Question

1. what are the measures of the missing angles? m \\( \angle x= \\)

1. what are the measures of the missing angles? m \\( \angle y= \\)

Explanation:

Step1: Find the measure of angle \( z \)

The sum of angles in a triangle is \( 180^{\circ} \). For the larger triangle with angles \( 50^{\circ},84^{\circ},z \), we have \( z=180-(50 + 84) \).

$$z=180 - 134=46^{\circ}$$

Step2: Use the exterior - angle property

The exterior - angle property states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
For the angle \( x \), assume it is an exterior angle. If we consider the smaller angle relationships (assuming the \( 13^{\circ}\) and \( z = 46^{\circ}\) are related to \( x \) in an exterior - interior angle sense). But if we assume \( x \) is formed by a linear pair with an adjacent angle. Wait, no, using the exterior - angle property for the overall triangle - related structure (assuming \( x \) is an exterior angle of a sub - triangle).
Wait, another approach:
The sum of angles around a point is \( 360^{\circ}\), but no. Wait, using the triangle angle sum and exterior - angle theorem properly.
Let's first find the third angle of the large triangle. Let the large triangle have angles \( A = 50^{\circ},B = 84^{\circ},C\). Then \(C=180-(50 + 84)=46^{\circ}\).
If we assume \( x\) is an exterior angle of a smaller triangle (with the \( 13^{\circ}\) angle and the angle adjacent to \( z\)).
Wait, using the exterior - angle theorem: \(x=13 + 46\) (assuming the non - adjacent interior angles of the exterior angle \(x\) are \(13^{\circ}\) and \(z = 46^{\circ}\))

$$x = 59^{\circ}$$

Step3: Find the measure of angle \( y\)

If we assume \(x=y + 13\) (using the exterior - angle property for a smaller sub - triangle where \(x\) is the exterior angle and \(y\) and \(13^{\circ}\) are the non - adjacent interior angles)
Since \(x = 59^{\circ}\), then \(y=x - 13\)

$$y=59-13 = 46^{\circ}$$

Answer:

For \(m\angle x\): \(59\)
For \(m\angle y\): \(46\)