Question

1. m∠2 = 49 + x

Explanation:

Step1: Identify triangle type (isosceles)

The triangle has two equal sides (marked with ticks), so it's isosceles. Thus, the base angles are equal. One base angle is \(43^\circ\), so the other (angle 2's related angle? Wait, no—wait, the triangle: the right angle? Wait, no, the triangle has a right angle? Wait, the horizontal and vertical sides are perpendicular? So it's a right triangle? Wait, no, the horizontal and vertical sides have a right angle (since one is horizontal, one vertical, so \(90^\circ\)). Wait, no, the triangle: angles sum to \(180^\circ\). Wait, the triangle has angles: \(43^\circ\), right angle (\(90^\circ\)), and angle 2? Wait, no, the diagram: the horizontal line, vertical line (right angle), and the slant side. Wait, the two sides (vertical and horizontal) are equal? Wait, the ticks: vertical side and horizontal side? No, the ticks are on the vertical side (angle 2's side) and the horizontal side? Wait, no, the diagram: the triangle has two sides with ticks, so those two sides are equal, making it an isosceles triangle. So the angles opposite those sides are equal. Wait, the horizontal side (with tick) and vertical side (with tick) are equal, so the angles opposite them are equal. The angle opposite the horizontal side is angle 2, and the angle opposite the vertical side is \(43^\circ\)? Wait, no, maybe it's a right triangle? Wait, the horizontal and vertical lines are perpendicular, so that angle is \(90^\circ\). Wait, no, the triangle: angles sum to \(180^\circ\). Let's re-express:

Wait, the triangle has angles: \(43^\circ\), \(90^\circ\), and angle 2? No, that can't be, because \(43 + 90 + \angle2 = 180\) would give \(\angle2 = 47\), but the formula is \(m\angle2 = 49 + x\). Wait, maybe the triangle is isosceles with two equal angles, not a right triangle. Wait, the ticks are on the two legs (the sides forming angle 2 and the \(43^\circ\) angle's side). Wait, maybe the triangle is isosceles, so the two base angles are equal? Wait, no, the diagram: the horizontal line and the slant line form a \(43^\circ\) angle, and the vertical line and slant line form angle 2. The horizontal and vertical lines are equal (ticks), so the triangle is isosceles with legs equal, so the base angles (the angles opposite the equal sides) are equal? Wait, no, in a triangle, equal sides have equal opposite angles. So if the two legs (horizontal and vertical) are equal, then the angles opposite them (angle 2 and the \(43^\circ\) angle) are equal? Wait, no, the \(43^\circ\) angle is at the base, and angle 2 is at the top. Wait, maybe the triangle is a right triangle? Wait, no, the horizontal and vertical lines are perpendicular, so that angle is \(90^\circ\). Wait, I think I made a mistake. Let's start over.

Wait, the triangle: one angle is \(43^\circ\), another angle is angle 2, and the third angle is \(90^\circ\) (since horizontal and vertical are perpendicular). So sum of angles in a triangle is \(180^\circ\). So:

\(43^\circ + 90^\circ + m\angle2 = 180^\circ\)

Wait, no, that would be \(133 + m\angle2 = 180\), so \(m\angle2 = 47^\circ\). But the formula is \(m\angle2 = 49 + x\). Wait, maybe the triangle is isosceles, so two angles are equal. The ticks are on the two sides (vertical and horizontal), so those sides are equal, so the angles opposite them are equal. The angle opposite the horizontal side is angle 2, and the angle opposite the vertical side is \(43^\circ\)? No, that would mean angle 2 = \(43^\circ\), but then \(49 + x = 43\), which would give \(x = -6\), which doesn't make sense. Wait, maybe the triangle is not a right triangle. Wait, the horizontal line is a transversal? No, the diagram: a triangle with a horizontal side, vertical side (right angle), and slant side. The horizontal side and slant side form a \(43^\circ\) angle, vertical side and slant side form angle 2. The vertical and horizontal sides are equal (ticks), so it's an isosceles right triangle? No, isosceles right triangle has angles \(45^\circ, 45^\circ, 90^\circ\). But here we have \(43^\circ\). Wait, maybe the \(43^\circ\) is an external angle? No, the diagram shows the angle inside the triangle. Wait, maybe I misread the diagram. Let's look again: the triangle has a horizontal side (with a tick), a vertical side (with a tick), and a slant side. The angle between the horizontal side and the slant side is \(43^\circ\), and the angle between the vertical side and the slant side is angle 2. So since the two sides (horizontal and vertical) are equal, the triangle is isosceles with those two sides as legs, so the base angles (the angles opposite the equal sides) are equal? Wait, no, in a triangle, the sides opposite equal angles are equal. So if the two legs (horizontal and vertical) are equal, then the angles opposite them (angle 2 and the \(43^\circ\) angle) are equal? Wait, no, the angle opposite the horizontal leg is angle 2, and the angle opposite the vertical leg is the \(43^\circ\) angle. So if the legs are equal, then angle 2 = \(43^\circ\)? But then \(49 + x = 43\) → \(x = -6\), which is odd. Alternatively, maybe the triangle is a right triangle, and the two angles (angle 2 and \(43^\circ\)) are complementary? Wait, no, in a right triangle, the two acute angles are complementary (sum to \(90^\circ\)). So if one angle is \(43^\circ\), the other (angle 2) is \(90 - 43 = 47^\circ\). Then \(m\angle2 = 49 + x = 47\) → \(x = 47 - 49 = -2\). No, that's still negative. Wait, maybe the diagram is different. Wait, the problem says \(m\angle2 = 49 + x\), and the triangle has a \(43^\circ\) angle, and the two sides (vertical and horizontal) are equal, so it's an isosceles triangle, so the base angles are equal. Wait, maybe the triangle is isosceles with angle 2 and the angle adjacent to \(43^\circ\) being equal? Wait, no, let's use the angle sum property. Sum of angles in a triangle is \(180^\circ\). Let's assume the triangle is isosceles with two equal angles: angle 2 and the angle opposite the equal side. Wait, the two equal sides are the ones with ticks, so the angles opposite them are equal. Let's call the equal sides \(a\) (horizontal) and \(b\) (vertical), so \(a = b\). Then the angles opposite them are angle 2 (opposite \(a\)) and the angle opposite \(b\) (let's call it \(\angle A\)). So \(\angle A = m\angle2\). The third angle is \(43^\circ\). So sum: \(m\angle2 + m\angle2 + 43^\circ = 180^\circ\) → \(2m\angle2 = 137^\circ\) → \(m\angle2 = 68.5^\circ\). Then \(49 + x = 68.5\) → \(x = 19.5\). But that doesn't seem right. Wait, maybe the triangle is a right triangle, and the \(43^\circ\) is one acute angle, angle 2 is the other, so they sum to \(90^\circ\). So \(43 + (49 + x) = 90\). Let's solve that:

Step1: Set up angle sum (complementary)

Since the triangle has a right angle (horizontal and vertical are perpendicular), the two acute angles are complementary (sum to \(90^\circ\)). So:
\(43^\circ + m\angle2 = 90^\circ\)

Step2: Substitute \(m\angle2 = 49 + x\)

Substitute \(m\angle2\) into the equation:
\(43 + (49 + x) = 90\)

Step3: Simplify and solve for \(x\)

Combine like terms:
\(92 + x = 90\)
Subtract 92 from both sides:
\(x = 90 - 92\)
\(x = -2\)

Wait, but that gives a negative \(x\). Maybe the triangle is not a right triangle. Wait, maybe the \(43^\circ\) is an external angle? No, the diagram shows it inside. Alternatively, maybe the two sides with ticks are the base and one leg, making it isosceles with base angles equal. Wait, the base is the horizontal side, and the leg is vertical, with ticks. So the base angles (at the base) are equal. So angle 2 is a base angle, and the other base angle is \(43^\circ\), so angle 2 = \(43^\circ\), but then \(49 + x = 43\) → \(x = -6\). No. Alternatively, maybe the triangle is isosceles with angle 2 being the vertex angle, and the two base angles are \(43^\circ\) each. Then sum: \(43 + 43 + m\angle2 = 180\) → \(m\angle2 = 94\). Then \(49 + x = 94\) → \(x = 45\). But why would the vertical and horizontal sides be equal then? Wait, maybe the diagram is a right triangle with legs equal (isosceles right triangle), but the angle given is \(43^\circ\), which is not \(45^\circ\). This is confusing. Wait, maybe the problem has a typo, or I misread the diagram. Wait, the original problem: \(m\angle2 = 49 + x\), and the triangle has a \(43^\circ\) angle, and two sides with ticks (so equal). Let's assume that the triangle is isosceles, so the two angles opposite the equal sides are equal. Let's say the equal sides are the ones forming angle 2 and the \(43^\circ\) angle. Wait, no, in a triangle, the sides opposite the angles. So if two sides are equal, their opposite angles are equal. So if the vertical side (opposite \(43^\circ\)) and horizontal side (opposite angle 2) are equal, then \(43^\circ = m\angle2\). But then \(49 + x = 43\) → \(x = -6\). Alternatively, maybe the triangle is a right triangle, and the angle \(43^\circ\) is adjacent to angle 2, so they are supplementary? No, in a triangle, angles are less than \(180^\circ\). Wait, maybe the diagram is a triangle with a horizontal line, a vertical line, and a slant line, forming a triangle where the horizontal and vertical lines are equal (ticks), so it's an isosceles triangle, and the angle between the slant line and horizontal line is \(43^\circ\), so the angle between slant line and vertical line (angle 2) is equal to \(43^\circ\) because it's isosceles. But then \(49 + x = 43\) → \(x = -6\). Alternatively, maybe the sum of angle 2 and \(43^\circ\) is \(180 - 90 = 90\) (right triangle), so \(43 + (49 + x) = 90\) → \(x = -2\). Maybe that's the case. Let's go with that.

Answer:

\(x = -2\)