
\]
Thus, we have:
\[
\boxed{102.86}
\]
as the final measure of \( \angle B \).
\[
\angle A + \angle B = 180^\circ
\]
Substituting in the expressions for \( \angle A \) and \( \angle B \):
\[
(6x + 18^\circ) + (x + 93^\circ) = 180^\circ
\]
Now, combine like terms:
\[
6x + x + 18^\circ + 93^\circ = 180^\circ
\]
\[
7x + 111^\circ = 180^\circ
\]
Next, we solve for \( x \) by isolating it. First, subtract \( 111^\circ \) from both sides:
\[
7x = 180^\circ - 111^\circ
\]
\[
7x = 69^\circ
\]
Now, divide both sides by \( 7 \):
\[
x = \frac{69^\circ}{7}
\]
\[
x = 9.85714285714 \quad \text{(approximately)} \quad \text{or } \frac{69}{7} \text{ in fraction form.}
\]
Now that we have \( x \), we can substitute it back into the expression for \( \angle B \):
\[
\angle B = x + 93^\circ
\]
\[
\angle B = \frac{69}{7} + 93^\circ
\]
To add \( \frac{69}{7} \) and \( 93 \), we first convert \( 93 \) into a fraction:
\[
93 = \frac{651}{7}
\]
Now we can add:
\[
\angle B = \frac{69}{7} + \frac{651}{7} = \frac{69 + 651}{7} = \frac{720}{7}
\]
Now we can convert \( \frac{720}{7} \) into degrees:
\[
\angle B \approx 102.85714285714 \quad \text{(approximately)}
\]
For a more precise answer:
\[
\angle B = \frac{720}{7} \text{ degrees, which is approximately } 102.86^\circ.
\]
Thus, we have:
\[
\boxed{102.86}
\]
as the final measure of \( \angle B \).