Question

which of the following equations could be used to find the value of z?
options:

• ( z + 168 = 53 )
• ( z + 53 = 180 )
• ( z + 168 = 180 )
• ( 168 = z + 53 )

(there is a triangle image with angles 53°, ( z° ), and 168°)

Explanation:

Step1: Recall the triangle angle sum property

The sum of the interior angles of a triangle is \( 180^\circ \). But wait, here we have an exterior angle? Wait, no, looking at the triangle, the three angles involved: one is \( 53^\circ \), one is \( z^\circ \), and the other is related to \( 168^\circ \). Wait, actually, if we consider the angle adjacent to \( 168^\circ \), it should be \( 180 - 168 = 12^\circ \)? No, wait, maybe the triangle has angles \( 53^\circ \), \( z^\circ \), and the angle supplementary to \( 168^\circ \). Wait, no, the correct approach: the sum of angles in a triangle is \( 180^\circ \). But if one of the angles is \( 168^\circ \)? No, that can't be, because the sum of angles in a triangle is \( 180^\circ \), so \( 168^\circ \) must be an exterior angle. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, no, let's re - examine.

Wait, the triangle has angles: \( 53^\circ \), \( z^\circ \), and the angle that forms a linear pair with \( 168^\circ \). The angle forming a linear pair with \( 168^\circ \) is \( 180 - 168=12^\circ \). Then, by the triangle angle sum, \( 53 + z+12 = 180 \), which simplifies to \( z + 65 = 180 \)? No, that's not matching. Wait, maybe the problem is that the three angles given (considering the \( 168^\circ \) as an interior angle? But that would make the sum exceed \( 180^\circ \) with \( 53^\circ \) and \( z^\circ \)). Wait, no, the correct equation: if we consider that the sum of \( z \) and \( 168^\circ \) is equal to \( 180^\circ \)? Wait, no, let's look at the options. The options are:

1. \( z + 168 = 53 \)
2. \( z + 53 = 180 \)
3. \( z + 168 = 180 \)
4. \( 168 = z + 53 \)

Wait, maybe the triangle has an angle \( z \), an angle \( 53^\circ \), and the angle \( 168^\circ \) is not an interior angle. Wait, the correct reasoning: in a triangle, the sum of angles is \( 180^\circ \). If we assume that one of the angles is \( 168^\circ \) (but that's impossible for a triangle). So the only way is that \( z\) and \( 168^\circ \) are related such that \( z + 168=180 \), which would mean that the angle supplementary to \( 168^\circ \) and \( z \) sum up? No, maybe the problem is simpler. The sum of angles in a triangle is \( 180^\circ \). If we have two angles: \( z \) and \( 168^\circ \), and the third angle is \( 53^\circ \)? No, that doesn't make sense. Wait, the correct option is \( z + 168 = 180 \) because if we consider that \( z\) and \( 168^\circ \) are two angles that add up to \( 180^\circ \) (maybe they are supplementary). So the equation \( z+168 = 180 \) is the correct one (the second option from the left, \( z + 168 = 180 \)).

Answer:

The equation \( z + 168 = 180 \) (the second option from the left)