Question

given the figure below, find the values of x and z.
(12x + 77)°
(7x + 46)°
z°
x =
z =

Explanation:

Step1: Identify Vertical Angles

The angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are vertical angles, so they are equal. Thus, we set up the equation:
\(12x + 77 = 7x + 46\)

Step2: Solve for \(x\)

Subtract \(7x\) from both sides:
\(12x - 7x + 77 = 46\)
\(5x + 77 = 46\)

Subtract 77 from both sides:
\(5x = 46 - 77\)
\(5x = -31\) Wait, that can't be right. Wait, maybe I misidentified the angles. Wait, actually, the angle \(z\) and \((7x + 46)^\circ\) might be supplementary? Wait, no, let's re-examine. Wait, maybe the angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are actually supplementary? Wait, no, vertical angles are equal. Wait, maybe I made a mistake. Wait, let's check the problem again. Wait, the two angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are vertical angles? Wait, no, when two lines intersect, vertical angles are equal. Wait, maybe the angle \((12x + 77)^\circ\) and \(z^\circ\) are supplementary, and \((7x + 46)^\circ\) and \(z^\circ\) are also... Wait, no, let's do it correctly.

Wait, actually, the correct approach: when two lines intersect, vertical angles are equal. So if \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are vertical angles, then:

\(12x + 77 = 7x + 46\)

Subtract \(7x\) from both sides: \(5x + 77 = 46\)

Subtract 77: \(5x = -31\) → \(x = -6.2\). That doesn't make sense for an angle. So maybe the angles are supplementary? Wait, maybe the angle \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are adjacent and form a linear pair? Wait, no, the figure shows two intersecting lines, so vertical angles are equal. Wait, maybe the problem has a typo, or I misread the angles. Wait, let's check again. The angle labeled \((12x + 77)^\circ\) and the angle labeled \((7x + 46)^\circ\) – maybe they are vertical angles, but maybe I made a mistake in the equation. Wait, no, vertical angles are equal. Wait, maybe the angle \(z\) is equal to \((7x + 46)^\circ\)? No, \(z\) is adjacent to \((12x + 77)^\circ\) as a linear pair. Wait, let's start over.

Correct approach: When two lines intersect, adjacent angles are supplementary (sum to \(180^\circ\)), and vertical angles are equal. So, the angle \((12x + 77)^\circ\) and \(z^\circ\) are supplementary, and \((7x + 46)^\circ\) and \(z^\circ\) are also... No, wait, the angle \((12x + 77)^\circ\) and \((7x + 46)^\circ\) – maybe they are vertical angles, but that gives a negative \(x\), which is impossible. So maybe the angle \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are actually supplementary? Let's try that.

If \((12x + 77) + (7x + 46) = 180\)

Combine like terms: \(19x + 123 = 180\)

Subtract 123: \(19x = 57\)

Divide by 19: \(x = 3\)

Ah, that makes sense. So maybe I misidentified the angles. So the two angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are adjacent and form a linear pair (supplementary), not vertical angles. That must be the case. So let's proceed with that.

Step1: Set up Supplementary Angles

The angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) are supplementary (form a linear pair), so:
\(12x + 77 + 7x + 46 = 180\)

Step2: Solve for \(x\)

Combine like terms:
\(19x + 123 = 180\)

Subtract 123 from both sides:
\(19x = 180 - 123\)
\(19x = 57\)

Divide both sides by 19:
\(x = \frac{57}{19}\)
\(x = 3\)

Step3: Find \(z\)

Now, \(z^\circ\) is equal to \((7x + 46)^\circ\) (vertical angles), or equal to \((12x + 77)^\circ\)? Wait, no, \(z\) is vertical to \((7x + 46)^\circ\)? Wait, no, when two lines intersect, vertical angles are equal. So if we have two intersecting lines, the angle \(z\) and \((7x + 46)^\circ\) are vertical angles? Wait, no, let's look at the figure. The angle \((12x + 77)^\circ\) and \(z^\circ\) are adjacent (linear pair), so they are supplementary. Wait, no, let's clarify:

After finding \(x = 3\), we can find \((12x + 77)^\circ\):

\(12(3) + 77 = 36 + 77 = 113^\circ\)

Then, since \(z\) and \((12x + 77)^\circ\) are supplementary (linear pair), \(z = 180 - 113 = 67^\circ\). Alternatively, check \((7x + 46)^\circ\):

\(7(3) + 46 = 21 + 46 = 67^\circ\), which is equal to \(z\) (vertical angles). So that works.

So let's redo the steps correctly:

Step1: Identify Linear Pair (Supplementary Angles)

The angles \((12x + 77)^\circ\) and \((7x + 46)^\circ\) form a linear pair, so their sum is \(180^\circ\):
\(12x + 77 + 7x + 46 = 180\)

Step2: Solve for \(x\)

Combine like terms:
\(19x + 123 = 180\)

Subtract 123:
\(19x = 57\)

Divide by 19:
\(x = 3\)

Step3: Find \(z\)

The angle \(z\) is equal to \((7x + 46)^\circ\) (vertical angles) or supplementary to \((12x + 77)^\circ\). Let's calculate \((7x + 46)^\circ\) with \(x = 3\):

\(7(3) + 46 = 21 + 46 = 67^\circ\), so \(z = 67^\circ\).

Answer:

\(x = 3\), \(z = 67\)