Given that triangles \( \triangle XYZ \) and \( \triangle RST \) are similar, the corresponding angles of the triangles are equal.
We know the following angles:
- \( m\angle X = 50^\circ \)
- \( m\angle S = 79^\circ \)
To find the measure of angle \( T \), we first need to calculate the measure of the remaining angle in triangle \( XYZ \):
The sum of angles in a triangle is \( 180^\circ \). Thus, we can find \( m\angle Y \) in triangle \( XYZ \):
\[ m\angle Y = 180^\circ - m\angle X - m\angle Z \]
Since the triangles are similar, we can identify the corresponding angles. Let \( m\angle Y \) correspond with \( m\angle R \) and \( m\angle T \) with \( m\angle Z \).
First, we calculate \( m\angle Z \):
- Calculate \( m\angle Y \): \[ m\angle Y = 180^\circ - 50^\circ - m\angle Z \]
Next, in triangle \( RST \): \[ m\angle R + m\angle S + m\angle T = 180^\circ \]
Substituting \( m\angle S = 79^\circ \): \[ m\angle R + 79^\circ + m\angle T = 180^\circ \]
Now, we can also express \( m\angle R \): \[ m\angle R = 180^\circ - 79^\circ - m\angle T \]
Thus, we have: \[ m\angle Y + m\angle R + m\angle S + m\angle T = 180^\circ \] This means we need:
To get both angles for similar triangles, use the calculated \( m\angle Y \):
Given that \( m\angle Y \) and \( m\angle R \) are corresponding angles, we need:
Continuing with \( m\angle X + m\angle S + m\angle T = 180^\circ \).
Let: \[ m\angle T = 180^\circ - m\angle X - m\angle S \] Now we can plug in values: \[ m\angle T = 180^\circ - 50^\circ - 79^\circ \]
Calculating, \[ m\angle T = 180^\circ - 129^\circ = 51^\circ \]
Thus, the measure of angle \( T \) is: \[ \boxed{51^\circ} \]