Question

given the figure below, find the values of x and z.
(figure of two intersecting lines with angles 93°, (8x + 45)°, and z°)
x =
z =

Explanation:

Step1: Identify Vertical Angles

Vertical angles are equal. So, \( 83^\circ \) and \( (8x + 45)^\circ \) are not vertical angles? Wait, no, actually, when two lines intersect, vertical angles are equal. Wait, looking at the figure, the angle of \( 83^\circ \) and \( z^\circ \) – no, wait, the angle \( 83^\circ \) and \( (8x + 45)^\circ \) – wait, maybe adjacent angles are supplementary? Wait, no, vertical angles: when two lines intersect, opposite angles (vertical angles) are equal, and adjacent angles are supplementary (sum to \( 180^\circ \)). Wait, let's correct: if two lines intersect, then vertical angles are equal, and linear pairs (adjacent angles) are supplementary.

Wait, in the figure, the angle \( 83^\circ \) and \( (8x + 45)^\circ \) – are they vertical angles? Wait, no, maybe \( 83^\circ \) and \( z^\circ \) are adjacent? Wait, no, let's re-examine. Wait, the problem is to find \( x \) and \( z \). Let's assume that \( 83^\circ \) and \( (8x + 45)^\circ \) are supplementary? No, wait, vertical angles: if two lines intersect, then vertical angles are equal. Wait, maybe the angle \( 83^\circ \) and \( z^\circ \) are vertical angles? No, wait, the angle \( (8x + 45)^\circ \) and \( z^\circ \) – no, let's think again.

Wait, when two lines intersect, the vertical angles are equal. So, if one angle is \( 83^\circ \), its vertical angle is also \( 83^\circ \), and the adjacent angle (linear pair) is \( 180 - 83 = 97^\circ \)? No, wait, no: linear pair angles sum to \( 180^\circ \). Wait, maybe the angle \( 83^\circ \) and \( (8x + 45)^\circ \) are supplementary? Wait, no, that can't be. Wait, maybe I made a mistake. Let's look at the equation. Wait, maybe the angle \( 83^\circ \) and \( (8x + 45)^\circ \) are equal? Wait, no, that would give \( 83 = 8x + 45 \), solving for \( x \): \( 8x = 83 - 45 = 38 \), \( x = 38 / 8 = 4.75 \), which doesn't seem right. Wait, maybe the angle \( 83^\circ \) and \( z^\circ \) are supplementary? No, wait, maybe the angle \( (8x + 45)^\circ \) and \( 83^\circ \) are supplementary. Wait, let's check:

Wait, linear pair: adjacent angles on a straight line sum to \( 180^\circ \). So, if \( 83^\circ \) and \( (8x + 45)^\circ \) are adjacent (forming a linear pair), then \( 83 + (8x + 45) = 180 \). Let's solve that:

Step1: Set up the equation for supplementary angles

\( 83 + 8x + 45 = 180 \)

Step2: Simplify the left side

\( 8x + 128 = 180 \)

Step3: Solve for \( x \)

\( 8x = 180 - 128 = 52 \)

\( x = 52 / 8 = 6.5 \)? Wait, no, 52 divided by 8 is 6.5? Wait, 8*6=48, 52-48=4, so 6.5? That seems odd. Wait, maybe the angle \( 83^\circ \) and \( (8x + 45)^\circ \) are vertical angles. Then \( 83 = 8x + 45 \). Then \( 8x = 83 - 45 = 38 \), \( x = 38 / 8 = 4.75 \). Still odd. Wait, maybe \( z \) is equal to \( (8x + 45)^\circ \), and \( 83^\circ \) and \( z^\circ \) are supplementary? Wait, no, let's try again.

Wait, maybe the angle \( 83^\circ \) and \( z^\circ \) are vertical angles, so \( z = 83 \). Then the angle \( (8x + 45)^\circ \) and \( 83^\circ \) are supplementary, so \( 8x + 45 + 83 = 180 \). Let's solve that:

\( 8x + 128 = 180 \)

\( 8x = 180 - 128 = 52 \)

\( x = 52 / 8 = 6.5 \). Wait, but 6.5 is 13/2. Alternatively, maybe the angle \( (8x + 45)^\circ \) and \( 83^\circ \) are vertical angles. Then \( 8x + 45 = 83 \), so \( 8x = 38 \), \( x = 38/8 = 4.75 \), and \( z \) is supplementary to \( 83^\circ \), so \( z = 180 - 83 = 97 \). Ah, that makes sense! Because when two lines intersect, vertical angles are equal, and linear pairs (adjacent angles) are supplementary. So, if \( 83^\circ \) and \( (8x + 45)^\circ \) are vertical angles, then \( 8x + 45 = 83 \). Then \( z \) is adjacent to \( 83^\circ \), so \( z = 180 - 83 = 97 \). Let's verify:

Step1: Solve for \( x \) using vertical angles

Vertical angles are equal, so \( 8x + 45 = 83 \)

Subtract 45 from both sides: \( 8x = 83 - 45 = 38 \)

Divide by 8: \( x = 38 / 8 = 4.75 \) (or \( 19/4 \))

Step2: Solve for \( z \) using linear pair

Adjacent angles (linear pair) sum to \( 180^\circ \), so \( z + 83 = 180 \)

Subtract 83: \( z = 180 - 83 = 97 \)

Answer:

\( x = 4.75 \) (or \( \frac{19}{4} \))
\( z = 97 \)