Question

find the value of each variable. 15. diagram: two intersecting lines, one angle 120°, another (2x − 10)° 16. diagram: intersecting lines with a vertical line, angles (2x)° and (4x + 108)° 17. diagram: intersecting lines with a vertical line, angles (2x + 25)°, y°, (3x − 10)°

Explanation:

Problem 15

Step1: Identify vertical angles

The angle \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? No, wait, actually, adjacent angles on a straight line are supplementary. Wait, no, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Wait, the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, looking at the diagram, the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, actually, when two lines intersect, vertical angles are equal. Wait, maybe the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, let's think again. If two lines intersect, the vertical angles are equal. Wait, maybe the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, perhaps the \(120^\circ\) and \((2x - 10)^\circ\) are supplementary? Wait, no, if they are adjacent, they would be supplementary, but if they are vertical, they are equal. Wait, looking at the diagram, the angle \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, maybe I made a mistake. Wait, actually, when two lines intersect, the vertical angles are equal. So if one angle is \(120^\circ\), the vertical angle should also be \(120^\circ\)? Wait, no, the angle \((2x - 10)^\circ\) is vertical to the angle adjacent to \(120^\circ\)? Wait, no, let's re-express. The sum of adjacent angles on a straight line is \(180^\circ\). Wait, the \(120^\circ\) and the angle adjacent to \((2x - 10)^\circ\) are supplementary? Wait, maybe the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles. Wait, let's check: if two lines intersect, vertical angles are equal. So \(2x - 10 = 120\)? Wait, no, that would make \(2x = 130\), \(x = 65\), but let's check. Wait, maybe the \(120^\circ\) and \((2x - 10)^\circ\) are supplementary? Wait, no, if they are adjacent, they would be supplementary. Wait, the diagram shows two intersecting lines, so the angle \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles? Wait, no, maybe the \(120^\circ\) and \((2x - 10)^\circ\) are vertical angles. Wait, let's solve \(2x - 10 = 120\). Then \(2x = 130\), \(x = 65\). Wait, but let's check: \(2(65) - 10 = 130 - 10 = 120\), which matches. So yes, vertical angles are equal.

Step1: Set vertical angles equal

\(2x - 10 = 120\)

Step2: Solve for x

Add 10 to both sides: \(2x = 120 + 10 = 130\)

Divide by 2: \(x = \frac{130}{2} = 65\)

Step1: Identify supplementary angles

The angles \((2x)^\circ\) and \((4x + 108)^\circ\) are supplementary because they form a linear pair (they are adjacent and on a straight line with the vertical line). Wait, no, the vertical line and the other line intersect, so the angles \((2x)^\circ\) and \((4x + 108)^\circ\) are supplementary? Wait, no, actually, the sum of angles on a straight line is \(180^\circ\), but also, the vertical line makes a straight angle, so the two angles \((2x)^\circ\) and \((4x + 108)^\circ\) are supplementary? Wait, no, let's see: the vertical line and the other line intersect, so the angles \((2x)^\circ\) and \((4x + 108)^\circ\) are adjacent to the right angle? Wait, no, the diagram shows a vertical line (up and down) and another line intersecting it, forming angles \((2x)^\circ\) and \((4x + 108)^\circ\). Wait, actually, the sum of \((2x)^\circ\) and \((4x + 108)^\circ\) should be \(180^\circ\) because they are on a straight line (the horizontal line? No, the vertical line and the other line form a linear pair? Wait, no, the two angles \((2x)^\circ\) and \((4x + 108)^\circ\) are adjacent and form a linear pair, so their sum is \(180^\circ\).

Step1: Set up the equation

\(2x + (4x + 108) = 180\)

Step2: Combine like terms

\(6x + 108 = 180\)

Step3: Subtract 108 from both sides

\(6x = 180 - 108 = 72\)

Step4: Divide by 6

\(x = \frac{72}{6} = 12\)

Step1: Identify vertical angles

The angles \((2x + 25)^\circ\) and \((3x - 10)^\circ\) are vertical angles? Wait, no, looking at the diagram, the angle \((2x + 25)^\circ\) and \((3x - 10)^\circ\) are vertical angles? Wait, no, actually, when two lines intersect, vertical angles are equal. Wait, the angle \((2x + 25)^\circ\) and \((3x - 10)^\circ\) are vertical angles? Wait, let's check: if they are vertical angles, then \(2x + 25 = 3x - 10\).

Step1: Set vertical angles equal

\(2x + 25 = 3x - 10\)

Step2: Solve for x

Subtract \(2x\) from both sides: \(25 = x - 10\)

Add 10 to both sides: \(x = 25 + 10 = 35\)

Step3: Find y

Now, the angle \(y^\circ\) is supplementary to either \((2x + 25)^\circ\) or \((3x - 10)^\circ\) because they form a linear pair. Let's compute \((2x + 25)^\circ\) with \(x = 35\): \(2(35) + 25 = 70 + 25 = 95^\circ\). Then \(y + 95 = 180\) (since they are on a straight line), so \(y = 180 - 95 = 85\). Alternatively, compute \((3x - 10)^\circ\): \(3(35) - 10 = 105 - 10 = 95^\circ\), same as above. So \(y = 180 - 95 = 85\).

Answer:

\(x = 65\)

Problem 16