Question

given ( m parallel n ), find the value of ( x ).
diagram showing lines ( m ), ( n ), and a transversal with angles ( (7x + 3)^circ ) and ( (8x - 10)^circ )
answer attempt 1 out of 2
( x = ) blank
submit answer

Explanation:

Step1: Identify supplementary angles

Since \( m \parallel n \), the angles \( (8x - 10)^\circ \) and \( (7x + 3)^\circ \) are supplementary (they form a linear pair), so their sum is \( 180^\circ \).
\[
(8x - 10) + (7x + 3) = 180
\]

Step2: Combine like terms

Combine the \( x \)-terms and the constant terms:
\[
8x + 7x - 10 + 3 = 180 \\
15x - 7 = 180
\]

Step3: Solve for \( x \)

Add 7 to both sides:
\[
15x = 180 + 7 \\
15x = 187
\]
Wait, that seems incorrect. Wait, actually, the angles \( (8x - 10)^\circ \) and \( (7x + 3)^\circ \) are same - side interior angles? No, wait, looking at the diagram, since \( m\parallel n \), and the transversal, the two angles \( (8x - 10) \) and \( (7x + 3) \) are actually same - side interior angles? No, wait, no. Wait, the two angles \( (8x - 10) \) and \( (7x + 3) \) are adjacent and form a linear pair? Wait, no, maybe they are alternate interior angles? Wait, no, let's re - examine.

Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, no, in the diagram, the two angles \( (8x - 10) \) and \( (7x + 3) \) are same - side interior angles? Wait, no, maybe they are consecutive interior angles. Wait, no, let's do the correct approach.

Wait, the two angles \( (8x - 10)^\circ \) and \( (7x + 3)^\circ \) are supplementary because they are same - side interior angles (since \( m\parallel n \)). So:

\[
(8x - 10)+(7x + 3)=180
\]

Combine like terms:

\( 8x+7x-10 + 3=180\)

\(15x-7 = 180\)

Add 7 to both sides:

\(15x=180 + 7=187\)

Wait, that gives \(x=\frac{187}{15}\approx12.47\), which seems odd. Wait, maybe I made a mistake in identifying the angle relationship.

Wait, maybe the two angles are alternate interior angles? No, alternate interior angles are equal. Wait, if \( m\parallel n \), and the transversal, maybe the two angles \( (8x - 10) \) and \( (7x + 3) \) are equal? Wait, let's check the diagram again.

Wait, the two angles \( (8x - 10)^\circ \) and \( (7x + 3)^\circ \) are vertical angles? No, vertical angles are equal. Wait, maybe the correct relationship is that they are same - side interior angles, but maybe I messed up.

Wait, let's start over. The two angles \( (8x - 10) \) and \( (7x + 3) \) are adjacent and form a linear pair? No, a linear pair sums to \( 180^\circ \). Wait, maybe the correct equation is \( (8x - 10)+(7x + 3)=180 \). Let's solve it again:

\(8x+7x-10 + 3 = 180\)

\(15x-7=180\)

\(15x=187\)

\(x=\frac{187}{15}\approx12.47\). But this seems unusual. Wait, maybe the angles are corresponding angles? No, corresponding angles are equal. Wait, maybe the diagram is such that the two angles are same - side interior angles, but maybe I made a mistake in the sign.

Wait, maybe the correct equation is \( (8x - 10)=(7x + 3) \)? Let's try that.

\(8x-10 = 7x + 3\)

Subtract \(7x\) from both sides:

\(x-10 = 3\)

Add 10 to both sides:

\(x = 13\)

Ah, maybe the two angles are alternate interior angles. Let's check: if \( m\parallel n \), and the transversal, the two angles \( (8x - 10) \) and \( (7x + 3) \) are alternate interior angles, so they are equal.

So, let's redo the steps with the correct relationship (assuming alternate interior angles):

Step1: Set angles equal (alternate interior angles)

Since \( m\parallel n \), the alternate interior angles are equal, so:
\[
8x - 10=7x + 3
\]

Step2: Solve for \( x \)

Subtract \(7x\) from both sides:
\[
8x-7x-10=7x - 7x+ 3\\
x - 10=3
\]
Add 10 to both sides:
\[
x=3 + 10\\
x = 13
\]

Answer:

\(x = 13\)