Question

13)
99° (5x + 1)°

Explanation:

Step1: Identify angle relationship

The angles \(99^\circ\), \((5x + 1)^\circ\), and the right angle (from the vertical and horizontal lines) should sum to \(180^\circ\) (straight line). Wait, actually, the vertical line and horizontal line form a right angle? No, looking at the diagram, the three angles (the \(99^\circ\), the \((5x + 1)^\circ\), and the right angle? Wait, no, the vertical and horizontal lines are perpendicular, so the angle between them is \(90^\circ\)? Wait, no, the straight line (horizontal) has a \(99^\circ\) angle, then a \((5x + 1)^\circ\) angle, and the vertical line. Wait, actually, the sum of \(99^\circ\), \((5x + 1)^\circ\), and the right angle? No, maybe the \(99^\circ\) and \((5x + 1)^\circ\) are adjacent to a right angle? Wait, no, the horizontal line is straight, so the sum of \(99^\circ\), \((5x + 1)^\circ\), and the angle between vertical and horizontal (which is \(90^\circ\))? No, that can't be. Wait, maybe the two angles \(99^\circ\) and \((5x + 1)^\circ\) are complementary to the right angle? Wait, no, looking at the diagram, the vertical line and horizontal line form a right angle, so the angle between them is \(90^\circ\). Then the \(99^\circ\) angle, the \((5x + 1)^\circ\) angle, and the right angle? No, that would be more than \(180^\circ\). Wait, maybe the \(99^\circ\) and \((5x + 1)^\circ\) are on a straight line, so \(99 + (5x + 1) + 90 = 180\)? No, that would be \(99 + 5x + 1 + 90 = 180\) → \(5x + 190 = 180\), which is negative. That can't be. Wait, maybe the vertical line is perpendicular to the horizontal line, so the angle between vertical and horizontal is \(90^\circ\), and the \(99^\circ\) angle and \((5x + 1)^\circ\) angle are such that \(99 + (5x + 1) = 90\)? No, that would be \(5x + 100 = 90\), \(5x = -10\), which is wrong. Wait, maybe I misread the diagram. Wait, the horizontal line is straight, so the sum of \(99^\circ\) and \((5x + 1)^\circ\) should be \(90^\circ\)? No, that doesn't make sense. Wait, maybe the \(99^\circ\) angle and \((5x + 1)^\circ\) angle are adjacent to a right angle, so \(99 + (5x + 1) = 90\)? No. Wait, maybe the diagram has a right angle (90 degrees) between the vertical and horizontal, and the \(99^\circ\) and \((5x + 1)^\circ\) are on the other side? Wait, no, let's re-express. The straight line (horizontal) has three angles? No, the diagram shows a horizontal line, a vertical line, with a \(99^\circ\) angle between the left horizontal and the vertical, and a \((5x + 1)^\circ\) angle between the vertical and the right horizontal. Wait, that makes sense! So the left horizontal to vertical is \(99^\circ\), vertical to right horizontal is \((5x + 1)^\circ\), and since vertical and horizontal are perpendicular, the sum of \(99^\circ\) and \((5x + 1)^\circ\) should be \(90^\circ\)? No, that can't be, because \(99 > 90\). Wait, that's impossible. So maybe the vertical line is not perpendicular? Wait, no, the arrow is vertical and horizontal, so they are perpendicular (right angle, \(90^\circ\)). Then the sum of \(99^\circ\), \((5x + 1)^\circ\), and \(90^\circ\) should be \(360^\circ\)? No, that's a full circle. Wait, I think I made a mistake. Let's look again. The horizontal line is straight, so the sum of the angles on one side of the straight line is \(180^\circ\). So the \(99^\circ\) angle, the \((5x + 1)^\circ\) angle, and the angle between vertical and horizontal (which is \(90^\circ\))? No, that would be \(99 + 5x + 1 + 90 = 180\) → \(5x + 190 = 180\) → \(5x = -10\), which is wrong. Wait, maybe the vertical line is not perpendicular, so the angle between vertical and horizontal is \((5x + 1)^\circ\), and the \(99^\circ\) angle and \((5x + 1)^\circ\) angle are supplementary to the straight line? Wait, no, the straight line is \(180^\circ\), so \(99 + (5x + 1) = 180\)? Wait, that would be \(5x + 100 = 180\) → \(5x = 80\) → \(x = 16\). Wait, that makes sense! Because the two angles \(99^\circ\) and \((5x + 1)^\circ\) are on a straight line, so they are supplementary (sum to \(180^\circ\)). Oh! I see, the vertical line is a red herring? No, wait, the diagram has a vertical line and a horizontal line, with the \(99^\circ\) angle between the left horizontal and vertical, and \((5x + 1)^\circ\) between vertical and right horizontal. So the vertical line is between the two horizontal segments, so the sum of \(99^\circ\) and \((5x + 1)^\circ\) should be \(90^\circ\)? No, that can't be. Wait, no, if the vertical line is perpendicular to the horizontal line, then the angle between vertical and horizontal is \(90^\circ\), so \(99^\circ\) is outside that? No, the diagram must have the two angles \(99^\circ\) and \((5x + 1)^\circ\) on a straight line, so their sum is \(180^\circ\). Wait, that must be it. So:

\(99 + (5x + 1) = 180\)

Step2: Solve for x

Simplify the equation:

\(99 + 5x + 1 = 180\)

\(5x + 100 = 180\)

Subtract 100 from both sides:

\(5x = 180 - 100\)

\(5x = 80\)

Divide both sides by 5:

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

\(x = 16\)

Answer:

\(x = 16\)