Question

1. there are three parallel lines and two transversal lines, with angle expressions ( x + 56 ) and ( x + 146 ) marked.

Explanation:

Step1: Identify angle relationship

The two angles \( x + 56 \) and \( x + 146 \) are same - side interior angles? No, wait, actually, looking at the parallel lines, the angle \( x + 146 \) and \( x + 56 \) should be supplementary? Wait, no, let's think again. Wait, if we consider the two angles, since the lines are parallel, the angle \( x + 146 \) and \( x + 56 \) should satisfy the relationship that \( (x + 146)+(x + 56)=180 \)? Wait, no, maybe it's a linear pair or supplementary. Wait, let's set up the equation.

Wait, actually, the two angles \( x + 146 \) and \( x + 56 \) are supplementary? Wait, no, let's see. If we have two parallel lines cut by a transversal, same - side interior angles are supplementary. Wait, maybe the angle \( x + 146 \) and \( x + 56 \) are supplementary. So:

\( (x + 146)+(x + 56)=180 \)? Wait, no, that would give \( 2x+202 = 180\), which would be negative. So maybe it's a different relationship. Wait, maybe the angle \( x + 146 \) is equal to \( 180-(x + 56) \)? Wait, no, let's re - examine.

Wait, the two angles \( x + 146 \) and \( x + 56 \): if we consider that the sum of a linear pair is 180, but maybe the angle \( x + 146 \) and \( x + 56 \) are such that \( (x + 146)-(x + 56)=90 \)? No, that's 90. Wait, \( (x + 146)-(x + 56)=90 \), which is always 90, that can't be. Wait, maybe the angle \( x + 146 \) and \( x + 56 \) are supplementary? Wait, no, let's do the math.

Wait, let's set up the equation correctly. Let's assume that the two angles are supplementary (because of parallel lines and transversal, same - side interior angles). So:

\( (x + 146)+(x + 56)=180 \)

Step2: Solve the equation

Combine like terms: \( 2x+202 = 180 \)

Subtract 202 from both sides: \( 2x=180 - 202=- 22 \)

Wait, that gives a negative value, which is impossible. So maybe the relationship is that \( (x + 146)-(x + 56)=90 \)? No, that's 90. Wait, maybe I made a mistake in the angle relationship. Wait, maybe the angle \( x + 146 \) is equal to \( 180-(x + 56) \)? Wait, no, let's try another approach.

Wait, maybe the angle \( x + 146 \) and \( x + 56 \) are such that \( x + 146=180-(x + 56) \)

Expand the right - hand side: \( x + 146 = 180 - x - 56\)

\( x + 146=124 - x\)

Add \( x \) to both sides: \( 2x+146 = 124\)

Subtract 146 from both sides: \( 2x=124 - 146=-22\)

Still negative. That can't be. Wait, maybe the angle \( x + 146 \) and \( x + 56 \) are vertical angles? No, they are not. Wait, maybe the two angles are supplementary in a different way. Wait, perhaps the angle \( x + 146 \) and \( x + 56 \) are adjacent and form a linear pair? No, the diagram shows parallel lines.

Wait, maybe I misread the angles. Let's look again. The angle \( x + 146 \) and \( x + 56 \): if we consider that the angle \( x + 146 \) is an exterior angle and \( x + 56 \) is an interior angle. Wait, another approach: the difference between \( x + 146 \) and \( x + 56 \) is 90? Wait, \( (x + 146)-(x + 56)=90 \), which is always 90. So maybe the angle \( x + 56 \) and \( x + 146 \) are such that \( x + 146 = 180-(x + 56) \) is wrong. Wait, maybe the correct equation is \( (x + 146)+(x + 56)=360 \)? No, that's for a full circle.

Wait, I think I made a mistake. Let's start over. Let's assume that the two angles are supplementary. So:

\( (x + 146)+(x + 56)=180\)

\( 2x+202 = 180\)

\( 2x=180 - 202=-22\)

\( x=-11\)

Wait, that gives a negative angle, which is not possible. So maybe the angle \( x + 146 \) is equal to \( x + 56+90 \)? Wait, \( x + 146=x + 56 + 90\), which simplifies to \( 146 = 56+90\), \( 146 = 146\). Oh! Wait, that's an identity. So maybe the problem is to find \( x \) such that the two angles are related by a right angle? Wait, no, maybe the angle \( x + 146 \) is \( x + 56 + 90\), so:

\( x + 146=x + 56+90\)

Subtract \( x \) from both sides: \( 146 = 56 + 90\), which is true. But that doesn't help. Wait, maybe the diagram has three parallel lines cut by two transversals, and the two angles are same - side interior angles with respect to one transversal and the other transversal forms a right angle?

Wait, maybe the correct equation is \( (x + 146)-(x + 56)=90\), which is always 90, so that's not helpful. Wait, maybe the angle \( x + 56 \) and \( x + 146 \) are supplementary in the sense that \( (x + 146)+(x + 56)=180\) is wrong, and actually, the angle \( x + 146 \) is equal to \( 180-(x + 56) \) is wrong. Wait, maybe I made a mistake in the angle relationship.

Wait, let's consider that the two angles \( x + 56 \) and \( x + 146 \) are such that \( x + 146 = 180-(x + 56) \) is incorrect, and instead, the angle \( x + 146 \) is an obtuse angle and \( x + 56 \) is an acute angle, and their sum is 180. Wait, but when we solved \( 2x+202 = 180\), we got \( x=-11\), which is negative. That can't be. So maybe the diagram is different. Wait, maybe the angle \( x + 146 \) and \( x + 56 \) are vertical angles? No. Wait, maybe the two angles are supplementary in the other way, like \( (x + 146)+(x + 56)=360\)? No, that would be \( 2x + 202=360\), \( 2x=158\), \( x = 79\). Wait, that gives positive angles. Wait, maybe I made a mistake in the angle relationship.

Wait, if the two angles are around a point, their sum is 360? No, that's for a full circle. Wait, maybe the two angles are supplementary in the sense that they are on a straight line but on opposite sides. Wait, let's try \( (x + 146)+(x + 56)=360\)

\( 2x+202 = 360\)

\( 2x=360 - 202 = 158\)

\( x = 79\)

Let's check: \( x + 56=79 + 56 = 135\), \( x + 146=79+146 = 225\). No, 135 + 225=360, which is a full circle. That doesn't make sense.

Wait, maybe the correct relationship is that the angle \( x + 146 \) and \( x + 56 \) are supplementary, but I had the equation wrong. Wait, no, supplementary angles add up to 180. If \( x + 146\) and \( x + 56\) are supplementary, \( x+146+x + 56 = 180\), \( 2x+202 = 180\), \( 2x=-22\), \( x=-11\). The angle \( x + 56=-11 + 56 = 45\), \( x + 146=-11+146 = 135\). And 45+135 = 180. Oh! Wait, 45 and 135 are supplementary. So even though \( x\) is negative, the angles are positive. So let's go with that.

Step2: Solve the equation

We have the equation:

\( (x + 146)+(x + 56)=180\)

Combine like terms:

\( 2x+202 = 180\)

Subtract 202 from both sides:

\( 2x=180 - 202=-22\)

Divide both sides by 2:

\( x=-11\)

Answer:

\( x=-11\)