Question

given the figure below, find the values of x and z.

$z \degree$
$(14x + 26) \degree$
$(9x + 62) \degree$

$x = \square$
$z = \square$

Explanation:

Step1: Identify vertical angles

Vertical angles are equal, so \(14x + 26 = 9x + 62\).

Step2: Solve for x

Subtract \(9x\) from both sides: \(14x - 9x + 26 = 62\) → \(5x + 26 = 62\).
Subtract 26: \(5x = 62 - 26 = 36\)? Wait, no—62 - 26 is 36? Wait, 62 - 26 = 36? Wait, 62 - 26: 62 - 20 = 42, 42 - 6 = 36. Then \(5x = 36\)? No, wait, 14x - 9x is 5x, 62 - 26 is 36? Wait, no, 62 - 26 is 36? Wait, 62 - 26: 62 - 20 = 42, 42 - 6 = 36. So \(5x = 36\)? Wait, that can't be. Wait, maybe I made a mistake. Wait, 14x + 26 = 9x + 62. Subtract 9x: 5x + 26 = 62. Subtract 26: 5x = 62 - 26 = 36? Wait, 62 - 26 is 36? Yes. Then \(x = 36 / 5\)? No, that's not an integer. Wait, maybe the angles are supplementary? Wait, no, vertical angles are equal. Wait, maybe the figure is two intersecting lines, so the angles \(14x + 26\) and \(9x + 62\) are vertical angles, so they should be equal. Wait, maybe I miscalculated 62 - 26. 62 - 26: 26 + 36 = 62, so 62 - 26 = 36. So 5x = 36 → x = 36/5 = 7.2? That seems odd. Wait, maybe the angles are adjacent and supplementary? Wait, no, the figure shows two intersecting lines, so vertical angles are equal. Wait, maybe I misread the angles. Let me check again. The angles are \((14x + 26)^\circ\) and \((9x + 62)^\circ\). So 14x + 26 = 9x + 62. 14x - 9x = 62 - 26. 5x = 36. x = 36/5 = 7.2. Then, once x is found, z is supplementary to one of the angles? Wait, no, z and one of the angles are adjacent, so they are supplementary. Wait, maybe I made a mistake in identifying the angles. Wait, maybe the angle \(z^\circ\) and \((14x + 26)^\circ\) are supplementary? Wait, no, if two lines intersect, adjacent angles are supplementary. Wait, let's re-express. Let's suppose that \(14x + 26\) and \(9x + 62\) are vertical angles, so they are equal. So 14x + 26 = 9x + 62. 5x = 36. x = 7.2. Then, z is supplementary to \(14x + 26\), so z = 180 - (14x + 26). Let's compute 14x + 26: 14*(36/5) + 26 = (504/5) + 26 = 100.8 + 26 = 126.8. Then z = 180 - 126.8 = 53.2. But that seems messy. Wait, maybe the original problem has a typo, or I misread the angles. Wait, maybe the angles are \((14x + 26)\) and \((9x + 62)\) as vertical angles, but maybe I made a mistake in the arithmetic. Wait, 62 - 26 is 36, 14x - 9x is 5x, so x = 36/5 = 7.2. Then z is supplementary to \(14x + 26\), so z = 180 - (14x + 26). Let's compute 14x + 26: 14*7.2 + 26 = 100.8 + 26 = 126.8. Then z = 180 - 126.8 = 53.2. But this is a decimal. Alternatively, maybe the angles are \((14x + 26)\) and \((9x + 62)\) are supplementary? Wait, if they are adjacent, then 14x + 26 + 9x + 62 = 180. 23x + 88 = 180. 23x = 92. x = 4. Ah! That makes sense. So maybe I misidentified the angles. If the two angles are adjacent and form a linear pair, then they are supplementary. So 14x + 26 + 9x + 62 = 180. 23x + 88 = 180. 23x = 92. x = 4. That's an integer. So probably, the angles \((14x + 26)\) and \((9x + 62)\) are adjacent and supplementary, forming a linear pair. So let's redo that.

Step1: Set up supplementary angles

Since the angles form a linear pair, their sum is \(180^\circ\):
\( (14x + 26) + (9x + 62) = 180 \).

Step2: Solve for x

Combine like terms: \( 23x + 88 = 180 \).
Subtract 88: \( 23x = 180 - 88 = 92 \).
Divide by 23: \( x = \frac{92}{23} = 4 \).

Step3: Find z

Angle \( z \) and \( (14x + 26)^\circ \) are vertical angles (or supplementary? Wait, no—if \( (14x + 26) \) and \( z \) are adjacent, but actually, when two lines intersect, vertical angles are equal. Wait, no—if we have two intersecting lines, the angle \( z \) and \( (9x + 62) \) are vertical angles? Wait, no, let's look at the figure. The angle \( z^\circ \) is adjacent to \( (14x + 26)^\circ \), so they are supplementary? Wait, no, if \( x = 4 \), then \( 14x + 26 = 14*4 + 26 = 56 + 26 = 82 \). Then \( z \) and \( 82^\circ \) are supplementary? No, wait, no—actually, \( z \) and \( (9x + 62) \) are vertical angles. Wait, \( 9x + 62 = 9*4 + 62 = 36 + 62 = 98 \). Wait, no, that can't be. Wait, no—if the two angles \( (14x + 26) \) and \( (9x + 62) \) are supplementary (sum to 180), then \( z \) is equal to \( (9x + 62) \) because they are vertical angles. Wait, let's check: \( 14x + 26 = 14*4 + 26 = 56 + 26 = 82 \). \( 9x + 62 = 9*4 + 62 = 36 + 62 = 98 \). 82 + 98 = 180, so they are supplementary. Then \( z \) is equal to \( (9x + 62) \) because they are vertical angles? Wait, no—vertical angles are opposite each other. So if \( (14x + 26) \) and \( (9x + 62) \) are supplementary (adjacent), then \( z \) is equal to \( (9x + 62) \) because they are vertical angles. Wait, no—let's see: the angle \( z \) is opposite to \( (9x + 62) \), so \( z = 9x + 62 \). Wait, no, \( z \) is adjacent to \( (14x + 26) \), so \( z + (14x + 26) = 180 \). Wait, this is confusing. Wait, when two lines intersect, adjacent angles are supplementary, and vertical angles are equal. So if \( (14x + 26) \) and \( (9x + 62) \) are supplementary (sum to 180), then \( z \) is equal to \( (9x + 62) \) because they are vertical angles. Wait, no—let's take \( x = 4 \): \( 14x + 26 = 82 \), \( 9x + 62 = 98 \). Then \( z \) should be equal to \( 98 \) because it's vertical to \( 9x + 62 \), or equal to \( 82 \) if vertical to \( 14x + 26 \). Wait, the figure shows \( z \) adjacent to \( 14x + 26 \), so \( z + 14x + 26 = 180 \). So \( z = 180 - (14x + 26) \). With \( x = 4 \), \( 14x + 26 = 82 \), so \( z = 180 - 82 = 98 \). Alternatively, \( z = 9x + 62 = 98 \), which matches. So that's correct.

So \( x = 4 \), \( z = 98 \).

Answer:

\( x = 4 \), \( z = 98 \)