Question

1. find the value of x.

diagram: two intersecting lines, one angle labeled 78°, another angle labeled (5x − 2)°
a. 58
b. 113
c. 16
d. 23

Explanation:

Step1: Identify angle relationship

The two angles (78° and \( (5x - 2)^\circ \)) are supplementary (they form a linear pair, so their sum is 180°). So, we set up the equation: \( 78 + (5x - 2) = 180 \).

Step2: Simplify the equation

First, combine like terms: \( 78 - 2 + 5x = 180 \) → \( 76 + 5x = 180 \).

Step3: Solve for x

Subtract 76 from both sides: \( 5x = 180 - 76 \) → \( 5x = 104 \)? Wait, no, wait. Wait, 180 - 78 is 102? Wait, no, wait, the angle with 78° and the angle \( (5x - 2)^\circ \) and the right angle? Wait, no, looking at the diagram, actually, the 78° angle and the \( (5x - 2)^\circ \) angle are adjacent to a straight line? Wait, no, maybe I made a mistake. Wait, actually, the two angles (78° and \( (5x - 2)^\circ \)) and the right angle? No, wait, the diagram shows two intersecting lines, one horizontal and one vertical? Wait, no, the horizontal line and vertical line intersect, so the angle between horizontal and vertical is 90°, but the 78° is between horizontal and the other line, and \( (5x - 2)^\circ \) is between vertical and the other line? Wait, no, maybe the 78° and \( (5x - 2)^\circ \) are complementary? Wait, no, let's re-examine.

Wait, the correct relationship: if there's a horizontal line and a vertical line (perpendicular), then the angle between horizontal and the slant line is 78°, and the angle between vertical and the slant line is \( (5x - 2)^\circ \). Since horizontal and vertical are perpendicular (90°), then 78° + \( (5x - 2)^\circ \) = 90°? Wait, that makes more sense. Oh! I made a mistake earlier. So the two angles are complementary (sum to 90°) because they are adjacent to a right angle (since horizontal and vertical are perpendicular). So correct equation: \( 78 + (5x - 2) = 90 \).

Step4: Solve the correct equation

Simplify: \( 78 - 2 + 5x = 90 \) → \( 76 + 5x = 90 \). Subtract 76: \( 5x = 90 - 76 = 14 \)? No, that can't be. Wait, no, maybe the 78° and \( (5x - 2)^\circ \) are supplementary? Wait, let's look at the diagram again. The two lines intersect: one is horizontal, one is vertical? No, the diagram has a horizontal line, a vertical line, and a slant line? Wait, the original diagram: the horizontal line, vertical line, and a slant line crossing both. So the angle between horizontal and slant is 78°, and the angle between vertical and slant is \( (5x - 2)^\circ \). Since horizontal and vertical are perpendicular (90°), then 78° + \( (5x - 2)^\circ \) = 90°? Wait, no, 78 + (5x - 2) = 90? Let's check the answer options. If x=16, 5*16 -2=78, 78+78=156, no. If x=23, 5*23 -2=113, 78+113=191, no. Wait, maybe the two angles (78° and \( (5x - 2)^\circ \)) are supplementary (sum to 180°) because they are on a straight line. Wait, 78 + (5x - 2) = 180. Then 5x +76=180 → 5x=104 → x=20.8, not an option. Wait, maybe the angle \( (5x - 2)^\circ \) is equal to 180 - 78 - 90? No, this is confusing. Wait, the answer options: A.58, B.113, C.16, D.23. Let's test each option.

Option C: x=16. Then 5*16 -2=78. 78° and 78°: are they vertical angles? Wait, if the horizontal line and slant line form 78°, and the vertical line and slant line form 78°, then maybe the 78° and \( (5x - 2)^\circ \) are equal? Wait, if x=16, 5x-2=78, so 78°=78°, which would mean they are vertical angles or alternate interior angles. Wait, maybe the 78° and \( (5x - 2)^\circ \) are equal because they are vertical angles? Wait, no, vertical angles are opposite each other. Wait, the diagram: the horizontal line, vertical line, and a slant line. So the angle between horizontal (left) and vertical (up) is 90°, but the 78° is between horizontal (left) and slant (right-down), and \( (5x - 2)^\circ \) is between vertical (down) and slant (right-down). Wait, maybe the 78° and \( (5x - 2)^\circ \) are complementary to the same angle? No, let's try x=23: 5*23 -2=113. 78+113=191, no. x=16: 78, 78+78=156, no. x=58: 5*58-2=288, too big. x=113: 5*113-2=563, too big. Wait, maybe I messed up the angle relationship. Wait, the two angles (78° and \( (5x - 2)^\circ \)) are supplementary because they are on a straight line (the vertical line? No, the horizontal line? Wait, the horizontal line is straight, so the angle on one side is 78°, and the angle on the other side with the slant line is \( (5x - 2)^\circ \), but there's also the vertical line. Wait, maybe the correct equation is 78 + 90 + (5x - 2) = 180? No, that would be 166 +5x=180 → 5x=14 → x=2.8, not an option. Wait, maybe the diagram is two intersecting lines, not including the vertical? Wait, the original diagram: a horizontal line, a vertical line, and a slant line crossing both. So the angle between horizontal (left) and slant (right) is 78°, and the angle between vertical (down) and slant (right) is \( (5x - 2)^\circ \). Then, the angle between horizontal (left) and vertical (up) is 90°, so the sum of 78° and \( (5x - 2)^\circ \) should be 90°? Wait, 78 + (5x - 2) = 90 → 5x +76=90 → 5x=14 → x=2.8, no. Alternatively, the angle \( (5x - 2)^\circ \) is supplementary to 78°, so 78 + (5x - 2) = 180 → 5x +76=180 → 5x=104 → x=20.8, not an option. Wait, the answer options must be correct, so I must have misinterpreted the diagram. Wait, maybe the two angles (78° and \( (5x - 2)^\circ \)) are adjacent and form a linear pair with a right angle? No, let's check the answer options again. Option D: x=23. 5*23 -2=113. 180 -78=102, no. Option C: x=16. 5*16 -2=78. 78=78, so maybe the 78° and \( (5x - 2)^\circ \) are equal because they are vertical angles? Wait, if the slant line intersects the horizontal line, then the vertical angle to 78° would be equal, but the \( (5x - 2)^\circ \) is on the vertical line. Wait, maybe the diagram is a transversal cutting a horizontal and vertical line (perpendicular), so the angle of 78° and \( (5x - 2)^\circ \) are such that 78 + (5x - 2) = 90? No, that gives x=2.8. Wait, maybe the vertical line is not perpendicular? No, vertical and horizontal are perpendicular. Wait, maybe the problem is that the two angles (78° and \( (5x - 2)^\circ \)) are supplementary, so 78 + (5x - 2) = 180. Then 5x = 180 -78 +2=104. 104/5=20.8, not an option. Wait, the answer options include 16, 23, etc. Wait, maybe I made a mistake in the angle relationship. Let's try x=16: 5*16 -2=78. So 78° and 78°: maybe they are alternate interior angles? If the slant line is a transversal, and the horizontal and vertical lines are perpendicular, then 78° and \( (5x - 2)^\circ \) would be equal if they are alternate interior angles. So 5x -2=78 → 5x=80 → x=16. Ah! That works. So the correct equation is 5x -2=78, because the two angles are equal (alternate interior angles or vertical angles). So:

Step1: Set angles equal

Since the angles are equal (vertical angles or alternate interior angles), we have \( 5x - 2 = 78 \).

Step2: Solve for x

Add 2 to both sides: \( 5x = 78 + 2 = 80 \).

Step3: Divide by 5

\( x = \frac{80}{5} = 16 \).

Answer:

C. 16